TABLES COMPLEX HYPERBOLIC AND CIRCULAR FUNCTIONS KENNELLY UNIVERSITY OF CALIFORNIA AT LOS ANGELES 65.66i = 0.29712 / 65.39'.4 o". sinh [2] TABLE I. HYPERBOLIC SINES, sinh = r /y. CONTINUED 0.6 0.7 0.8 0.9 i.o 45 46 47 48 49 0.60042 48.440 0.59918 49-437 0-59793 50.434 0.59667 51.426 0.59542 52.414 0.70094 49.676 0.69894 50.679 0.69695 51.676 0.69497 52.666 0.69299 53.652 0.80184 S^oS 0.79885 52-112 0.79587 53-109 0.79291 54-099 0.78996 55-o82 0.90327 52.728 0.89904 53.735 0.89482 54.734 0.89060 55.725 0.88640 56.707 1-00553 54-531 0-99975 55-540 0-99394 56.550 0.98816 57-543 0.98242 58.525 50 5 1 52 53 54 0.59418 53.398 0.59295 54-379 0.59174 55-355 0.59053 56-326 0.58932 57.293 0.69102 54-632 0.68907 55.606 0.68713 56.574 0.68521 57.537 0.68331 58.493 0.78703 56-058 0.78412 57-026 0.78124 57.987 0.77838 58.940 0-77555 59-886 0.88224 57.679 0.87810 58.642 0.87400 59.595 0.86993 60.538 0.86590 61.472 0.97672 59-495 0.97105 60.453 0.96543 61.399 0.95986 62.333 0-95435 63.255 55 56 57 58 59 0.58814 58.256 0.58698 59.215 0.58583 60.171 0.58469 61.122 0.58357 62.069 0.68144 59-445 0.67959 60.391 0.67776 61.331 0.67595 62.265 0.67419 63.193 0.77275 60.824 0.76999 61.755 0.76727 62.678 0.76459 63.593 0.76195 64.502 0.86192 62.396 0.85800 63.312 0.85414 64.218 0.85034 65.114 0.84660. 66.000 0.94890 64.166 0-94353 65.065 0.93825 65.952 0.93305 66.827 0.92795 67.691 6o 61 62 63 64 0.58249 63.013 0.58142 63.953 0.58037 64.889 0.57934 65.821 0.57834 66.749 0.67247 64.117 0.67078 65.036 0.66912 65.951 0.66749 66.859 0.66591 67.762 0.75938 65.405 0.75686 66.300 0-75439 67.189 0.75197 68.070 0.74962 68.944 0.84295 66.878 0.83937 67.747 0.83587 68.607 0.83244 69.458 0.82909 70.300 0.92295 68.544 0.91805 69.385 o-9 I 3 2 5 70-215 0.90856 71.033 0.90400 71.841 65 66 67 68 69 0-57737 67.674 0.57643 68.596 0-57553 69.515 0.57465 70.430 0-57379 7I-342 0.66437 68.661 0.66288 69.554 0.66145 70.444 0.66005 71.329 0.65870 72.209 0-74733 69.812 0.74512 70.674 0.74298 7i.5 2 9 0.74091 72-379 0.73891 73-223 0.82585 71.136 0.82270 71.962 0.81967 72.780 0.81672 73.590 0.81387 74.392 0.89957 72-637 0.89527 73-424 0.89111 74.201 0.88708 74.968 0.88320 75.723 70 72 73 74 0.57297 72.251 0.57219 73.157 0.57H5 74-061 0.57074 74-962 0.57006 75.860 0.65740 73-o85 0.65616 73.957 0.65498 74.825 0-65385 75-689 0.65278 76.550 0.73698 74-061 0.73513 74.894 0-73337 75-722 0.73169 76.544 0.73009 77-361 0.81114 75.187 0.80853 75.975 0.80602 76.756 0.80363 77.530 0.80137 78.298 0.87947 76-469 0.87589 77-207 0.87247 77.936 0.86921 78.656 0.86612 79.368 75 76 77 78 79 0.56941 76-756 0.56881 77.649 0.56824 78.540 0.56772 79.429 0.56724 80.317 0.65176 77.408 0.65081 78.263 0.64992 79.114 0.64909 79.962 0.64832 80.808 0.72858 78.174 0.72716 78.982 0.72583 79-787 0.72459 80.588 0.72345 81.385 0.79924 79.059 0.79723 79-8i5 0-79535 80.566 0-79359 81.312 0.79197 82.053 0.86320 80.072 0.86045 80.769 0.85788 81.458 0.85549 82.141 0.85328 82.818 80 81 82 83 84 0.56679 81.203 0.56638 82.087 0.56602 82.970 0.56570 83.852 0.56542 84.732 0.64761 81.652 0.64697 82.493 0.64640 83.332 0.64589 84.169 0.64545 85.005 0.72241 82.179 0.72146 82.969 0.72061 83.757 0.71985 84.543 0.71919 85.327 0.79048 82.789 0.78913 83.522 0.78792 84.251 0.78685 84.976 0.78592 85.699 0.85125 83.489 0.84940 84.156 0.84774 84-817 0.84628 85.474 0.84501 86.128 85 86 8? 88 89 0.56518 85.612 0.56498 86.490 0.56483 87.368 0.56473 88.246 0.56466 89.123 0.64507 85.839 0.64476 86.673 0.64452 87.506 0.64435 88.338 0.64425 89.169 0.71863 86.109 0.71817 86.889 0.71781 87.668 0.71755 88.446 0.71740 89.223 0.78513 86.420 0.78448 87.138 0.78397 87.855 0.78361 88.571 0.78339 89.286 0.84393 86.779 0.74305 87.426 0.84236 88.071 0.84186 88.715 0.84156 89.358 9P 0.56464 90.000 0.64422 90.000 0.71736 90.000 0.78333 90.000 0.84147 OO.OPO Examples, sinh (i.o /go") = 0.84147 /go . sinh" 1 (0.87947 /76.469) = 1.0/70. t3] TABLE I. HYPERBOLIC SINES, sinh (p IS) = r {y. CONTINUED 1.4 i-S 45 1.1089 56.519 1.2138 58.692 1.3205 61.034 1.4297 63-568 1-5418 66.262 46 I.IOI2 57-543 1.2037 59.726 1.3078 62.092 1.4138 64.639 1.5222 67-355 47 1-0935 58.555 I-I937 60.748 1.2951 63.128 1-3979 65.689 1.5027 68.426 48 1.0858 59-553 1.1838 6i.753 1.2824 64.142 1.3822 66.717 1.4834 69.474 49 1.0782 60.536 I-I739 62.741 1.2699 65-I37 1.3665 67.723 1.4642 70.496 50 1.0706 61.506 1.1641 63.712 1.2574 66.113 1-3509 68.707 I-445I 71.492 5i 1.0630 62.461 I-I543 64.666 1.2450 67.068 1-3355 69.668 1.4262 72.462 52 1-0556 63.401 1.1446 65.603 1.2327 68.004 1.3202 70.605 1.4075 73-405 53 1.0482 64.327 1-1350 66.523 1.2206 68.919 1-3051 7i.5i9 1.3889 74-321 54 1.0409 65-239 1.1256 67-425 1.2086 69.814 1.2902 72.409 1.3706 75.210 55 1.0336 66.136 1.1162 68.310 1.1967 70.688 1.2754 73-275 I.3525 76.072 56 1.0265 67.019 1.1070 69.178 1.1850 7I-542 1.2609 74.117 1-3347 76.907 57 1.0195 67.888 1.0979 70.028 I-I735 72.376 1.2466 74-936 1.3172 77.713 58 1.0126 68.742 1.0890 70.860 1.1622 73.188 1.2325 75-730 1.3000 78.491 59 1.0058 69.581 1.0802 7I-675 1.1511 73.980 1.2187 76.500 1.2831 79.240 6o 0.99920 70.406 1.0716 72.474 1.1403 74-752 1.2052 77.246 1.2665 79.964 61 0.99269 71.218 1.0632 73-255 1.1296 75-502 1.1919 77.967 1-2503 80.657 62 0.98633 72.016 1.0550 74-oi9 1.1192 76.232 1.1790 78.663 1-2345 81.321 63 0.98013 72.800 1.0470 74.767 1.1091 76.942 1.1664 79-335 1.2191 81.956 64 0.97409 73-570 1.0392 75-497 1.0992 77.632 1.1541 79-983 1.2040 82.562 65 0.96821 74.3?7 1.0316 76.211 1.0895 78.301 1.1421 80.607 1.1894 83.140 66 0.96251 75-07I 1.0242 76.909 1.0802 78.950 1-1305 81.207 1.1752 83.690 67 0.95698 75-8oi 1.0171 77-590 1.0712 79.580 1.1193 81.783 1.1615 84.211 68 0.95165 76-519 I.OIO2 78.257 1.0625 80.191 1.1084 82.335 1.1482 84.704 69 0.94650 77.225 1.0035 78.907 1.0540 80.783 1.0979 82.865 I.I354 85.169 70 0.94156 77-9i8 0.99712 79-543 1.0459 Si-357 1.0879 83.373 1.1231 85.607 71 0.93682 78.600 0.99099 80.164 1.0382 81.912 1.0783 83.858 1.1113 86.018 72 0.93229 79.271 0.98514 80.771 1.0308 82.450 1.0691 84.322 1. 1000 86.402 73 0.92798 79-931 0-97957 81-365 1.0237 82.971 1.0603 84-765 1.0893 86.761 74 0.92388 80.580 0.97428 81.946 1.0170 83.476 1.0520 85.188 1.0791 87.096 75 0.92001 81.220 0.96927 82.514 1.0107 83.966 1.0441 85-59I 1.0694 87.406 76 0.91637 81.850 0.96456 83.069 1.0047 84.440 1.0367 85-975 1.0603 87.693 77 0.91296 82.471 0.96015 83.614 0.99916 84.900 1.0298 86.342 1.0518 87.958 78 0.90978 83.084 0.95605 84.148 0.99397 85-346 1.0233 86.692 1.0439 88.201 79 0.90685 83.689 0.95226 84.672 0.98917 85-780 1.0173 87.026 1.0366 88.425 80 0.90416 84.286 0.94878 85-187 0.98477 86.202 1.0119 87.346 1.0299 88.630 81 0.90172 84.877 0.94562 85-693 0.98077 86.614 1.0069 87.651 1.0238 88.818 82 0-89953 85.462 0.94279 86.191 0.97715 87.016 1.0024 87.944 . 1.0183 88.991 83 0.89759 86.041 0.94029 86.683 0.97400 87.408 0.99845 88.227 1.0135 89.149 84 0.89590 86.616 0.93811 87.169 0.97124 87-793 0.99502 88.499 1.0093 89.294 85 0.89447 87.186 0.93626 87.649 0.96890 88.171 0.99210 88.762 1.0057 89.429 86 0.89330 87-753 0-93474 88.124 0.96698 88.544 0.98971 89.018 1.0027 89-554 87 0.89238 88.317 0.93356 88.596 0.96548 88.912 0.98785 89.269 1.0004 89-673 88 0.89173 88.879 0.93272 89.065 0.96441 89.277 0.98652 89-515 0.99880 89-785 89 0.89134 89.440 0.93221 89-533 0.96377 89.639 0.98572 89.758 0.99782 89.893 DO 0.89121 90.000 0.93204 90.000 0.96356 90.000 0.98545 90.000 0.99749 90.000 Examples. sinh (i. i/45) = 1.1089 /j ;6.si9 = 1.1089 /s6.3i'.o8'. sinh" 1 (1.1084 /82.335) = 1.4 /68. [4] TABLE I. HYPERBOLIC SINES, sinh (p /5) = r /% CONTINUED 1.6 1.8 1.9 45 I-6575 69.117 1.7776 72.133 1.9029 75.292 2-0343 78.590 2.1726 82.016 46 1-6338 70.241 1-7493 73-288 1.8693 76.486 1.9947 79.829 2.1266 83-304 47 1.6103 71-339 1.7210 74.418 I-8359 77.651 1-9554 81.037 2.0809 84.560 48 1.5868 72.409 1.6929 75-5I5 1.8027 78.784 1.9165 82.2IO 2.0356 85.778 49 I-563S 73-451 1.6651 76.582 1.7697 79.883 1.8779 83.346 1.9907 86.958 5 1.5404 74-465 I-6375 77.618 1-7370 80.946 1.8396 84.444 1.9462 88.098 51 I-5I75 75-449 1.6102 78.620 1.7046 81.974 1.8016 85.503 1.9022 89.196 52 1.4949 76.402 1-5831 79-590 1.6726 82.966 1-7643 86.524 1.8588 90-253 53 I-4725 77-325 1-5563 80.527 1.6410 83.921 1-7273 87.505 1.8160 91.267 54 1.4504 78.218 1-5299 81.429 1.6098 84.839 1.6909 88.444 1-7737 92.236 55 1.4285 79.079 1-5039 82.296 I-579I 85-718 1.6550 89.342 1.7320 93.160 56 1.4070 79.910 1.4782 83.128 1.5488 86.558 1.6196 90.197 1.6911 94.039 57 1-3859 80.709 1-4530 83-924 1.5190 87.358 1.5848 9I.OO8 1.6509 94.869 58 1-3651 Si-475 1.4282 84.683 1.4898 88.116 i.S5o6 91-773 1.6114 95-650 59 1-3447 82.209 1.4039 85.406 1.4611 88.834 1-5171 92.493 1.5726 96.382 60 I-3247 82.910 1.3800 86.091 1-4330 89.510 1.4843 93.167 1-5347 97.062 61 1-3051 83.578 1-3567 86.738 1-4055 90.142 1-4523 93.792 1.4976 97-688 62 1.2860 84.212 1-3339 87-347 1.3787 90.730 1.4210 94.368 1.4614 98.261 63 1.2674 84.813 1.3117 87.917 I.3525 91.274 3904 94.893 1.4260 98.778 64 1.2492 85-380 1.2901 88.447 1.3270 Qi-774 .3606 95.368 i-39 J 5 99.238 65 1.2316 85-9I3 1.2690 88.938 1.3022 92.228 33i6 95-792 i.358o 99.639 66 1.2145 86.413 1.2486 89.390 1.2781 92.636 3035 96.162 I.325S 99.980 t5 7 1.1979 86.879 1.2288 89.802 1.2548 92.997 .2762 96.478 1.2940 100.261 68 1.1819 87.311 1.2097 90.175 1.2322 93-3*2 .2499 96.741 1.2635 100.479 69 1.1664 87.710 1.1913 90.508 1.2104 93-58o .2244 96.949 1.2340 100.634 70 1.1516 88.076 1.1736 90.8OI 1.1895 93.802 .1999 97.102 1.2056 100.725 7i I-I373 88.410 1.1566 9 LOSS 1.1694 93.976 .1764 97.199 1.1783 100.750 72 1.1237 88.712 1.1403 91.271 1.1502 94.104 1539 97.241 1.1521 100.709 73 1.1107 88.982 1.1248 91.448 1.1318 94.187 1324 97.227 1.1271 100.602 74 1.0984 89.221 I. HOI 91.588 1.1144 94-224 1.1119 97.I58 1.1033 100.428 75 1.0867 89.431 1.0961 91.692 1.0979 94-215 1.0925 97-034 1.0807 100.187 76 1-0757 89.613 1.0829 91.760 1.0823 94.162 1-0743 96.856 1.0594 99.880 77 1.0654 89.767 1.0706 91.794 1.0677 94.068 1.0571 96.625 1-0394 99.507 78 1.0558 89.894 I.059I 9!-795 1.0541 93-933 1.0411 96.343 1.0207 99.070 79 i .0469 89.997 1.0485 9I-765 1-0415 . 93-7S8 1.0262 96.OI2 1.0033 98.571 80 1.0388 90.076 1.0388 91-705 1.0299 93-545 1.0126 95-633 0.98729 98.011 81 1.0314 90.134 1.0299 91.618 1.0194 93.298 I.OOO2 95.209 0.97271 97-394 82 1.0248 90.171 I.O2I9 9I-505 1.0099 93.019 0.98899 94-744 0.95958 96.722 83 1.0189 90.191 I.OI49 91.370 1.0015 92.711 0.97907 94.240 0.94792 96.000 84 1.0138 90.194 1.0088 91.214 0.99422 92-374 0.97043 93-702 0-93775 95-233 85 1.0095 90.184 I.OO36 91.039 0.98802 92.015 0.96309 93-133 0.92911 94.426 86 i. 0060 90.161 0.99929 90.850 0.98293 91.636 0.95706 92-539 0.92201 93-584 87 1.0032 90.129 0.99596 90.649 0.97896 91.241 0.95236 91.923 0.91646 92.713 88 I.OOI2 90.090 0-99357 90.438 0.97612 90.834 0.94899 91.291 0.91248 91.821 89 I.OOOO 90.046 0.99214 90.220 0.97442 90.419 0.94697 90.648 0.91009 90.914 90 Q-99957 90.000 0.99166 90.000 0.97385 90.000 0.94630 90.000 0.90930 90.000 Examples, sinh (1.6 /6o) = 1.3247 /82.9io = 1.3247 /82.S4'.36V sinh" 1 (1.1999 /97.iQ2) = 1.9 /7Q. [Si TABLE I. HYPERBOLIC SINES, sinh (p /5) = r /jy. CONTINUED 2-3 2.4 2-5 45 2.3190 85.558 2-4745 89.205 2.6404 92.946 2.8177 96.769 3.0079 100.661 46 2.2658 86.905 2.4135 90.613 2.5707 94.419 2.7386 98.312 2.9185 102.278 47 2.2131 88.213 2-3530 91.981 2.5017 95- 8 5i 2.6603 99.813 2.8301 103.852 48 2.1608 89.482 2.2936 93-307 2-4334 97.241 2.5829 101.271 2.7429 105-383 49 2.1090 90.711 2.2337 94.592 2.3659 98.588 2.5065 102.685 2.6568 106.869 SO 2.0577 91.898 2.1750 95.834 2.2992 99.890 2.4311 104.053 2.5720 108.309 Si 2.0071 93.042 2.1171 97.031 2.2334 101.146 2.3568 105.375 2.4885 109.701 52 I-957I 94.142 2.0600 98.181 2.1685 102.354 2.2836 106.648 2.4064 111.044 S3 1.9078 95- I 97 2.0037 99.284 2.1046 103.514 2.2117 107.871 2-3257 1I2 -337 54 1.8592 96.205 1.9483 100.338 2.0418 104.623 2.1410 109.042 2.2465 "3-578 55 1.8114 97-i65 1.8938 101.342 1.9801 105.680 2.0714 110.160 2.1687 114.765 56 1.7644 98.076 1.8402 102.294 I-9J95 106.683 2.0032 111.223 2.0925 115.896 57 1.7182 98.935 1.7876 103.193 1. 8600 107.629 1.9364 112.228 2.0178 116.970 58 1.6729 99.742 1.7360 104.035 1.8016 108.518 1.8709 "3-174 1.9447 117-984 59 1.6284 100.494 1.6854 104.820 1-7445 109.347 i. 8068 114.059 1-8733 118.936 60 1.5849 101.191 I-6359 105.546 1.6886 110.114 1.7441 114.880 1.8035 119.823 61 I-5424 101.830 I.5875 106.210 1.6340 110.816 1.6829 115-634 1-7353 120.642 62 1.5008 102.410 1.5402 106.811 1.5807 111.451 1.6232 116.319 1.6688 121.391 63 1.4603 102.929 1.4941 107.345 1.5287 112.016 1.5649 116.931 1.6040 122.067 64 1.4208 103.386 1.4492 107.811 1.4780 112.509 1.5081 117.467 1.5409 122.665 65 1.3824 103.777 1-4055 108.207 1.4286 112.926 1.4528 117.924 1.4794 123.182 66 J -345i . 104.101 1.3630 108.529 1.3806 113.264 I-399I 118.297 1.4196 123.613 67 1.3089 104.357 1.3218 108.775 I-334I "3-519 1.3469 118.583 1.3616 123-954 68 1.2738 104.542 1.2819 108.943 1.2890 113.688 1.2963 118.777 1-3052 124.198 69 1.2399 104.655 1-2433 109.029 1-2453 113.767 1-2473 118.874 1.2506 124.341 70 1.2072 104.694 1.2060 109.030 1.2031 "3-752 1.1999 118.868 1.1977 124.376 7i 1.1758 104.656 1.1701 108.944 1.1624 113.638 1.1541 118.754 1.1466 124.296 72 I-I457 104.541 I-I356 108.769 1.1232 113.422 1.1099 118.526 1.0972 124.092 73 1.1168 104.348 1.1026 108.501 1.0857 113.100 1.0675 118.177 1.0497 123-755 74 1.0893 104.074 1.0711 108.138 1.0498 112.667 1.0268 117.701 1.0040 123.277 75 1.0632 103.719 1.0411 107.678 i-oi55 112.118 0.98795 117.091 0.96019 122.648 76 1-0385 103.283 1.0126 107.119 0.98292 111.449 0.95090 116.341 0.91831 121.857 77 1-0153 102.766 0.98581 106.459 0.95214 110.659 0.91576 II5-443 0.87843 120.891 78 0-99353 102.168 0.96065 105.699 0.92320 109.744 0.88261 114.390 0.84063 119.740 79 0.97332 101.491 0.93722 104.839 0.89615 108.701 0.85152 113.178 0.80503 118.392 80 0.95468 100.736 0.91557 103.879 0.87109 107-531 0.82256 111.803 0.77173 116.836 81 0-93765 99907 0-89575 102.822 0.84808 106.233 0.79589 110.261 0.74083 115.063 82 0.92229 99.008 o 87784 101.671 0.82719 104.810 0.77158 108.550 0.71251 113.065 83 0.90862 98.043 0.86187 100.432 0.80853 103.267 0.74974 106.676 0.68692 110.841 84 0.89668 97-oiS 0.84789 99.111 0.79215 101.611 0-73051 104.643 0.66422 108.390 85 0.88652 95-933 0.83596 97-7I5 0.77813 99.851 0.71396 102.462 0.64461 105.723 86 0.87817 94.806 0.82613 96.254 0.76655 98.000 0.70026 100.147 0.62827 102.855 87 0.87164 93-639 0.81845 94.740 0-75747 96.071 0.68948 97.717 0.61536 99.811 88 0.86696 92.443 0.81293 93-183 0.75095 94.082 0.68171 95-197 0.60602 96.624 89 0.86414 91.226 0.80961 91-599 0.74702 92.051 0.67703 92.615 0.60036 93-338 90 0.86321 90.000 0.80850 90.000 0-74571 90.000 0.67546 90.000 0.59847 90.000 Examples, sinh (2.5 775 = 0.96019 /i22.648 = 0.96019 /i22 .38'.53 g . sinh" 1 (0.60036 /93-338) = 2.5 789. [6J TABLE I. 2.6 HYPERBOLIC SINES, sinh (p IS) = r /T. CONTINUED 2.7 2.8 2.9 3.0 o o o o o 4S 3.2121 104.613 3-43i8 108.614 3-6685 112.653 3-9236 116.721 4.1986 120.814 46 3-HI5 106.307 3-3*91 110.386 3-5426 114.506 3-7832 118.658 4.0426 122.832 47 3.0123 107.957 3.2079 112. Il6 3.4186 116.317 3-6453 120.551 3-8895 124.808 48 2.9144 109.564 3-0985 II3.80I 3.2966 118.084 3-5098 122.400 3-7394 126.741 49 2.8179 111.126 2.9908 115.442 3.1767 119.806 3.3768 124.205 3-5923 128.630 5 2.7229 112.641 2.8849 117.037 3.0590 121.483 3-2465 125.965 3-4483 130.474 5i 2.6295 114.109 2.7809 118.585 2.9436 123.113 3.1189 127.679 3.3076 132.272 S 2 2.5378 115.528 2.6789 120.084 2.8306 124.694 2.9941 129.345 3.1701 134.024 53 2.4478 116.897 2-5789 121.532 2.720O 126.226 2.8721 130.963 3-0359 135.729 54 2-3595 118.214 2.4809 122.929 2.6118 127.708 2.7529 132.532 2.9051 137.386 55 2.2729 119.476 2.3850 124.274 2.5060 129.138 2.6366 134-050 2.7777 138.994 56 2.1882 120.684 2.2913 125.563 2.4027 130.515 2.5232 135-517 2.6536 140.553 57 2-1053 121.834 2.1996 126.797 2.3019 131-837 2.4127 136.932 2-5330 142.062 58 2.0242 122.925 2.IIOI 127.973 2.2036 133-103 2-3051 138.293 2.4157 143-519 59 1.9450 123.954 2.0228 129.088 2.1077 134-3" 2.2004 139.598 2.3018 144.924 60 1.8677 124.918 1-9377 130.140 2.0144 135-459 2.0986 140.847 2.1912 146.276 61 1-7923 125.816 I.8547 I3I.I28 1.9236 136.544 1.9997 142.037 2.0839 147-574 62 1.7187 126.644 1-7739 132.047 1-8352 137-565 1.9036 143.167 1.9798 148.817 63 1.6470 127.399 1-6952 132.895 1-7493 138.519 1.8103 144.235 1.8789 150.004 64 1-5772 128.077 1.6187 133.669 1.6658 I39-403 1.7198 145-239 1.7812 151-133 65 1-5094 128.674 1-5442 I34-364 I-5847 140.214 1.6319 146.176 1.6865 152.204 66 1-4435 129.185 1.4719 134.978 1.5060 140.948 1-5467 147.044 1.5948 I53- 2I 4 67 1-3794 129.606 1.4016 I35-504 I-4295 141.599 1.4641 147.838 1.5061 154.161 68 1.3171 129.930 J-3334 135-935 1-3553 142.164 1.3840 148.555 1.4202 I55-043 69 1.2568 130.151 1.2673 136.266 1-2833 142.634 1.3063 149.190 1-3370 155-858 70 1-1983 130.262 1.2031 136.489 1.2136 I43-005 1.2310 149.738 1.2565 156.602 7i 1.1417 130.254 1.1409 136.596 1.1460 143.267 1.1581 150.191 1.1785 157.271 72 1.0870 130.117 1.0807 136.576 1.0804 143.411 1.0874 150.541 1.1030 157.860 73 1.0341 129.841 1.0226 136.415 1.0169 143.424 1.0189 150.780 1.0298 158.363 74 0.98316 129.414 0.96631 I36.IOI 0.95550 143.292 0.95251 150.894 0.95893 158.772 75 0.93420 128.822 0.91207 I35-6I7 0.89606 142.996 0.88819 150.867 0.89022 159.077 76 0.88720 128.049 0.85985 134-943 0.83862 142.516 0.82587 150.680 0.82356 159-265 77 0.84227 127.079 0.80965 134.058 0.78319 141.827 0.76551 150.309 0.75887 I59-3 I 9 78 0.79948 125.893 0.76157 I32-935 0.72979 140.897 0.70706 149.721 0.69605 159.216 79 0.75891 124.471 0.71568 I3I-S42 0.67847 139.686 0.65054 148.876 0.63503 158.925 80 0.72068 122.793 0.67215 129.847 0.62934 138.145 0-59597 147.721 0-57573 158.404 81 0.68499 120.837 0.63109 I27.8l2 0.58252 136.217 0.54340 146.183 0.51809 I57.592 82 0.65203 118.583 0.59275 125.396 0.53822 I33-833 0.49294 144.169 0.46213 156.404 83 0.62196 116.016 0-55739 122.554 0.49674 130.908 0.44484 141-553 0.40789 154.709 84 0.59507 113.126 0.52536 119.252 0.45849 127-351 0.39946 138.168 0.35548 152-313 85 0.57164 109.910 0.49705 II5-458 0.42399 123.066 0.35738 133-795 0.30521 148-903 86 0.55191 106.382 0.47291 III.l64 0-39394 117.968 0.31946 128.168 0.25775 143-974 87 0.53621 102.571 0-45344 106.390 0.36915 112.015 0.28691 121.005 0.21433 136.701 88 0.52479 98.524 0.43912 IOI.I94 0.35055 105.244 0.26143 112. IIO 0.17731 125.843 89 0.51783 94-307 0.43033 95-682 0-33893 97.809 0.24496 101.576 0.15092 110.091 90 0.51550 90.000 0.42738 90.000 0.33499 90.000 0.23925 90.000 0.14112 90.000 Examples, sinh (3.0 /5o) = 3.4483 / sinh"" 1 (0.15092 /iio.Q9i) SQ-474 = 3-4483 /i3Q-28 / .26 > . - 3-Q /8 9 - [7] TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /_y 0-3 0.4 45 I.OOOOI 0.287 I.OOOI3 1.148 1.00067 2.578 I.O02IO 4.578 1.00519 7.141 46 0.99983 0.287 0-99943 1.146 0.99910 2-577 0-99933 4-584 1.00085 7-159 47 0.99966 0.286 0.99873 1.144 0-99753 2.576 0.99655 4-584 0.99649 7.166 48 0.99948 0.285 0.99804 1.141 0-99597 2-571 0.99378 4.578 0.99214 7-165 49 0.99931 0.283 0-99735 1.136 0.99441 2.562 0.99101 4.566 0.98780 7-155 5 0.99914 0.282 0.99666 1.131 0.99285 2-551 0.98824 4-551 0.98347 7-137 51 0.99897 0.280 0-99597 1.123 0.99131 2-536 0.98547 4-528 0.97917 7.109 52 0.99880 0.278 0.99529 1-115 0.98977 2-519 0.98274 4.500 0.97490 7-073 S3 0.99863 0.275 0.99462 1.105 0.98825 2.498 0.98003 4.467 0.97065 7.028 54 0.99846 0.272 0-99395 1.094 0.98674 2-474 0-97734 4427 0.96644 6-973 55 0.99830 0.269 0.99329 1.081 0.98525 2.447 0.97468 4-382 0.96226 6.910 56 0.99814 0.265 0.99263 1.067 0.98377 2.417 0.97205 4-332 0.95814 6.838 57 0.99798 0.261 0.99199 1.052 0.98232 2-383 0.96945 4.276 0.95406 6.756 58 0.99782 0-257 0.99135 1.036 0.98089 2-347 0.96690 4.214 0.95005 6.666 59 0.99766 0-253 0.99073 1.018 0.97948 2.308 0.96438 4.147 0.94609 6.567 60 0.99751 0.249 0.99012 0.999 0.97810 2.267 0.96191 4-075 0.94219 6.460 61 0.99736 0.244 0.98952 0.979 0.97674 2.221 0.95948 3-998 0.93838 6-343 62 0.99721 0.238 0.98893 0-957 0.97541 2-173 0.95711 3-9I4 0-93465 6.217 63 0.99707 0.232 0.98835 0-934 0.97411 2.123 0.85478 3-826 0.93099 6.083 64 0.99693 0.226 0.98780 0.910 0.97285 2.07O 0-95251 3-733 0.92741 5-941 65 0.99679 O.22O 0.98725 0.885 0.97163 2.014 0.95031 3-635 0.92393 5-789 66 0.99666 O.2I4 0.98672 0.859 0.97044 1-955 0.94816 3-532 0-92054 5-631 67 0.99653 0.207 0.98621 0.832 0.96927 1.894 0.94608 3-423 0.91725 5-463 68 0.99641 O.2OO 0.98571 0.804 0.96814 1.830 0.94407 3-3io 0.91407 5.288 69 0.99629 0.193 0.98523 0-775 0.96706 1.764 0.94213 3-!93 0.91100 5.106 70 0.99617 0.185 0.98477 0.744 0.96601 1.696 0.94026 3.072 0.90805 4.916 7i 0.99606 0.177 0.98433 0.713 0.96500 1.626 0.93846 2.946 0.90521 4.718 72 0.99596 0.169 0.98391 0.681 0.96404 1-553 0.93674 2.816 0.90248 4-5I3 73 0.99586 0.161 0.98350 0.648 0.96313 1.479 0.93510 2.682 0.89989 4.302 74 0.99576 0.153 0.98312 0.614 0.96227 1.402 0-93354 2-544 0.89742 4.085 75 0.99567 0.144 0.98276 0.580 0.96145 1-324 0.93207 2.403 0.89508 3-86i 76 0.99559 0-135 0.98242 0-545 0.96068 1.244 0.93069 2.258 0.89288 3-631 77 0-9955 1 0.126 0.98210 0.509 0.95996 1.162 0.92938 2. Ill 0.89081 3-396 78 0-99544 0.117 0.98181 0.472 0.95929 1.078 0.92816 1.960 0.88889 3-155 79 0-99537 0.108 0.98154 0-435 0.95866 0-993 0.92705 1.807 0.88711 2.910 80 0.99531 0.099 0.98128 0-397 0.95808 0.908 0.92603 1.652 0.88548 2.660 81 0.99525 0.090 0.98105 0-359 0.95756 0.821 0.92509 1.494 0.88399 2.406 82 0.99520 0.080 0.98085 0.320 0.95710 0.732 0.92425 1-333 0.88266 2.149 83 0-995 1 5 0.070 0.98067 0.281 0.95670 0.643 0.62351 1.170 0.88147 1.888 84 0.99511 0.060 0.98051 0.242 0-95635 0-552 0.92287 i. 006 0.88044 1.624 85 0.99508 0.050 0.98037 0.203 0.95605 0.461 0.92233 0.841 0.87957 1-357 86 0.99506 0.040 0.98026 0.163 0.95580 0.369 0.92189 0.675 0.87886 1.087 87 0.99504 0.030 0.98018 0.123 0.95560 0.277 0.92154 0.508 0.87830 0.816 88 0.99502 0.02O 0.98012 0.082 0-95545 0.185 0.92128 0.340 0.87790 0-544 89 0.99501 O.OIO 0.98009 0.041 0-95537 0.093 0.92112 0.171 0.87766 0.272 90 0.99500 o.ooo 0.98007 o.ooo 0.95534 o.ooo 0.92106 o.ooo 0.87758 o.ooo Note. cosh (o/5) = i.o /o^for all values of d. Examples, cosh (0.5 /8i) = 0.88399 /2.4o6 = 0.88399 /2.24 / .22 r . cosh" 1 (0.97810 /2.267) = 0.3 /6o. [8] TABLE II. HYPERBOLIC COSINES, cosh (p /8) = r /T. CONTINUED 0.6 0.7 0.8 0.9 45 1.01070 10.254 1.01982 13.890 1.03360 18.010 1.05333 22.567 1.08031 27.487 46 1.00449 10.291 1.01136 13.960 1.02263 18.132 1.03959 22.755 1.06358 27.762 47 0.99825 10.315 1.00289 14.013 1.01164 18.231 1.02583 22.919 1 .04680 28.011 48 0.99199 10.327 0.99441 14.050 1.00063 18.309 1.01203 23-059 1.03000 28.235 49 0-98575 10.326 0.98594 14.071 0.98963 18.368 0.99822 23.176 1.01315 28.433 50 0-97953 10.313 0.97748 14-075 0.97864 18.405 0.98443 23.267 0.99632 28.603 Si 0-97333 10.287 0.96906 14.061 0.96768 18.421 0.97064 23-332 0.97950 28.743 52 0.96716 10.248 0.96066 14.031 0.95675 18.414 0.95690 23-372 0.96270 28.854 53 0.96103 10.196 0.95232 13.982 0.94587 18.384 0.94320 23-383 0.94596 28.933 54 0-95495 10.131 0.94404 / 13.916 0.93506 18.332 0.92957 23-367 0.92928 28.981 55 0.94893 10.053 0.93582 13-831 0.92432 18.256 0.91603 23.321 0.91267 28.994 56 0.94296 9.961 0.92768 13.728 0.91367 18.156 0.90257 23.246 0.89615 28.974 57 0.93706 9-856 0.91962 13.606 0.90312 18.031 0.88922 23.140 0.87976 28.917 58 0.93125 9.738 0.91166 13-465 0.89270 17.881 0.87602 23.003 0.86350 28.823 59 0.92552 9.606 0.90382 13.306 0.88240 17-705 0.86294 22.833 0.84739 28.689 6o 0.91987 9.461 0.89607 13.127 0.87222 17.505 0.85001 22.631 0.83142 28.518 61 0.91433 9-303 0.88846 12.929 0.86221 17.278 0.83727 22.394 0.81564 28.303 62 0.90889 9- J 3i 0.88100 12.713 0.85237 17.024 0.82471 22.122 0.80008 28.047 63 0.90357 8-945 0.87369 12.476 0.84271 16.743 0.81237 2I.8I4 0.78475 27.745 64 0.89838 8.747 0.86653 12. 22O 0.83324 16.435 0.80025 21.471 0.76966 27.396 65 0.89332 8.536 0.85954 "945 0.82398 16.100 0.78837 21.091 0.75484 26.999 66 0.88838 8.312 0.85272 11.652 0.81494 15-738 0.77675 20.673 0.74029 26.555 67 0.88358 8.076 0.84609 n-339 0.80613 15-349 0.76540 2O.2I8 0.72603 26.061 68 0.87894 7.827 0.83966 11.007 0.79757 I4-93I 0-75435 19.722 0.71212 25-5 I 3 69 0.87445 7.565 0.83344 10.656 0.78927 14.486 0.74362 I9.I88 0.69857 24.911 70 0.87012 7.292 0.82744 10.287 0.78125 14.014 0.73322 I8.6I5 0.68539 24-254 71 0.86596 7.007 0.82166 9.900 0.77352 I3-5I5 0.72317 iS.OOI 0.67261 23-54I 72 0.86197 6.711 0.81611 9.496 0.76610 12.990 0.71347 I7.348 0.66026 22.772 73 0.85816 6.404 0.81081 9-075 0.75899 12.438 0.70416 16.656 0.64836 21.946 74 0.85454 6.086 0.80576 8.637 0.75220 11.860 0.69527 15.926 0.63692 21.061 75 0.85 in 5.758 0.80097 8.183 0.74575 11.257 0.68681 I5-I56 0.62599 20.115 76 0.84787 5.420 0.79646 7.7I3 0.73965 10.630 0.67878 14-347 0.61560 19.111 77 0.84484 5-073 0.79222 7.228 0.73392 9-979 0.67121 I3.502 0.60577 18.049 78 0.84201 4.718 0.78826 6.729 0.72856 9.306 0.66412 12.621 0.59652 16.929 79 0.83939 4-355 0.78459 6.217 0.72359 8.612 0.65753 II.7O6 0.58790 15-753 80 0.83698 3-984 0.78121 5-694 0.71901 7.897 0.65145 10-755 0.57991 14.521 81 0.83479 3.606 0.77814 5-!59 0.71484 7.164 0.64589 9.776 0.57259 I3-237 82 0.83282 3.221 0.77538 4-613 0.71108 6.413 0.64087 8.768 0.56596 11.904 83 0.83108 2.831 0.77293 4-057 0.70774 5-647 0.63641 7-733 0.56005 10.526 84 0.82957 2.436 0.77079 3-493 0.70483 4.867 0.63252 6.674 0.55487 9.105 85 0.82828 2.037 0.76898 2.922 0.70236 4-075 0.62921 5-595 0.55046 7.647 86 0.82722 1.634 0.76750 2-345 0.70033 3.272 0.62648 4.498 0.54683 6.157 87 0.82640 1.228 0.76634 1.763 0.69875 2.462 0.62435 3.386 0.54399 4.642 88 0.82581 0.820 0.76551 1.177 0.69761 1.645 0.62283 2.263 0.54195 3.106 89 0.82546 0.410 0.76501 0.589 0.69693 0.823 0.62192 I-I33 0.54072 i-SS^ 9 0.82534 o.ooo 0.76484 o.ooo 0.69671 o.ooo 0.62161 o.ooo 0.54030 o.ooo Examples, cosh (0.7 771) = 0.82166 /g.goo = 0.82166 /g cosh" 1 (0.69857 /24.9ii) = 1.0/69. [9] TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /jy. CONTINUED 1.4 45 46 47 48 49 I-H57 1.0959 1-0759 1-0559 J-0359 32.686 33-o67 33-424 33-754 34-056 I. 1608 1.1376 I. 1143 1.0910 1.0676 38.076 38.582 39.063 39-517 39-943 1.2163 1.1897 1.1630 1.1363 1.1095 43-570 44.210 44.827 45-421 45.989 1.2830 1.2529 .2227 .1926 .1624 49.084 49.864 50-625 51-365 52.083 1.3616 1-3279 1.2941 1.2604 1.2267 54-550 55-471 56.378 57-266 58.134 50 Si 52 53 54 1.0158 0.99578 0-97577 0.95580 0.93589 34.328 34-570 34-780 34-957 35-!00 1.0443 1.0209 0.99757 0.97428 0.95104 40.341 40.709 41.045 41-349 41.618 1.0828 1.0561 1.0294 1.0027 0.97613 46.529 47.041 47-525 47-978 48.400 .1322 .IO2I .072O .O42I I.OI22 52-777 53-446 54.090 54.708 55-299 1.1931 1.1596 1.1262 1.0929 1.0598 58.982 59.8lO 6o.6l8 61.407 62.175 55 56 57 58 59 0.91605 0.89630 0.87666 0.85716 0.83781 35.206 35-275 35-305 35-^93 35-238 0.92788 0.90481 0.88186 0.85902 0.83633 41.850 42.045 42.201 42.314 42.383 0.94964 0.92325 0.89697 0.87082 0.84482 48.788 49.141 49.458 49.736 49-973 0.98242 0.95277 0.92327 0.89393 0.86474 55-862 56-396 56.900 57-372 57-Sii 1.0268 0.99406 0.96149 0.92912 0.89696 62.921 63.645 64-347 65.025 65.681 60 61 62 63 64 0.81862 0.79961 0.78081 0.76223 0.74390 35.I39 34.992 34-794 34-545 34.242 0.81379 0.79144 0.76929 0.74735 0.72565 42.406 42.379 42.299 42.164 41.970 0.81898 0.79331 0.76783 0.74256 0.71752 50.168 50-315 50.412 50.457 50.446 0-83573 0.80692 0.77831 0.74991 0.72173 58.214 58.578 58.902 59-185 59.421 0.86502 0.83332 0.80186 0.77065 0.73969 66.313 66.919 67.497 68.048 68.571 65 66 67 68 69 0.72584 0.70808 0.69063 0.67352 0.65677 33-881 33-459 32.974 32-423 31-804 0.70420 0.68304 0.66218 0.64164 0.62146 4i-7i3 41.389 40.993 40.520 39.966 0.69271 0.66815 0.64385 0.61985 0.59618 50-373 50.235 50.026 49-739 49.368 0-69377 0.66605 0.63858 0.61136 0.58441 59.608 59-742 59.8i7 59.828 59-769 o. 70899 0.67855 0.64836 0.61844 0.58877 69.064 69.524 69.950 70.340 70.689 70 7i 72 73 74 0.64043 0.62452 0.60907 0.59410 0.57966 31.112 30.344 29.498 28.570 27-558 0.60166 0.58227 0.56334 0.54489 0.52697 39-324 38.589 37-754 36.812 35.756 0.57283 0.54984 0.52724 0.50506 0.48335 48.905 48-342 47.668 46.870 45-937 0-55775 0.53137 0.50529 0-47955 0.45414 59-633 59.410 59.090 58-659 58-104 0-55937 0.53022 0.50134 0.47271 0-44434 70.995 7I-253 7L458 7I.6O2 71.677 75 76 77 78 79 0.56579 0-55252 0-53990 0.52796 0-51675 26.458 25.269 23.989 22.616 21.151 0.50963 0.49292 0.47688 0.46159 0.44710 34.578 33-271 31.828 30.242 28.507 0.46215 0.44151 0.42149 0.40216 0.38360 44.856 43-6io 42.180 40.548 38.692 0.42910 0.40446 0.38028 0.35658 0-33345 57405 56.542 55.487 54.206 52-659 0.41622 0.38836 0.36075 0-33339 0.30630 71.672 71-573 71.361 71.015 70.501 80 81 82 83 84 0.50631 0.49669 0.48794 0.48009 0.47320 19-594 17.946 16.210 14-391 12.495 0.43349 0.42083 0.40919 0.39867 0.38935 26.616 24-567 22.359 19-993 17-475 0.36591 0.34921 0.33362 0.31928 0.30635 36-59I 34-223 3!-569 28.608 25-333 0.31099 0.28928 0.26848 0.24877 0.23039 50.800 48.566 45-888 42.681 38.853 0.27951 0.25304 0.22691 O.2OI22 0.17608 69.774 68.774 67-4I3 65-561 63.018 85 86 87 88 89 0.46730 0.46241 0-45857 0.45581 0.45415 10.529 8.501 6.422 4-304 2-159 0.38129 0.37457 0.36927 0.36545 0.36314 14.813 I2.O22 9-I2I 6.132 3.081 0.29499 0.28538 0.27770 0.27208 0.26865 21.739 17.838 13-659 9-247 4.668 0.21368 0.19902 0.18684 0.17765 0.17193 34-303 28.949 22.746 15-729 8.051 0.15169 0.12853 O.IO7IO 0.08871 0.07570 59-474 54.420 47.029 36-059 20.198 90 0.45360 o.ooo 0.36236 O.OOO 0.26750 o.ooo 0.16997 o.ooo 0.07074 o.ooo Examples, cosh (1.3 /73) = 0.50506 /46.87o = 0.50506 /46.52'.i2" cosh" 1 (0.07074 /o) = 1.5 /9Q- [10] TABLE II. HYPERBOLIC COSINES. 1.6 1.7 i cosh (p ^S) = r /y. CONTINUED 8 1.9 2.0 o o o 45 1-4524 59.916 I-5556 65.149 1.6714 70.229 1.7999 75.I52 I.94I3 79.922 46 .4149 60.974 1.5141 66.336 1.6257 71.536 1.7496 76.569 1.8861 81.437 47 3774 62.021 1.4727 67.516 1.5802 72.837 1.6997 77-979 1-8313 82.947 48 .3400 63-051 1.4316 68.681 1-5350 74.125 1.6502 79-378 1.7771 84-445 49 .3028 64.066 1.3906 69-833 1.4901 75.403 i. 6012 80.768 1-7235 85-935 50 .2657 65.065 1-3499 70.973 1.4456 76.671 1-5526 82.149 1.6706 87.417 .2287 66.049 I-3095 72.IOO 1.4015 77.929 1.5046 83-522 1.6184 88.890 52 .1919 67.018 1.2693 73-217 1-3579 79.179 1-4572 84.890 1.5669 90-356 53 1554 67.971 1.2295 74-324 I.3I47 80.422 1.4104 86.252 1.5162 91.817 54 .1191 68.909 1.1900 75-421 1.2719 81.659 1-3643 87.609 1.4664 93-275 55 1.0831 69-833 1.1509 76.508 1.2297 82.892 1.3188 88.963 1.4174 94.731 56 1-0473 70.744 I.II2I 77.589 1.1879 84.122 1.2740 90.318 1-3693 96.188 57 1.0118 71.640 1.0737 78.663 1.1467 85-350 1.2299 91.674 1.3221 97.646 58 0-97653 72.522 1-0358 79.730 1.1061 86.576 1.1865 93.032 1.2758 99.108 59 0.94160 73-392 0.99826 80.795 i. 066 1 87.808 I-I439 94-398 1-2305 100.577 60 0.90699 74-249 0.96117 81.859 1.0266 89.045 1.IO2O 95-773 1.1861 102.057 61 0.87268 75-094 0.92451 82.919 0.98772 90.287 1.0609 97.I58 1.1427 103.548 62 0.83871 75.926 0.88831 83-982 0.94946 91.540 I.O207 98.559 1.1003 105.054 63 0.80508 76.748 0.85256 85-048 0.91182 92.807 0.98119 99.978 1.0589 106.580 64 0.77177 77o6i 0.81727 86.121 0.87480 94.092 0.94254 101.420 1.0186 108.129 65 0.73879 78.365 0.78245 87.204 0.83842 95.400 0.90472 102.891 0.97928 109.707 66 0.70615 79.161 O.748IO 88.300 0.80268 96.734 0.86773 104.394 0.94100 111.318 67 0.67383 79-951 0.71420 89.415 0.76759 98.101 O.86l6o 105-937 0.90378 112.968 68 0.64184 80.736 0.68078 90.551 o.733i4 99.508 0.79631 107.527 0.86764 114.663 69 0.61019 81.519 0.64782 91.717 0.69935 100.962 0.76190 109.170 0.83259 116.410 70 0.57887 82.300 0.61533 92.917 0.66624 102.473 0.72838 110.875 0.79865 118.215 0.54786 83.084 0.58331 94.161 0.63379 104.050 0.69576 112.653 0.76583 120.087 72 0-51715 83.872 0.55175 95-458 0.60202 105.706 0.66406 114.514 0.73414 122.035 73 0.48674 84.668 0.52065 96.819 0.57095 107.453 0.63329 116.471 0.70361 124.068 74 0.45663 85-477 0.49003 98.258 0.54061 109.308 0.60349 "8-537 0.67428 126.197 75 0.42680 86.304 0.45988 99-793 0.51100 111.291 0.57470 120.726 0.64618 128.432 76 0.39724 87.156 0.43021 101.445 0.48217 "3-423 0.54696 123.058 0.61934 130.785 77 0.36794 88.041 0.40104 103.241 0.45415 "5-732 0.52030 125.550 0.59381 133.267 78 0.33888 88.972 0.37238 105.215 0.42701 118.248 0.49480 128.224 0.56965 135.890 79 0.31006 89.963 0.34428 107.409 0.40082 121.008 0.47055 131.100 0.54691 138.664 80 0.28146 91.036 0.31679 109.880 0-37567 124.055 0.44764 134.201 0.52568 141.601 81 0.25309 92.221 0.28997 112.701 0-35167 127.438 0.42617 137-553 0.50603 144.713 82 0.22494 93-56i 0.26395 "5-971 0.32897 131.210 0.40626 141.172 0.48805 148.003 83 0.19700 95.128 0.23887 119.816 0.30779 135-433 0.38806 145.082 0.47184 151.474 84 0.16926 97-033 0.21497 124.413 0.28838 140.167 0.37174 149.292 0-4575 155-127 85 0.14177 99-473 O.I926I 129.984 0.27102 145-463 0-35749 153.806 0.44514 158.954 86 0.11467 102.829 O.I723I 136.806 0.25603 151-356 0.34550 158.614 0.43486 162.943 87 0.08809 107.920 0.15476 145.176 0.24378 157-847 0-33593 163.691 0.42674 167.072 88 0.06261 116.850 0.14093 I55-306 0.23469 164.877 0.32895 168.991 0.42088 171.315 89 0.04025 136-057 O.I3I95 167.116 0.22911 172.324 0.32472 174-453 0.41734 175-637 90 0.02920 180.000 0.12884 180.000 0.22720 180.000 0.32329 180.000 0.41615 180.000 Examples, cosh (2.0 /go ) = 0.41615 /i8o. cosh" 1 (0.54691 /i38.664) = 2.0 779. TABLE II. HYPERBOLIC COSINES, cosh (p {8) = r /_V CONTINUED 2.2 2-3 2.4 2-5 45 2.0958 84-55I 2.2636 89.050 2-4449 93.438 2.6403 97.730 2.8502 101.944 46 2.0350 86.159 2.1966 90.740 2.3711 95.202 2-5589 99.564 2.7603 103.842 47 1-9749 87-755 2.1306 92.417 2.2984 96.952 2.4788 101.380 2.672O 105.720 48 I-9I55 89-341 2-0655 94.081 2.2269 98.687 2.4001 103.179 2.5853 107.578 49 1.8569 90.917 2.0013 95-734 2.1567 100.408 2.3229 104.962 2.5004 109.417 5 1.7992 92.484 1.9382 97.376 2.0877 102.116 2.2473 106.729 2.4173 111.238 51 1.7424 94.043 1-8763 99.009 2.O2OO 103.812 2.1732 108.482 2.3361 113.041 52 1.6865 95-594 I-8I55 100.633 1-9537 105.498 2.1008 IIO.220 2.2568 114.826 53 1.6316 97.140 1-7559 102.249 1.8889 107.173 2.0301 III.944 2-1795 "6.593 54 1-5777 98.681 1.6976 103.858 1-8255 108.838 1.9612 113.656 2.IO42 118.344 55 I-5249 IOO.22O 1.6406 105.462 1.7637 110.495 1.8940 "5.356 2.0310 120.079 56 1-4732 101.758 1.5848 107.063 1-7034 112.145 1.8286 117.045 1-9599 121.799 57 1.4226 103.296 I-5303 108.662 1.6447 113.789 1.7651 118.724 1.8909 123.504 58 i-373i 104.836 1.4772 110.261 1.5876 115-430 1-7035 120-394 1.8241 125-195 59 1.3248 106.382 1.4256 111.862 1.5322 117.069 1.6438 122.057 1-7594 126.874 60 1.2776 107.936 1-3754 113.467 1.4784 118.708 1-5859 123.715 1.6969 128.541 61 1.2317 109.501 1.3266 115.079 1.4262 120.347 1.5299 125.369 1.6366 130.198 62 1.1870 III.O79 1.2792 116.700 1.3758 121.991 1.4758 I27.O20 1.5785 131.846 63 I-I435 112.673 1.2332 118.333 1.3270 123.641 1-4237 128.671 1.5226 133.486 64 I.IOI2 114.288 1.1887 119.982 1.2799 125.300 I.3736 130.322 1.4689 135.120 65 I. O6O2 115.928 I-I457 121.650 1-2345 126.970 1-3254 I3L977 I.4I73 136.749 66 I.02O4 H7-597 1.1042 123-339 1.1908 128.654 1.2791 I33-638 1.3679 138.374 67 0.98193 119.299 1.0642 125-055 1.1488 130.355 1-2347 I35-306 1.3207 139.998 68 0.94474 I2I.O4I 1.0256 126.801 1. 1086 132.077 1.1923 136.985 1.2756 141.622 69 0.90886 122.827 0.98860 128.581 1.0700 133-823 1.1518 138.676 1.2327 143-247 70 0.87429 124.662 0.95308 130.400 1-0332 135-595 1.1132 140.382 1.1919 144.876 7i 0.84105 126.554 0.91908 132.263 0.99804 137-397 1.0765 I42.IO6 i- I 532 146.511 72 0.80915 128.509 0.88663 134.174 0.96464 I39-233 1.0417 143.850 1.1166 148.153 73 0.77861 130.533 0-85573 136.137 0.93297 141.106 i. 0088 I45.6l6 1.0821 149.803 74 Q-74945 132.634 0.82638 138.158 0.90303 143.019 0.97785 147407 1.0497 151.464 75 0.72171 134.818 0.79862 140.242 0.87481 144.976 0.94877 149.225 1.0193 I53-I37 76 0.69541 137.094 0.77246 142.393 0.84833 146.980 0.92156 I5I.07I 0.99093 154-823 77 0.67060 139.468 0.74789 144.615 0.82360 149-033 0.89622 152.948 0.96459 156.524 78 0.64732 141.946 0.72497 146.911 0.80062 151-136 0.87276 I54-857 0.94026 158.241 79 0.62558 144-534 0.70372 149.285 0-77940 153-291 0.85117 156.798 0.91793 159-974 80 0.60544 147.237 0.68416 I5I-738 0.75996 155-499 0-83145 158.771 0.89757 161.723 81 0.58697 150.058 0.66632 154-271 0.74231 157.760 0.81361 160.778 0.87918 163.490 82 0.57021 152.997 0.65024 156.883 0.72646 160.074 0.79764 162.817 0.86276 165.274 83 0.55523 156.054 0.63595 159-572 0.71244 162.438 0-78354 164.888 0.84829 167.074 84 0.54209 I59-225 0.62349 162.334 0.70025 164.848 0.77131 166.987 0-83575 168.889 85 0-53085 162.503 0.61287 165.164 0.68990 167.301 0.76095 I69.II2 0.82518 170.717 86 0.52156 165.876 0.60413 168.054 0.68141 169.793 0.75247 171.260 0.81650 172-558 87 0.51428 169.329 0-59731 170.994 0.67479 172.316 0.74587 I73.427 0.80979 174.409 88 0-50905 172.849 0.59242 173-973 0.67006 174.863 0.74116 I75.6IO 0.80499 176.269 89 0.50590 176.413 0.58948 176.980 0.66722 177.427 0.73834 177.803 0.80207 178.134 90 0.50485 iSo.OOO 0.58850 180.000 0.66628 180.000 0.73740 iSo.OOO 0.80114 180.000 Examples, cosh (2.2 745) = 2.2636 /89.O5o = 2.2636 /89.o3'.oo < '. cosh" 1 (1.0821 /I49.8o3) = 2.5 773. [12] TABLE II. HYPERBOLIC COSINES, cosh (p /5) = r /T. CONTINUED 2.6 2.7 2.9 45 3-753 106.093 3-3I63 110.190 3-5741 114.248 3-8497 118.275 4-1443 122.282 46 2.9758 I08.O5I 3.2O62 112.207 3-4523 116.322 3-7I46 I 20.406 3-9945 124.469 47 2.8783 109.987 3-0984 114.199 3-3329 118.368 3-5826 122.506 3-8483 126.623 48 2.7827 III.9OI 2.9928 116.166 3.2162 ' 120.387 3-4536 124-575 3-7057 128.743 49 2.6891 113-793 2.8896 118.108 3.1023 122.378 3.3278 I26.6I4 3-5667 130.828 5 2-5977 115.663 2.7889 120.025 2.9913 124.340 3-2053 128.621 3-43J5 132.879 5i 2.5085 II7.5I2 2.6908 121.917 2.8832 126.273 3.0861 130.595 3.3001 134.894 52 2.4216 119.340 2-5952 123.784 2.7780 128.177 2.9703 132.536 3.1726 ' 136.873 53 2.3369 I2I.I47 2.5023 125.626 2.6759 130.052 2.8580 134-444 3.0490 I38.8I5 54 2-2545 122.931 2.4I2O 127.442 2.5769 131.898 2.7492 136.320 2-9294 I4O.72O 55 2.I74S 124.695 2.3245 129.233 2.4809 I33.7I5 2.6439 138.162 2.8137 142.587 56 2.0970 I26.44O 2.2397 131.000 2.3880 I35-502 2.5421 139.969 2.7020 I44-4I5 57 2.0219 128.165 2.1577 132.742 2.2983 I37-259 2-4439 141.742 2-5943 146.203 58 1.9492 129.872 2.0785 134.460 2. 2Il8 138.987 2.3492 143-479 2.4905 I47-95I 59 1.8790 131.560 2.0021 136-154 2.1284 140.685 2.2580 145.180 2.3907 149.659 60 1.8113 I33- 2 3I 1.9284 137.824 2.0481 142.354 2.1703 146.846 2.2949 I 5 I -32S 61 1.7460 134.885 1.8575 I39.47I 1.9709 143-993 2.0861 148.476 2.2030 152.949 62 1.6832 136.523 1.7894 141.095 1.8968 145.601 2.0053 150.071 2.1148 I 54-53Q 63 1.6229 138.146 I.724I 142.696 1.8259 147.179 1.9280 151.627 2.0304 156.067 64 1.5650 139-754 I.66I5 144.276 1.7580 148.728 1.8541 I53-I46 1.9498 I 57-559 65 1.5096 141.349 1. 6oi6 I45-834 1.6931 150.247 1-7835 154.628 1.8729 159.005 66 1.4566 142.932 1-5445 I47-37I I.63II I5L737 1.7162 156.071 1.7997 160.406 67 1.4060 144.504 1.4900 148.887 I.572I I53.I98 1.6522 157-477 1.7301 161.759 68 1-3578 146.066 1.4382 150-384 I.5l6l 154.628 I-59I4 158.844 1.6640 163.064 69 1.3120 I47.62O 1.3889 151.863 1.4629 156.030 1-5338 I6O.I72 1.6013 164.321 70 1.2685 149.164 1.3422 I53-322 I.4I26 157.404 1-4793 161.460 1.5420 165.528 7i 1.2273 I50-705 1.2981 154.764 1.3650 158.749 1.4278 162.709 1.4861 166.685 72 1.1885 152.239 1-2565 156.189 1.3202 160.066 1-3793 163.919 1-4335 167.792 73 i-^Q I53-770 1.2174 157-599 1.2782 I6I.355 1.3338 165.090 1.3840 168.848 74 1.1176 I55-298 1.1807 158.994 1.2388 162.617 1.2912 166.222 1-3377 169.853 75 1.0855 156.825 1.1465 160.374 I.2O2O 163.853 1.2514 167.315 1.2946 170.806 76 1.0556 I5 8 -35I 1.1147 161.742 I.I677 165.063 1.2144 168.370 1.2545 171.708 77 1.0278 159.878 1.0852 163.096 I.I360 166.248 1. 1802 169.387 1.2174 172.560 78 I.OO22 161.406 1.0580 164.440 1.1069 167.409 1.1486 170.368 1.1832 173.362 79 0.97880 162.937 1-0331 165.773 1 .0803 168.547 I.I2OO I7I-3I3 1.1519 174.116 80 0-95745 164.471 1.0105 167.096 1.0561 169.664 1.0938 172.225 J-^SS 174.823 81 0.93821 I66.008 0.99006 168.411 1.0342 170.760 I.O7O2 173.104 1.0979 175-485 82 0.92106 167.549 0.97189 169.718 I.OI47 171.837 I.O492 173-953 1.0751 176.104 83 0.90596 169.094 0.95592 171.018 0.99765 172.897 1.0307 174-774 1-0550 176.685 84 0.89290 170.643 0.94211 172.312 0.98287 173.942 I.OI48 175-571 1.0377 177.229 85 0.88l87 172.196 0.93045 173.601 0.97041 174-973 I.OOI4 176.344 1.0231 177.742 86 0.87286 I73.752 0.92093 174.886 0.96025 175-993 0.99042 177.099 I.OII2 178.228 87 0.86587 I75.3II 0.91354 176.168 0.95235 177.003 0.98190 177.838 I.OOI9 178.691 88 0.86089 176.873 0.90827 177-447 0.94671 178.006 0.97582 178.566 0.99528 179.137 89 0.8 5 78 9 178.436 0.90512 178.724 0-94334 179.004 0.97217 I79-285 0.99130 I7.9-S7I 90 0.85689 iSo.OOO 0.96407 180.000 O.94222 iSo.OOO 0.97096 iSo.OOO 0.98999 180.000 Examples, cosh (2.8 785) = 0,97041 /I74.g73 = 0.97041 /i74.g8 / .23* cosh" 1 (i.5420/i65.528) = 3.0 /7Q. [13] TABLE III. HYPERBOLIC TANGENTS, tanh (p [$) = r o-3 0.4 o-S 45 4 6 47 4 8 49 o.ioooo 44.812 o.ioooi 45.812 O.IO002 46.812 0.10003 47.812 0.10004 48.813 0.19997 44-235 0.20006 45.236 0.20015 46.237 0.20024 47-239 0.20034 48.242 0.29981 43.282 0.30012 44.281 0.30043 45.280 0.30075 46.283 0.30107 47.290 0.39921 41-954 0-39995 42.945 0.40069 43.942 0.40143 44-942 0.40217 45-947 0.49757 40.250 0.49902 41.229 0.50047 42.219 0.50192 43.213 0.50340 44.213 50 Si 52 53 54 0.10006 49.813 0.10007 50.813 0.10008 51.814 0.10009 52.816 o.iooio 53.818 0.20043 49-245 0.20053 50.25 1 0.20062 51.256 0.20071 52.262 0.20081 53.268 0.30138 48-297 0.30169 49.306 0.30201 50.315 0.30232 51.328 0.30263 52.344 0.40293 46.955 0.40370 47-969 0.40446 48.986 0.40521 50.005 0.40596 51.031 0.50490 45.220 0.50639 46.233 0.50789 47-252 0.50939 48.276 0.51089 49.308 55 56 57 58 59 o.ioon 54.820 0.10013 55.823 0.10014 56.825 0.10015 57.828 0.10016 58.830 0.20090 54.276 0.20099 55-285 0.20107 56.295 0.20115 57-306 0.20124 58.318 0.30294 53.362 0-30325 54-382 0-30355 55-404 0-30385 56.426 0,30414 57.452 0.40671 52.058 0.40746 53.089 0.40820 54.124 0.40892 55.164 0.40964 56.207 0.51239 50.344 0.51389 51.388 0-51538 52.439 0.51687 53.494 0.51835 54.556 6o 61 62 63 64 0.10017 59-833 0.10018 60.837 0.10019 61.841 O.I002O 62.845 O.IOO2O 63.849 0.20132 59.332 0.20140 60.345 0.20148 61.360 0.20156 62.375 0.20164 63.391 0.30444 58.479 0.30473 59-5 I o 0.30501 60.542 0.30528 61.575 0-30555 62.610 0.41037 57-255 0.41107 58.305 0.41176 59-359 0.41244 60.417 0.41311 61.479 0-51983 55-625 0.52128 56.700 0.52271 57.781 0.52412 58.867 0-52552 59-959 65 66 67 68 69 O.IOO2I 64.853 0.10022 65.857 O.IOO23 66.862 O.IOO24 67.866 O.IOO25 68.871 0.20171 64.408 0.20179 65.425 0.20186 66.443 0.20193 67.461 0.20199 68.480 0.30580 63.647 0.30605 64.686 0.30630 65.727 0.30655 66.769 0.30678 67.813 0.41376 62.544 0.41441 63.612 0.41504 64.685 0.41565 65.760 0.41623 66.837 0.52689 61.058 0.52824 62.164 0.52957 63.276 0.53085 64.391 0.53209 65.511 70 72 73 74 O.IOO26 69.877 0.10026 70.882 0.10027 71.888 0.10028 72.893 0.10028 73.898 0.20206 69.501 O.2O2I2 7O.522 0.20217 71.544 O.2O223 72.566 O.2O228 73.589 0.30701 68.859 0.30722 69.906 0.30742 70-955 0.30762 72.005 0.30781 73-056 0.41680 67.918 0.41735 69.002 0.41788 70.089 0.41838 71,179 0.41886 72.272 0-53330 66.638 0.53446 67.771 0.53560 68.909 0-53669 70-052 0-53773 71-200 75 76 77 78 79 O.IOO29 74.904 0.10029 75-9 10 0.10030 76.916 0.10030 77.922 0.10031 78.928 O.2O234 74.612 0.20239 75.635 0.20243 76.659 0.20246 77.684 0.2O25O 78.709 0.30799 74.108 0.30816 75.162 0.30831 76.217 0.30846 77-273 0.30860 78.329 0.41932 73.368 0.41975 74.466 0.42016 75.566 0.42054 76.668 0.42088 77.771 0.53872 72.351 0-53965 73-507 0.54054 74-667 0.54138 75.831 0.54215 76.999 80 81 82 83 84 0.10031 79-934 0.10032 80.940 0.10032 81.946 0.10032 82.953 0.10033 83.960 0.20254 79-734 0.20257 80.759 0.20260 81.785 0.20263 82.812 0.20265 83.838 0.30872 79-386 0.30884 80.445 0.30894 81.506 0.30904 82.567 0.30912 83.628 0.42120 78.876 0.42150 79-983 0.42177 81.092 0.42201 82.203 0.42222 83.315 0.54285 78.170 0-54349 79-344 0.54407 80.520 0.54459 81.699 0.54503 82.881 85 86 87 88 89- 0.10033 84.967 0.10033 85.974 0.10033 86.981 0.10033 87.988 0.10033 88.994 0.20267 84.864 0.20268 85.891 0.20270 86.918 0.20270 87.946 0.20271 88.973 0.30920 84.689 0.30926 85.751 0.30930 86.813 0.30933 87.875 0.30934 88.937 0.42239 84.427 0.42252 85.540 0.42264 86.654 0.42274 87.768 0.42279 88.883 0.54540 84.065 0.54571 85.251 0.54596 86.438 0.54616 87.626 0.54628 88.813 90 0.10033 90.000 0.20271 90.000 0.30934 90.000 0.42280 90.000 0.54631 90.000 Note. tanh (o /J) = o /j Examples. tanh (0.5 /6o) = 0.51983 /55.625 = 0.51983 /55-37'. ,?o". [14] TABLE III. HYPERBOLIC TANGENTS, tanh (p 5) = r /j. CONTINUED 0.6 0.7 0.8 0.9 45 0.59406 38.183 0.68732 35.786 0-77577 33-098 0-85756 30.161 0.93077 27.044 46 0.59650 39.146 0.69109 36.719 0.78117 33.980 0.86480 30.980 0.93909 27.784 47 0.50898 40.119 0.69495 37.663 0.78671 34.878 0.87229 31.815 0.94950 28.539 48 0.60149 4 I -99 0.69888 38.617 0.79240 35.790 0.88001 32.666 0.95938 29.308 49 0.60403 42.088 0.70287 39.581 0.79824 36.715 0.88799 33-531 0.96966 30.092 5 0.60660 43.085 0.70694 40-557 0.80421 37.653 0.89620 34.412 0.98032 30.892 5i 0.60920 44.092 0.71107 41-545 0.81031 38.605 0.90466 35.310 0.99136 31.710 52 0.61182 45.107 0.71527 42.543 0-81655 39-573 0.9*337 36.223 1.00282 32.545 53 0.61447 46.130 0.71952 43-555 0.82291 40.556 0.92231 37.155 1.01469 33.400 54 0.61713 47.162 0.72382 44-577 0.82940 41.554 0.93150 38.105 1.02697 34-274 55 0.61980 48.203 0.72817 45.612 0.83601 42.568 0.94094 39.075 1.03970 35.172 56 0.62248 49.254 0-73257 46.662 0.84274 43.599 0.95063 40.066 1.05287 36.091 57 0.62517 50.315 0.73700 47-725 0.84957 44-647 0.96056 41.078 1.06648 37-035 58 0.62785 51.384 0.74145 48.800 0.85649 45.712 0.97069 42.111 1.08054 38.004 59 0-63053 52-463 0.74593 49-888 0.86351 46.797 0.98106 43.167 1.09506 39.002 60 0.63322 53-552 0.75047 50.990 0.87063 47.900 0.99168 44.247 1.11009 40.026 61 0.63588 54-650 0-75499 52-107 0.87781 49.022 1.00251 45-353 1.12555 41.08,2 62 0-63852 55-758 0-75950 53-238 0.88504 50.165 1.01353 46.486 1.14144 42.168 63 0.64115 56.876 0.76400 54.383 0.89232 51.327 1.02471 47-645 I-I5777 43-289 64 0.64376 58.002 0.76848 55.542 0.89965 52.509 1.03604 48.831 I-I7454 44-445 65 0.64633 59.138 0.77294 56.716 0.90698 53.712 1-04753 50.044 1.19173 45-638 66 0.64886 60.284 0-77737 57-902 0.9U33 54.936 1.05916 51.289 1.20935 46.869 67 0-65135 61.439 0.78177 59.105 0.92166 56.180 1.07090 52.562 1.22737 48.140 68 0.65380 62.603 0.78609 60.322 0.92894 57.448 1.08268 53.868 1.24569 49-455 69 0.65618 63.777 0.79033 61.553 0.93616 58.737 1.09447 55.204 1.26429 50.812 70 0.65850 64.959 0.79450 62.798 0.94332 60.047 1.10627 56.572 1.28316 52.215 ?i 0.66075 66.150 0.79858 64.057 0.95037 61.379 1.11803 57.974 1.30221 53.666 72 0.66294 67.350 0.80256 65.329 0.95727 62.732 1.12972 59.408 1.32140 55.164 73 0.66505 68.559 0.80641 66.614 0.96402 64.106 1.14126 60.874 1.34063 56.710 74 0.66708 69.775 0.81014 67.913 0.97060 65.501 1.15260 62.372 I-35986 58-307 75 0.66902 70.998 0.81371 69.225 0.97697 66.917 1.16370 63.903 1.37894 59-957 76 0.67086 72.228 0.81713 70.550 0.98311 68.352 1.17450 65.468 1-39775 61.658 77 0.67261 73.466 0.82038 71.886 0.98898 69.808 1.18493 67.064 1.41620 63.409 78 0.67425 74-7" 0-82345 73.233 0-99455 71-282 1.19495 68.691 1.43412 65.212 79 0.67577 75.962 0.82632 74.591 0.99981 72.773 1.20447 70.347 1.45141 67.065 80 0.67718 77.219 0.82899 75.958 1.00473 74.282 1.21344 72.034 1.46790 68.968 81 0.67847 78.481 0.83144 77.334 1.00926 75.805 1.22179 73-746 1.48345 70-919 82 0.67964 79.749 0.83366 78.719 I-OI339 77-344 1.22946 75.483 1.49790 72.913 83 0.68068 81.021 0.83564 80.112 1.01710 78.896 1.23640 77-243 1.51110 74-948 84 0.68159 82.296 0.83738 81.512 1.02036 80.460 1.24253 79-025 1.52289 77-023 85 0.68236 83.575 0.83887 82.917 1.02316 82.034 1.24781 80.825 I-533H 79.I32 86 0.68299 84.856 0.84009 84.328 1.02548 83.617 1.25219 82.640 1.54170 81.269 87 0.68349 86.140 0.84105 85.743 1.02730 85.206 1.25566 84.468 1.54848 83.429 88 0.68385 87.426 0.84173 87.161 1.02860 86.801 1.25814 86.308 1-55339 85.609 89 0.68406 88.713 0.84214 88.580 1.02937 88.400 1.25963 88.153 I-55637 87.802 90 0.68413 90.000 0.84229 90.000 1.02960 90.000 1.26015 90.000 1-55740 90-000 Eaxmples. tanh (0.9 /77) = 1.18493 /67.o64 = 1.18493 /67-03'. 50*. (0.66708 = Q.6 /74. [15] TABLE III. HYPERBOLIC TANGENTS, tanh (p IS) = r y. CONTINUED i.i 1.2 1.4 45 0.99389 23-833 1.0457 20.616 1.0857 17.464 1. 1143 14.484 1-1323 11.712 46 1.0049 24.476 1.0582 21.144 1.0993 17.882 1.1284 14-775 1.1464 11.884 47 1.0164 25-131 1.0713 21.685 1.1136 18.301 1.1433 15.064 1.1613 12.048 48 1.0283 25-799 1.0851 22.236 1.1286 18.721 1.1590 15.352 1.1770 12.208 49 1.0408 26.480 1.0995 22.798 I-I445 19.148 1.1756 15.640 1.1936 12.362 5 1-0539 27.178 1.1147 23-371 1.1613 19.584 1.1932 I5-930 1.2112 12.510 1.0676 27.891 1.1307 23-957 1.1789 20.027 1.2118 16.222 1.2299 12.652 52 1.0818 28.621 1.1474 24-558 1.1976 20.479 1-2315 16.515 1.2498 12.787 S3 1.0967 29.370 1.1650 25-I74 1.2173 20.941 1.2524 16.811 1.2709 12.914 54 1.1122 30.139 1.1835 25.807 1.2381 21.414 1.2746 17.110 1-2933 13-035 55 I.I284 30.930 1.2030 26.460 1.2602 21.900 1.2982 I7-4I3 I.3I72 13-151 56 I-I453 3 J -744 1.2235 27-I33 1.2836 22.401 1.3234 17.721 L3427 13.262 57 1.1629 32-583 1.2450 27.827 1.3084 22.918 1.3502 18.036 1.3700 13.366 58 1.1813 33-449 1.2677 28.546 I.3346 23-452 1.3787 18.358 1.3992 13.466 59 1.2005 34-343 1.2916 29.292 1.3626 24.007 1.4093 18.689 I-430S 13-559 6o 1.2206 35-267 .3168 30.068 I.3923 24.584 1.4421 19.032 1.4642 13-651 <6i 1.2415 36.226 3434 30.876 1-4239 25.187 1.4772 19.389 I-5005 13-738 62 1.2632 37.222 3714 31.720 I.4576 25.820 1.5148 19.761 I-5396 13.824 63 1.2858 38-255 .4009 32.603 1-4935 26.485 1-5553 20.150 1.5819 13.908 64 1.3094 39.328 .4321 33-527 I-53I9 27.186 1.5990 20.562 1.6277 i3-99i 65 1-3339 40.446 1.4649 34.498 I-5729 27.928 1.6463 20.999 1.6776 14.076 66 1-3593 41.612 1-4995 35-520 1.6167 28.715 1.6974 21.465 I-73I9 14.166 67 1.3856 42.827 1-5359 36.597 1.6637 29-554 1.7528 21.966 I.79H 14.261 68 1.4129 44.096 1-5743 37-737 1.7140 30.452 1.8131 22.507 1.8566 14.364 69 1.4411 45.421 1.6148 38-941 1.7680 3I-4I5 1.8788 23.096 1.9285 14.480 70 1.4702 46.806 1.6573 40.219 1.8260 32-452 1.9506 23.740 2.0078 14.612 71 1.5001 48.256 1.7020 41-575 1.8882 33-570 2.0293 24.448 2.0959 14-765 72 1-5307 49-773 1.7488 43-017 I-955I 34-782 2.1158 25.232 2.1942 14.944 73 1.5620 51.361 .1-7977 44-553 2.0269 36.101 2.2III 26.106 2.3043 .15-159 74 I-5938 53-022 1.8488 46.190 2.1041 37-539 2.3164 27.084 2.4285 15-419 75 1.6261 54.762 1.9019 47-936 2.1869 39.110 2-4332 28.186 2.5694 15-734 76 1-6585 56.581 1.9568 49.798 2.2757 40.830 2.5631 29-433 2.7303 16.120 77 1.6910 58.482 2.0134 51-786 2.3706 42.720 2.7078 30-855 2-9*57 16.597 78 1.7232 60.468 2.0713 53.906 2.4716 44-798 2.8697 32.486 3-1312 17.186 79 I -755 62.538 2.1299 56.165 2.5786 47.088 3-0509 34-367 3-3843 17.924 80 1.7858 64.692 2.1887 58.571 2.6912 49.611 3.2536 36-546 3-6845 18.856 81 1-8155 66.931 2.2471 61.126 2.8085 52-391 3.4807 39.085 4.0459 20.044 82 1.8435 69.252 2.3040 63.832 2.9290 55-447 3-7337 42.056 4.4876 21.578 83 1.8696 71.650 2-3585 66.690 3.0506 58.800 4-0135 45-546 5-0365 23-588 84 1-8933 74.121 2.4094 69.694 3-I704 62.460 4.3188 49.646 5-73" 26.276 85 1.9142 76.657 2-4555 72.836 3-2845 66.432 4.6429 54-459 6.6297 29-955 86 1.9318 79.252 2-4955 76.102 3-3884 70.706 4-9729 60.069 7.8015 35-134 87 1.9460 81.895 2.5281 79-475 3-4767 75-253 5.2871 66.523 9-34I3 42.644 88 1.9564 84-575 2.5523 82.933 3-5446 80.030 5-5531 73-786 11.259 53-726 89 1.9627 87.281 2.5671 86.452 3-5875 84.971 5-7333 81.707 13.182 69-695 90 1.9648 90.000 2.5721 90.000 3.6021 90.000 5.9978 90.000 14.101 90.000 Examples. tanh (i 4 /6 4 ) = 1-5990 /20.562 = 1.5990 /20.33' 43'- tanh- 1 (1.7550 /62.538 = i.i /79. [16] TABLE III. HYPERBOLIC TANGENTS, tanh (p /8) = r /. CONTINUED 1.6 1.8 1.9 45 1.1413 9.201 1.1428 6.984 1-1385 5-063 .1302 3-438 1.1191 2.094 46 1.1548 9.267 I.I553 6.952 1.1499 4-950 .1401 3.260 1.1275 1.867 47 1.1691 9.318 1.1686 6.902 1.1618 4.814 1505 3-058 1-1363 1.613 48 1.1842 9-358 1.1826 6.834 I-I743 4-659 .1614 2.832 I-I455 1-333 49 I.2OO2 9-38S 1.1974 6-749 1.1876 4.480 .1728 2.578 I-I550 1-023 So .2171 9.400 1.2131 6.645 1.2016 4-275 .1848 2.295 1.1650 0.681 2351 9.400 1.2296 6.520 1.2163 4-045 1975 1.981 I-I754 0.306 52 2541 9-384 1.2472 6-373 1.2318 3-787 .2107 1.634 1.1863 0.103 53 .2744 9-354 1.2659 6.203 1.2483 3-499 2247 1-253 1.1977 0.550 54 .2960 9-309 1.2857 6.008 1.2657 3.180 2394 0-835 1.2096 1.039 55 .3190 9.246 1.3067 5.788 1.2842 2.826 2549 0-379 I.222O I-57I 56 3435 9.166 1-3292 5-539 1-3038 2.436 .2713 O.I2I 1.2350 2.149 57 -3698 9.069 1-3532 5.261 1-3247 2.008 1.2886 0.666 1.2487 2.777 58 3979 8-953 1.3788 4-953 1.3469 1.540 1.3069 1.259 1.2630 3.458 59 .4281 8.817 1.4063 4.611 1.3706 1.026 1.3263 1.905 1.2781 4-195 6o 1.4606 8.661 I-4358 4.232 1-3959 0.465 1-3470 2.606 1.2939 4-995 61 1.4956 8.484 I-4675 3.819 1-4230 0.145 1.3689 3.366 I.3I06 5.860 62 1-5334 8.286 1.5017 3.365 1.4521 0.810 1.3922 4.191 I.328I 6-793 63 1-5743 8.065 1.5386 2.869 1-4833 1-533 1.4170 5-085 1.3466 7.802 64 1.6187 7.819 I-5785 2.326 1.5169 2.318 1-4435 6.052 1.3662 8.891 65 1.6670 7-548 1.6218 1-734 I-553I 3.172 1.4718 7.099 1.3868 10.068 66 1.7198 7-252 1.6690 1.090 i-59~23 4.098 1.5022 8.232 1.4086 11-338 67 1.7777 6.928 1.7206 0.387 1-6347 5.104 1-5347 9-459 I-43I7 12.707 68 1.8413 6-575 1.7770 0.376 1.6807 6.196 1.5696 10.786 1.4562 14.184 69 1.9116 6.191 1-8389 1.209 1.7308 7.382 1.6070 12.221 1.4821 I5.776 70 1.9893 5-776 1.9072 2.116 1-7854 8.671 1.6474 13-773 I-5095 17.490 2.0759 5.326 1.9828 3.106 1.8451 10.074 1.6908 15-454 I-5385 19-337 72 2.1728 4.840 2.0667 4.187 1.9105 11.602 1.7376 17-273 I-5693 21.326 73 2.2819 4-3U 2.1603 5.371 1.9823 13.266 1.7881 19.244 I.60I9 23.466 74 2.4053 3-744 2.2652 6.670 2.0613 15.084 1.8425 21.379 1.6363 25.769 75 2.5461 3.127 2.3834 8.101 2.1484 17.076 1.9011 23.692 1.6725 28.245 76 2.7079 2-457 2.5172 9-685 2.2446 19.261 1.9641 26.202 I.7I05 30-905 77 2.8957 1.726 2.6696 11.447 2.3510 21.664 2.0317 28.925 I-7503 33-76o 78 3- IJ 57 0.922 2.8442 13-420 2.4685 24-3I5 2.1040 31.881 I.79I7 36.820 79 3.3766 0.034 3-0455 15.644 2.5984 27.250 2.1809 35.088 I-8344 40.093 80 3.6909 0.960 3.2790 18.175 2.7416 30.510 2.2621 38.568 . 1.8781 43-590 81 4-0755 2.087 3-55I9 21.083 2.8987 34.140 2.3469 42.344 1.9222 47-319 82 4-5558 3-390 24.466 3.0700 38.191 2-4344 46.428 1.9661 51.281 83 5.1722 4-937 4.2487 28.446 3-2540 42.722 2.5230 50.842 2.OO9O 55-474 84 5.9896 6.839 4.6927 33-199 3-4477 47.793 2.6104 55.590 2.0497 59.894 85 7.1203 9.289 5.2102 38.945 3-6459 53.448 2.6940 60.673 2.0872 64.528 86 8.7720 12.668 5-7994 45-956 3-8392 59.720 2.7701 66.075 2.I2O2 69-359 87 11.388 17.791 6.4356 54-527 4-oi57 66.606 2.8350 71.768 2.1476 74-359 88 I5-990 26.760 7-0503 64.868 4.1592 74-043 2.8849 77.700 2.1680 79-494 89 24.844 46.011 7-5I93 76.896 4-253I 81.905 2.9163 83.805 2.1807 84.723 90 34.232 90.000 7.6968 90.000 4.2863 90.000 2.9271 90.000 2.1850 90.000 Note. Negative quantities are in heavy type. Examples, tanh (1.6/54) = 1.2960 /9.309 = 1.2960 /9.i8'.32*. tanh (2.0/64^) = 1.3662 \8.89i = 1.3662 \8.53'.28'. tr 1 (1.4718 \7.o9 9 ) = 1.9/65. TABLE III. HYPERBOLIC TANGENTS, tanh (p [S) = r /jy. CONTINUED 2.3 2.4 2.5 45 1.1065 1.007 i .0932 0-155 1.0799 0.492 1.0672 0.961 1-0553 1.283 46 1.1134 0.746 i .0987 0.127 1.0842 0.783 1.0702 1.252 1-0573 1.564 47 1. 1206 0.458 i .1044 0.436 1.0884 I.IOI 1.0732 1.567 1.0592 1.868 48 1.1280 0.141 i .1102 0-774 1.0927 1.446 1.0762 1.908 i. 0610 2.195 49 I-I357 0.206 i .1161 1.142 1.0970 1.820 1.0790 2.277 1.0626 2.548 5 I-I437 0.586 i .1222 1.542 1.1013 2.226 1.0818 2.676 i .0640 2.929 5i 1.1519 I.OOI i .1284 1.978 1.1056 2.666 1.0845 3.107 1-0653 3-340 52 1.1604 1.452 i 1347 2.452 1.1099 3-144 1.0870 3.572 1.0663 3-782 S3 1.1693 1-943 i .1411 2.965 1.1142 3-659 1.0894 4.073 1.0671 4-256 54 1.1784 2.476 i 1477 3-520 1.1185 4.215 1.0917 4.614 1.0676 4.766 55 1.1878 3-055 i -1544 4.120 1.1227 4.815 1.0937 5.196 1.0678 5-314 56 1.1976 3.682 i .l6l2 4.769 1.1268 5-462 1-0955 5.822 1.0677 5-903 57 1.2078 4.361 i .1681 5-469 1.1309 6.160 1.0971 6.496 1.0672 6-534 58 1.2183 5-094 i -I75I 6.226 1.1348 6.912 1.0984 7.220 1.0662 7.211 59 1.2292 5.888 i .1822 7.042 1.1386 7.722 1.0993 7.998 1.0647 7.938 60 1.2405 6-745 i .1894 7.921 1.1422 8-594 1.0999 8.835 1.0628 8.718 61 1.2522 7.671 i .1967 8.869 1-1457 9-531 I.IOOI 9-735 i .0603 9.556 62 1.2644 8.669 i .2041 9.889 1.1490 10.540 1.0999 10.701 1.0572 10.455 63 1.2771 9-744 i .2Il6 10.988 1.1520 11.625 1.0992 11.740 1-0535 11.419 64 1.2903 10.902 i .2192 12.171 1.1548 12.791 1 .0980 12.855 i .0490 12.455 65 I-3039 12.151 i .2268 13-443 1-1573 14.044 1.0962 14-053 1.0438 13-567 66 1.3181 13-496 i -2344 14.810 1-1595 15-390 1.0939 15-341 1.0378 14-761 67 I-3329 14.942 i .2421 16.280 1.1613 16.836 1.0909 16.723 1.0310 16.044 68 1-3483 16.499 i .2499 17-858 1.1628 18.389 1.0872 18.208 1.0232 17.424 69 1.3642 18.172 i .2576 I9-552 1.1639 20.056 1.0829 19.802 1.0145 18.906 70 1.3808 19.968 i .2654 21.370 1.1645 21.843 1.0779 21.514 1.0049 20.500 7i 1.3980 21.898 i .2732 23-319 1.1647 23-759 1.0721 23-352 0.99427 22.215 72 I-4I59 23.968 i .2809 25-405 1.1644 25.811 1-0655 25-324 0.98265 24.061 73 1-4344 26.185 i -2885 27.636 i-.i637 28.006 1.0582 27-439 0.97006 26.048 74 1-4535 28.560 i .2961 30.020 1.1625 30.352 1.0501 29.706 0.95649 28.187 75 1-4732 31.099 i .3036 32.564 1. 1608 32.858 1.0413 32.134 0.94202 30.489 76 1-4934 33-8II i .3109 35-274 1.1587 35-531 1.0318 34-730 0.92671 32-966 77 1.5140 36.702 i .3l8l 38.156 1.1561 38.374 1.0218 37-505 0.91068 35-633 78 I-5348 39-778 i 3251 41.212 I-I53I 41.392 1.0113 40.467 0.89405 38.501 79 1-5559 43-043 1-3318 44-446 1.1498 44.590 1.0004 43.620 0.87701 41-582 80 1.5768 46.501 i. .3383 47-859 1.1462 47.968 0.98931 46.968 0.85979 44-887 81 1-5974 50.151 i 3444 51-449 1.1425 5I.527 0.97822 50-517 0.84264 48.427 82 1.6174 53.989 i 35oo 55-212 1.1387 55-264 0.96733 54-267 0.82584 52.209 83 1.6364 58.011 i, 3552 59.140 I-I349 59.I7I 0.95686 58.212 0.80977 56.233 84 1.6541 62.210 i, 3599 63.223 1.1312 63-237 0.94710 62.344 0.79476 60.499 85 1.6700 66.570 i, ,3640 67.449 1.1279 67-450 0.93826 66.650 0.78120 64.994 86 1.6837 71.070 i, 3674 71.800 1.1250 71-793 0.93061 71.113 0.76946 69.703 87 1.6949 75.690 i. 372 76.254 1.1225 76.245 0.92439 75-710 0.75991 74.598 88 1.7031 80.406 i, 3722 80.790 1.1207 80.781 0.91979 80.413 0.75283 79-645 89 1.7081 85.187 i. 3734 85.381 1.1196 85.376 0.91696 85.188 0.74849 84.796 90 1.7099 90.000 I -3738 90.000 1.1192 90.000 0.91601 90.000 0.74702 90.000 Note. Negative quantities are in heavy type. Examples. tanh (2.1 748) = 1.1280 /o.i4i = 1.1280 /o.o8'.28*. tanh (2.1 /49) = 1.1357 \o.2o6 = 1.1357 \o i2'.22*. (1.0318 \34-73o) = 2.4 776. [18] TABLE III. HYPERBOLIC TANGENTS, tanh 2.6 2.7 2.8 (p /$) = r /y. CONTINUED 2.9 3.0 o o o 45 1.0445 1.480 1.0348 1-576 1.0264 1-595 1.0192 1-554 1.0131 1.468 46 1.0456 1.744 I.0352 I.82I 1.0262 1.816 1.0185 1.748 I.OI2O 1.637 47 1.0465 2.030 1-0354 2.083 1.0257 2.051 1.0175 1-955 I.OI07 1.815 48 1-0473 2.337 1-0353 2.365 1.0250 2.303 1.0163 2.175 I.OOgi 2.OO2 49 1.0479 2.667 1.0350 2.666 1.0240 2.572 1.0148 2.409 I.OO72 2.198 50 1.0483 3.022 1.0344 2.988 1.0227 2.857 1.0129 2.656 I.OO49 2.405 51 1.0483 3.403 1-0335 3-332 I.02IO 3.160 1.0106 2.916 1.0023 2.622 52 1.0480 3.812 1.0323 3.700 I.OlSg 3-483 1.0080 3-191 0.99922 2.849 53 1-0475 4.250 1.0307 4.094 1.0164 3-826 1.0049 3.481 0.99572 3.086 54 1.0466 4.717 1.0286 4-513 i- OI 35 4.190 1.0013 3.788 0.99174 3-334 55 1.0452 5.219 I.O26O 4-959 I.OIOI 4-577 0.99722 4.112 0.98722 3-593 56 1.0434 5.756 1.0230 5-437 1. 006 1 4.987 0.99254 4-452 0.98211 3.862 57 1.0412 6.331 I.OI94 5-945 1.0015 5-422 0.98724 4.810 0.97637 4.141 58 1-0385 6.947 I.OI52 6.487 0.99625 5-884 0.98126 5-186 0.96995 4-432 59 1-0351 7.606 I.OIO4 7.066 0.99027 6.374 0-97454 5-582 0.96279 4-735 60 1.0311 8.313 I.OO48 7.684 0-98353 6.895 0.96701 5-999 0.95481 5-049 61 1.0265 9.069 0.99847 8-343 0-97597 7-449 0.95862 6-439 0-94597 5-375 62 i. 02 1 1 9.879 0.99129 9.048 0.96751 8.036 0.94930 6.904 0.93619 5-713 63 1.0149 10.747 0.98322 9.801 0.95807 8.660 0.93897 7-392 0.92540 6.063 64 1.0079 11.677 O.9742O 10.607 0.94760 9.325 0.92756 7.907 0.91352 6.426 65 0.99990 12.675 0.96414 11.470 0.93603 10.033 0.91500 8.452 0.90047 6.801 66 0.99099 13.747 0.95300 12.393 0.92328 10.789 0.90121 9.027 0.88618 7.192 67 0.98106 14.898 0.94070 13-383 0.90927 H.599 0.88612 9.639 0.87055 7.598 68 0.97006 16.136 0.92720 14.449 0.89396 12.464 0.86964 10.289 0.85350 8.021 69 0-95794 17.469 0.91243 15-597 0.87726 13-396 0.85169 10.982 0.83495 8.463 70 0.94467 18.902 0.89635 16.833 0.85912 14-399 0.83221 11.722 0.81482 8.926 71 0.93022 20.451 0.87893 18.168 0.83950 15-482 0.81113 12.518 0.79302 9.414 72 0.91458 22.122 O.860I3 19-613 0.81834 16.655 0.78839 I3.378 0.76946 9.932 73 0.89775 23.929 0.83994 21.184 0.79562 I7.93I 0.76394 14-310 0.74408 10.485 74 0.87976 25.884 0.8l839 22.893 0.77134 19-325 0-73774 15-328 0.71683 11.081 75 0.86066 28.003 0-79552 24-757 o.7455i 20.857 0.70977 16.448 0.68765 11.729 76 0.84053 30.302 0.77140 26.799 0.71815 22.547 0.68005 17.690 0.65650 12.443 77 0.81949 32-799 O.746I2 29.038 0.68938 24:421 0.64861 19.078 0.62338 13-241 78 0.79769 35.5I3 0.71985 31-505 0.65930 26.512 0.61551 20.647 0.58829 14.146 79 0-77535 38.466 0.69278 34-231 0.62807 28.861 0.58087 22.437 0-55129 15.191 80 0.75271 41.678 O.6652O 37-249 0.59596 3L5I9 0.54486 24.504 0.51245 16.419 81 0.73010 45-171 0.63743 40.599 0.56326 34-543 0.50773 26.921 0.47190 17-893 82 0.70790 48.966 0.60990 44-322 0.53039 38.004 0.46981 29.784 0.42986 19.700 83 0.68652 53.078 0.58311 48.464 0.49791 41.989 0.43156 33-221 0.38661 21.976 84 0.66645 57.5I7 0.55765 53.060 0.46648 46.591 0-39363 37.403 0.34257 24.916 85 0.64820 62.286 0.53421 58.143 0.43691 5L907 0.35689 42.549 0.29833 28.839 86 0.63230 67.370 0.51353 63.722 0.41024 58.025 0-32255 48.931 0.25491 34-254 87 0.61928 72.740 0.49636 69.778 0.38762 64.988 0.29220 56.833 0.21393 41.990 88 0.60959 78.349 0.48347 76.253 0.37028 72.762 0.26791 66.456 0.17815 53-294 89 0.60361 84.129 0-47544 83.042 0-35929 81.195 0.25198 77.709 0.15225 69.480 90 0.60160 9O.OOO 0.47273 90.000 0-35553 90.000 0.24641 90.000 0.14255 90.000 Note. Negative quantites are in heavy type. Examples tanh (2.6 765) = 0.99990 /i2.675 = 0.99990 /i2.40 / .3O* r . tanh" 1 (0.88618 \7.i92) = 3.0/66. [19] sinh 1ABLE, 1 V . ^UJX-IVC.^ . L r\ \j r /\v^ i u e '' '1 O.I 0.2 o-3 0.4 0-5 45 46 47 48 49 o i.ooooo 0.096 0-99993 o-95 0.99987 0.095 0.99981 0.094 0.99975 0.094 o i.ooooo 0.383 0.99975 0.382 0.99950 0.381 0.99930 0.380 0.99910 0.378 o i.ooooo 0.860 0.99950 0.858 0.99897 0.856 0.99847 0.854 0.99797 0.852 o 1.00013 1.532 0.99920 1.529 0.99828 1.526 0-99733 1-520 0.99640 1.513 o 1.00032 2.391 0.99888 2.388 0.99744 2.385 0.99598 2.378 0.99454 2.368 So Si 52 53 54 0.99970 0.094 0.99965 0.093 0.99960 0.092 0.99955 0.091 0.99950 0.090 0.99880 0.376 0.99860 0.374 0.99840 0.371 0.99815 0.367 0.99795 0.362 0.99743 0.848 0.99690 0.842 0.99640 0.834 0.99590 0.826 0.99540 0.818 0.99550 1.506 0.99460 1.497 0.99370 1.486 0.99280 1.472 0.99190 1.458 0.99312 2.357 0.99170 2.342 0.99028 2.325 0.98888 2.304 0.98748 2.281 55 56 57 58 59 0.99944 0.089 0.99939 0.088 0-99933 0.086 0.99928 0.085 0.99922 0.083 0-99775 0.357 0.99755 0.352 0.99730 0.347 0.99705 0.342 0.99685 0.336 0.99490 0.809 0.99443 0.799 0.99397 0.787 0.99347 0.773 0.99300 0.760 0.99103 1.440 0.99018 1.421 0.98930 1.400 0.98845 1.378 0.98763 1.354 0.98610 2.254 0.98476 2.226 0.98344 2.195 0.98212 2.l6o 0.98082 2.123 60 61 62 63 64 0.99917 0.082 0.99912 0.081 0.99907 0.079 0.99902 0.077 0.99897 0.075 0.99670 0.331 0.99645 0.324 0.99625 0.317 0.99605 0.309 0.99585 0.301 0.99257 0.746 0.99213 0.731 0.99170 0.715 0.99127 0.698 0.99083 0.680 0.98685 1.330 0.98603 1.302 0.98523 1.273 0.98448 1.243 0-98373 I - 212 0.97956 2.085 0.97832 2.042 0.97710 1.998 0.97590 1.950 0.97476 1.900 65 66 67 68 69 0.99893 0.073 0.99889 0.071 0.99885 0.069 0.99881 0.066 0.99877 0.064 0.99570 0.293 0-99555 0.284 0.99540 0.275 0.99520 0.265 0.99505 0.255 0.99040 0.661 0.99000 0.641 0.98963 0.621 0.98927 0.599 0.98891 0.577 0.98300 1.179 0.98232 1.144 0.98165 1.108 0.98100 1.070 0.98035 1.030 0.97364 1.848 0.97254 1.794 0.97150 1.738 0.97046 1.679 0.96946 1.617 7o 72 73 74 0.99873 0.062 0.99869 0.059 0.99865 0.057 0.99861 0.054 0.99858 0.051 0.99490 0.245 0.99475 0.235 0.99460 0.225 0.99445 0.214 0.99435 0.203 0.98857 0.555 0.98823 0.532 0.98790 0.508 0.98760 0.483 0.98733 0.458 0-97975 0.990 0.97918 0.948 0.97863 0.905 0.97808 0.861 0.97758 0.817 0.96852 1.554 0.96760 1.489 0.96676 1.422 0.96592 1.354 0.96514 1.284 75 76 77 78 79 0.99855 0.048 0.99852 0.045 0.99850 0.042 0.99847 0.039 0.99845 0.036 0.99425 0.192 0.99415 0.180 0.99405 0.168 0-99395 0.156 0.99385 0.144 0.98707 0.432 0.98680 0.406 0.98653 0.379 0.98633 0.351 0.98613 0.322 0.97710 0.771 0.97665 0.724 0.97625 0.676 0.97585 0.628 0-97545 0-578 0.96440 1. 212 0.96368 1.138 0.96304 I.O62 0.96246 0.986 0.96190 0.909 80 81 82 83 84 0.99843 0.033 0.99841 0.030 0.99839 0.026 0.99838 0.023 0.99837 O.O2O 0-99375 0.131 0.99365 0.118 0.99360 o.i 06 0.99355 0.093 0.99350 0.080 0.98593 0.294 0.98577 0.266 0.98563 0.238 0-98553 0.209 0.98543 0.180 0.97512 0.528 0.97483 0.477 0.97458 0.425 0-97433 0.373 0.97413 0.321 0.96134 0.830 0.96088 0.750 0.96046 0.669 0.96008 0.587 0.95974 0.505 85 86 87 88 89 0.99836 O.OI7 0.99835 0.014 0.99834 o.on 0.99833 0.008 0.99832 0.004 0-99345 0.067 0.99340 0.054 0-99335 0.041 0-99335 0.028 0-99335 0.014 0.98537 0.150 0.98530 O.I2O 0,98523 0.090 0.98517 O.66o 0.98510 O.O3O 0-97395 0.268 0.97380 0.215 0.97370 0.162 0.97365 0.108 0.97360 0.054 0.95944 0.422 0.95920 0.338 0.95904 0.254 0.95894 0.170 0.95890 0.085 90 0.99831 o.ooo 0-99335 o.ooo 0.98507 o.ooo 0-97355 o.ooo 0.95886 o.ooo Note. sinh h en = 0/5.. i.o wn 6 , Example. sinh (0.3 769) 0.98891 /o.577 o 98891 /o.34', ,37'. Q-3 /69 [20] 1AULK J.V. ^ UXS-IVC^ i 11\ \J J 6 JNTINUED 45 46 47 48 49 0.6 1.00070 3.440 0.99863 3.437 0.99655 3-434 0.99445 3426 0.99237 3-4I4 0.7 o 1.00134 4.676 0.99849 4.679 0.99564 4.676 0.99281 4 666 0.98999 4.652 0.8 o 1.00230 6.108 0.99856 6.112 0.99484 6.109 0.99114 6.099 0.98745 6.082 0.9 o 1.00363 7.728 0.99893 7.735 0.99425 7.734 0.98955 7.725 0.98488 7.707 1.0 o 1.00553 9-531 0.99975 9.546 0.90394 9-550 0.98816 9.543 0.98242 9.525 50 52 53 54 0.99030 3.398 0.98825 3.379 0.98623 3.355 0.98421 3.326 0.98220 3.293 0.98717 4.632 0.98439 4.606 0.98161 4-574 0.97887 4-537 0.97616 4.493 0.98379 6.058 0.98015 6.026 0-97655 5-987 0.97298 5.940 0.96944 5.886 0.98026 7.679 0.97567 7-642 0.97111 7.595 0.96659 7.538 0.96211 7.472 0.97672 9-495 0.97105 9.453 0.96543 9.399 0.95986 9.333 0-95435 9-255 55 56 57 58 59 0.98023 3.256 0.97830 3.215 0.97638 3.171 0.97448 3.122 0.97262 3.069 0.97349 4.445 0.97084 4.391 0.96823 4.331 0.96564 4.265 0.96313 4.193 0.96594 5.824 0.96249 5.755 0-95909 5-678 0-95574 5-593 0.95244 5.502 0.95769 7-396 0-95333 7-312 0.94904 7.218 0.94482 7.114 0.94067 7.000 0.94890 9.166 0-94353 9-065 0.93825 8.952 0.93305 8.827 0.92795 8.691 6o 61 62 63 64 0.97081 3.013 0.96903 2.953 0.96728 2.889 0.96557 2.821 0.96390 2.749 0.96067 4.117 0.95826 4.036 0.95589 3-951 0.95356 3-859 o-gs^o 3-762 0.94923 5.405 0.94608 5.300 0.94299 5.189 0.93996 5.070 0.93703 4-944 0.93661 6.878 0.93263 6.747 0.92874 6.607 0.92493 6.458 0.92121 6.300 0.92295 8.544 0.91805 8.385 0.91325 8.215 0.90856 8.033 0.90400 7.841 65 66 67 68 69 0.96228 2.674 0.96072 2.596 0.95922 2.515 0.95775 2.430 0.95632 2.342 0.94911 3.661 0.94697 3.554 0.94493 3.444 0.94293 3.329 0.94100 3.209 0.93416 4.812 0.93140 4.674 0.92873 4.529 0.92614 4.379 0.92364 4.223 0.91761- 6.136 0.91411 5.962 0.91074 5.780 0.90747 5.590 0.90430 5.392 0.89957 7.637 0.89527 7.424 0.89111 7-2OI 0.88708 6.968 0.88320 6.723 70 72 73 74 0.95495 2.251 0.95365 2.157 0.95242 2.061 0.95123 1.962 0.95010 1.860 0.93914 3.085 0.93737 2.957 0.93569 2.825 0.93407 2.689 0.93254 2.550 0.92123 4.061 0.91891 3.894 0.91671 3.722 0.91461 3.544 0.91261 3.361 0.90127 5.187 0.89837 4.975 0.89558 4.756 0.89292 4.530 0.89041 4.298 0.87947 6.469 0.87589 6.207 0.87247 5-936 0.86921 5.656 0.86612 5.368 75 76 77 78 79 0.94902 1.756 0.94802 1.649 0.94707 1.540 0.94620 1.429 0.94540 1.317 0.93109 2.408 0.92973 2.263 0.92846 2.114 0.92727 1.962 0.92617 i. 808 0.91073 3.174 0.90895 2.982 0.90729 2.787 0.90574 2.588 0.90431 2.385 0.88804 4.059 0.88581 3.815 0.88372 3.566 0.88177 3.312 0-87997 3.053 0.86320 5.072 0.86045 4.769 0.85788 4.458 0.85549 4-I4I 0.85328 3:818 80 81 82 83 84 0.94465 1.203 0.94397 1.087 0.94337 0.970 0.94283 0.852 0.94237 0.732 0.92516 1.652 0.92424 1.493 0.92343 1.332 0.92270 1.169 0.92207 1.005 0.90301 2.179 0.90183 1.969 0.90076 1.757 0.89981 1.543 0.89899 1.327 0.87831 2.789 0.87681 2.522 0.87547 2.251 0.87428 1.976 0.87324 1.699 0.85125 3.489 0.84940 3.156 0.84774 2.817 0.84628 2.474 0.84501 2.128 85 86 87 88 89 0.94197 0.612 0.94163 0.490 0.94138 0.368 0.94122 0.246 0.94110 0.123 0.92153 0.839 0.92109 0.673 0.92074 0.506 0.92050 0.338 0.92036 0.169 0.89829 1.109 0.89771 0.889 0.89726 0.668 0.89694 0.446 0.89675 0.223 0.87237 1.420 0.87164 1.138 0.87108 0.855 0.87068 0.571 0.87043 0.286 0.84393 1-779 0.84305 1.426 0.84236 1.071 0.84186 0.715 0.84156 0.358 90 0.94107 o.ooo 0.92031 o.ooo 0.89670 o.ooo 0.87037 o.ooo 0.84147 o.ooo E^p,, sinh (1.0/85) 0.84393 /i.779 -0 84101/1 46' .44". 1.0/85 [21] sinh d 1ABI I.I ,E IV. L,UKK,t 1.2 _HJNLr ? AC 1 UK. - T /y. L.ON i-3 i-4 TINUED 1-5 O o 45 1.0081 11.519 I.OII5 13.692 1.0158 16.034 1. 02 1 2 18.568 1.0279 21.262 46 I.OOII "543 I.003I 13.726 i. 0060 16.092 I.OO99 18.639 1.0148 21-355 47 0.99409 "555 0-99475 I3.748 0-99623 16.128 0.99850 18.689 I.OOlS 21.426 48 0.98709 "553 0.98650 13-753 0.98654 16.142 0.98729 18.717 0.98893 21-474 49 0.98018 0.97825 I3.74I 0.97685 16.137 0.97607 18.723 0.97613 2 1 .496 50 0.97327 11.506 0.97008 13.712 0.96722 16.113 0.96493 18.707 0.96340 21.492 51 0.96636 11.461 0.96192 13.666 0.95770 16.068 0-95393 18.668 0.95080 21.462 52 0.95964 11.401 0.95383 13.603 0.94822 16.004 0.94300 18.605 0.93833 21.405 53 0.95291 11.327 0-94583 I3.523 0.93892 I5-9I9 O.9322I 18.519 0.92593 21.321 54 0.94627 11.239 0.93800 I3.42S 0.92969 15.814 0.92157 18.409 0.91373 2I.2IO 55 0.93963 11.136 0.93017 I3.3IO 0.92054 15.688 O.9IIOO 18.275 0.90167 2I.O72 56 0.93318 11.019 0.92250 13.178 0.91154 I5.542 0.90064 18.117 0.88980 20.907 57 0.92682 10.888 0.91492 13.028 0.90269 I5-376 0.89043 17.936 0.87813 20.713 58 0.92054 10.742 0.90750 12.860 0.89400 15.188 0.88036 17-730 0.86667 2O.49I 59 0.91436 10.581 0.90017 12.675 0.88546 14.980 0.87050 17.500 0.85540 2O.240 6o 0.90836 10.406 0.89300 12.474 0.87715 14-752- 0.86086 17-246 0-84433 19.964 61 0.90244 10.218 0.88600 12.255 0.86892 14.502 0.85136 16.967 0-83353 I9-657 62 0.89666 10.016 0.87918 12.019 0.86092 14.232 0.84214 !6.66 3 0.82300 19.321 63 0.89103 9.800 0.87250 11.767 0.85315 13-942 0.83314 16.335 0.81273 18.956 64 0.88554 9-570 0.86600 11.497 0.84554 13.632 0.82436 15-983 0.80267 18.562 65 0.88019 9-327 0.85967 II. 211 0.83808 I3.30I 0.81579 15.607 0.79293 18.140 66 0.87501 9.071 0.85350 10.909 0.83094 12.950 0.80750 15.207 0.78347 17.690 67 0.86998 8.801 0.84758 10.594 0.82400 12.580 0.79950 14-783 0-77433 17.211 68 0.86514 8.519 0.84183 10.257 0.81731 12.191 0.79171 14-335 0.86547 16.704 69 0.86045 8.225 0.83625 9.907 0.81077 11.783 0.78421 13-865 0.75693 16.169 70 0.85596 7.918 0.83093 9-543 0.80454 "357 0.77707 13-373 0.74873 15.607 71 0.85165 7.600 0.82583 9.164 0.79862 10.912 O.77O2I 12.858 0.74087 I5.0I8 72 0.84754 7.271 0.82095 8-771 0.79292 10.450 0.76364 12.322 0-73333 14.402 73 0.84362 6.931 0.81631 8.365 0.78746 9.971 0.75736 11.765 0.72620 I3.76I 74 0.83989 6.580 0.81190 7.946 0.78231 9-476 11.188 0.71940 13.096 75 0.83637 6. 2 2O 0.80773 7.514 0.77746 8.966 0-74579 10.591 0.71293 12.406 76 0.83306 5.850 0.80380 7.069 0.77287 8.440 0.74050 9-975 0.70687 11.693 77 0.82996 5-471 0.80013 6.614 0.76858 7.900 0-73557 9-342 0.70120 10.958 78 0.82707 5.084 0.79671 6.148 0.76459 7-346 0.73093 8.692 0-69593 10.201 79 0.82441 4.689 0-79355 5.672 0.76090 6.780 0.72664 8.026 0.69107 9-425 80 0.82196 4.286 0.79065 5-187 0-75751 6.202 0.72279 7-346 0.68660 8.630 81 0.81975 3.877 0.78802 4-693 0-75444 5.614 0.71921 6.651 0.68253 7.8l8 82 0.81775 3.462 0.78566 4.191 0-75165 5.016 0.71600 5-944 0.67887 6.991 83 0.81599 3.041 0.78358 3-683 0.74923 4.408 0.71319 5.227 0.67567 6.149 84 0.81445 2.616 0.78175 3.169 0-747" 3-793 0.71073 4-499 0.67287 5-294 85 0.81315 2.186 0.78022 2.649 0-74531 3-i7i 0.70864 3-762 0.67047 4.429 86 0.81209 1-753 0.77895 2.124 0.74383 2-544 0.70694 3.018 0.66847 3-554 87 0.81125 1.317 0-77797 1.596 0.74268 1.912 0.70561 2.269 0.66693 2.673 88 0.81066 0.879 0.77727 1.065 0.74185 1.277 0.70466 1.515 0.66587 1-785 89 0.81031 0.440 0.77684 0-533 0.74136 0.639 0.70409 0.758 0.66521 0.893 90 0.81019 o.ooo 0.77670 o.ooo 0.74120 o.ooo 0.70389 o.ooo 0.66499 o.ooo E: camole. sinh (1.5 765) n ^rnn? / - o *7n / ?n-5 / T R <>':: /65 [22] 1ABLK IV. l^UJK.JK.Jl,V^lliNj 1.6 1.7 r-rt^lUJ*.. - m ' IJk CONTINUED 1.8 1.9 2.O o o b o 45 1-0359 24.117 1.0456 27.I33 1.0572 30.292 1.0707 33.590 1.0863 37.016 46 1. 02 1 1 24.241 1.0290 27.288 1.0385 30.486 1.0499 33.829 1-0633 37.304 47 I.O064 24-339 1.0124 27.418 1.0199 30.651 1.0292 34.037 1.0405 37.560 48 0.99175 24.409 0.99582 27.5I5 1.0015 30.784 1.0087 34-2io 1.0178 37.778 49 0.97719 24451 0.97947 27.582 0.98317 30.883 0.98837 34-346 0-99535 37.958 50 0.96275 24-465 0.96323 27.618 0.96500 30.946 0.96821 34-444 0.97310 38.098 Si 0.94844 24-449 0.94718 27.620 0.94700 30.974 0.94821 34-503 0.95110 38.196 52 0.93431 24.402 0.93123 27.590 0.92922 30.966 0.92858 34-524 0.92940 38.253 53 0.92031 24-325 0.91547 27.527 0.91167 30.921 0.90911 34.505 0.90800 38.267 54 0.90650 24.218 0.89994 27.429 0.89433 30-839 0.88995 34-444 0.88685 38.236 55 0.89281 24.079 0.88465 27.296 0.87728 30.718 0.87106 34-342 0.86600 38.160 56 0.87938 23.910 0.86953 27.128 0.86044 30.558 0.85242 34-197 0.84555 38.039 57 0.86619 23.709 0.85471 26.924 0.84389 30.358 0.83411 34.008 0.82545 37.869 58 0.85319 23-475 0.84012 26.683 0.82767 30.116 0.81611 33-773 0.80570 37-650 59 0.84044 23.209 0.82582 26.406 0.81172 29-834 0.79847 33-493 0.78630 37.382 60 0.82794 22.910 0.81176 26.091 0.79611 29.510 0.78121 33-I67 0.76735 37.062 61 0.81569 22.578 0.79806 25.738 0.78083 29.142 0.76437 32.792 0.74880 36.688 62 0.80375 22.212 0.78464 25-347 0.76594 28.730 0-74795 32-368 0.73070 36.261 63 0.79213 2I.8I3 0.77159 24.917 0.75139 28.274 0.73179 31-803 0.71300 35.778 64 0.78075 21.380 0.75888 24.447 0.73722 27.774 0.71611 3I-368 0-69575 35.238 65 0.76975 20.913 0.74647 23.938 0.72344 27.228 0.70084 30.792 0.67900 34.639 66 0.75906 20.413 0.73447 23.390 0.71006 26.636 0.68605 30.162 0.66275 33.980 67 0.74869 19.879 0.72282 22.802 0.69711 25-997 0.67168 29.478 0.64700 33-261 68 0.73869 19.311 0.71159 22.175 0.68455 25-3" 0.65779 28.741 0.63175 32.479 69 0.72900 I8.7IO 0.70076 21.508 0.67244 24.580 0.64421 27.949 0.61700 31-634 70 0.71975 18.076 0.69035 20.801 0.66083 23.802 0.63153 27.102 0.60280 30.725 0.71081 17.410 0.68035 20.055 0.64967 22.976 0.61916 26.199 0.58915 29-750 72 O.7023I 16.712 0.67076 19.271 0.63900 22.104 0.60731 25.241 0.57605 28.709 73 0.69419 15.982 0.66165 18.448 0.62878 21.187 0.59600 24.227 0-56355 27.602 74 0.68650 15.221 0.65300 17.588 0.61911 20.224 0.58526 23-158 o-55 l6 S 26.428 75 0.67919 I4.43I 0.64476 16.692 0.60994 19.215 0.57500 22.034 0-54035 25-187 76 0.67231 I3.6I3 0.63700 15.760 0.60128 18.162 0.56542 20.856 0.52970 23.880 77 0.66588 12.767 0.62976 14.794 0.59317 17.068 0.55637 19.625 0.51970 22.507 78 0.65988 11.894 0.62300 13-795 0.58561 15-933 0-54795 18-343 0-51035 21.070 79 0.65431 10.997 0.61676 12.765 0.57861 I4.758 0.54011 17.012 0.50165 19-571 80 0.64925 IO.O76 0.61106 11.705 0.57217 13-545 0.53295 I5-633 0.49364 18.01 1 81 0.64463 9- J 34 0.60582 10.618 0.56633 12.298 0.52642 14.209 0.48636 16.391 82 0.64050 8.171 0.60112 9-505 0.56106 11.019 0.52052 12-744 0.47979 14.722 83 0.63681 7.191 o.597oo 8.370 0.55639 9.711 0-51530 11.240 0.47396 13.000 84 0.63363 6.194 0-59341 7.214 0.55234 8-374 0.51075 9.702 0.46888 11.233 85 0.63094 5.184 0.59035 6.039 0.54890 7-oi5 0.50689 8-133 0.46455 9.426 86 0.62875 4.161 0.58782 4.850 0.54607 5-636 0.50372 6-539 0.46101 7.584 87 O.627OO 3.129 0.58586 3-649 0.54387 4.241 0.50124 4-923 0.45823 5-713 88 0.62575 2.090 0.58445 2.438 0.54229 2-834 0.49947 3.291 0.45624 3-821 89 O.6250O 1.046 0.58361 1.230 0.54135 1.419 0.49841 1.648 0.45505 1.914 90 0.62473 o.ooo 0.58333 o.ooo 0.54103 o.ooo 0.49805 o.ooo 0.45465 o.ooo Example. sinh (i 8/77) O CO? I 7 , ^I7.o68 = O.CQ?!? /I7.4' .05'. 1.8/77 < ~'oyo i / i [23] sinh 6 1ABLE IV. 2.1 UUiVKJl,^ JL 1IN (j 2.2 rA^iUK.. - - r 77. CON 2.3 2.4 TINUED 2-5 o o o o O 45 1.1043 40.558 I.I248 44.205 1.1480 47.946 1.1740 5L769 1.2032 55-661 46 1.0789 40.905 1.0970 44.613 1.1177 48.419 1.1411 52.312 1.1674 56.278 47 1.0538 41.213 1.0695 44.981 1.0877 48.851 1.1085 52.813 1.1320 56.852 48 1.0289 41.482 1.0423 45-307 1.0580 49.241 1.0762 53.27I 1.0972 57.383 49 1.0043 41.711 I.OI53 45-592 1.0286 49.588 1.0444 53-685 1.0627 57.869 50 0.97986 41.898 0.98864 45-834 0.99965 49.890 1.0130 54-053 J.0288 58.309 Si 0.95576 42.042 0.96232 46.031 0.97104 50.146 0.98200 54-375 0.99540 58.701 52 0.93195 42.142 0.93636 46.181 0.94283 50.354 0.95150 54-648 0.96256 59.044 53 0.90848 42.197 0.91077 46.284 0.91504 50.5H 0.92154 54-871 0.93028 59-337 54 0.88533 42.205 0.88559 46.338 0.88774 50.623 0.89208 55-042 0.89860 59-578 55 0.86257 42.165 0.86082 46.342 0.86091 50.680 0.86308 55.i6o 0.86748 59-765 56 0.84019 42.076 0.83646 46.294 0-83457 50.683 0.83467 55-223 0.83700 59.896 57 0.81819 41.935 0.81255 46.193 0.80869 50.629 0.80683 55-228 0.80712 59-970 58 0.79662 41.742 0.78909 46.035 0.78331 50.518 0-77954 55-174 0.77788 59.984 59 0-77543 41.494 0.76609 45.820 0.75848 50.347 0.75283 55-059 0.74932 59-936 60 0.75471 41.191 0-74359 45-546 o.734i7 50.114 0.72671 54.880 0.72140 59-823 61 0-73447 40.830 0.72159 45-2io 0.71043 49.816 0.70121 54.634 0.69412 59.642 62 0.71467 40.410 O.7OOO9 44.811 0.68725 49-451 0.67633 54-3I9 0.66752 59-39 1 63 0.69538 39.929 0.67914 44-345 0.66465 49.016 0.65204 53-931 0.64160 59.067 64 0.67657 39-386 0.65873 43.811 0.64261 48.509 0.62838 53-467 0.61636 58.665 65 0.65829 38.777 0.63886 43.207 0.62113 47.926 0.60533 52.924 0.59176 58.182 66 0.64053 38.101 0.61955 42.529 0.60026 47.264 0.58296 52-297 0.56784 57.613 67 0.62329 37-357 O.6OO82 41-775 0.58004 46.519 0.56121 51-583 0.54464 56.954 68 0.60657 36-542 0.58268 40.943 0.56043 45-688 0.54013 50.777 0.52208 56.198 69 0.59043 35-655 0.56514 40.029 0.54143 44.767 0.51971 49.874 0.50024 55-341 70 0.57487 34-694 0.54818 39.030 0.52309 43-752 0.49996 48.868 0.47908 54-376 71 0.55990 33-656 0.53186 37-944 0-50539 42.638 0.48088 47-754 0.45864 72 0-54557 32.541 0.51618 36.769 0.48835 41.422 0.46246 46.526 0.43888 52.092 73 0.53181 3L348 0.50118 35-501 0.47204 40.100 0.44479 45-177 0.41988 50.755 74 0.51871 30.074 0.48682 34.I38 0.45644 38-667 0.42783 43-701 0.40160 49-277 75 0.50629 28.719 '0.47323 32-678 0.44152 37.118 0.41165 42.091 0.38408 47.648 76 0-49453 27.283 0.46027 31.119 0.42736 35-449 0.39621 40.341 0.36732 45-857 77 0.48347 25.766 0.44810 29-459 0.41397 33-659 0.38157 38.443 0.35137 43-891 78 0.47311 24.168 0.43666 27.699 0.40135 31-744 0.36775 36.390 0.33625 41.740 79 0.46349 22.491 0.42601 25-839 0.38963 29.701 0.35480 34-I78 0.32201 39-392 80 0.45461 20.736 0.41617 23.879 0.37873 27.531 0.34273 31-803 0.30869 36.836 81 0.44650 18.907 0.40716 21.822 0.36873 25-233 0.33162 29.261 0.29633 34-063 82 0.43919 17.008 0.39902 19.671 0-35965 22.810 0.32149 26.550 0.28500 31-065 83 0.43268 15-043 0.39176 I7-432 0.35153 20.267 0.31239 23.676 0.27477 27.841 84 0.42699 13-015 0.38540 15.111 0.34441 17.611 0.30438 20.643 0.26569 24.390 85 0.42215 10.933 0.37998 12.715 0.33831 14.851 0.29748 17.462 0.25784 20.723 86 0.41818 8.806 0-37551 10.254 0.33328 I2.OOO 0.29178 14.147 0.25131 16.855 87 0.41507 6.639 O.372O2 7.740 0.32933 Q.07I 0.28728 10.717 0.24614 12.811 88 0.41284 4-443 0.36951 5-183 0.32650 6.082 0.28405 7.197 0.24241 8.624 89 0.41150 2.226 0.36800 2-599 0.32479 3-05I 0.28210 3-615 0.24014 4.338 90 0.41105 o.ooo 0.36750 o.ooo 0.32422 O.OOO 0.28144 o.ooo 0.23939 0.000 Example. sinh (2. 3/84!) = n 7/1/1/1 T , f T *7^ fkT T /I7 36'. ff 2.3/84 -->-i -" ' / * / *o v [24] 1ABLE J.V. 2.6 ^UJS.K.H,^ 1 1IN Vj 2-7 r A^ J. U.K.. 2.8 - r 77. CONTINUED 2.9 3.0 o e o o o 45 1.2354 59-6I3 1.2710 63.614 I.3I02 67.653 1-3530 7L72I 1-3995 75.8I4 46 1.1967 60.307 1.2293 64.386 1.2652 68.506 1.3046 72.658 1-3475 76-832 47 1.1586 60.957 1.1881 65.116 I.22O9 69.317 1.2570 73-551 1.2965 77.808 48 I.I209 61.564 1.1476 65.801 I.I774 70.084 1.2103 74-400 1.2465 78.741 49 1.0838 62.126 1.1077 66.442 I.I345 70.806 1.1644 75.205 1.1974 79-630 50 1.0473 62.641 1.0685 67.037 1.0925 71.483 1.1195 75.965 1.1494 80.474 Si I.OII3 63.109 1.0300 67.585 1.0513 72.113 1-0755 76.679 1.1025 81.272 52 0.97608 63.528 0.99219 68.084 I.OIO9 72.694 1.0324 77-345 1.0567 82.024 53 0.94145 63.897 0.95514 68.532 0.97143 73-226 0.99038 77-963 I.OI2O 82.729 54 0.90750 64.214 0.91881 68.929 0.93279 73-708 0.94928 78.532 0.96837 83.386 55 0.87419 64.476 0.88333 69.274 0.89500 74-138 0.90917 79.050 0.92590 83.994 56 0.84162 64.684 0.84863 69.563 0.85811 74.515 0.87007 79-5I7 0.88453 84.553 57 0.80973 64.834 0.81467 69.797 0.822II 74.837 0.83197 79-932 0.84433 85.062 58 0.77854 64.925 0.78152 69.973 0.78700 75.103 0.79487 80.293 0.80523 85.510 59 0.74808 64.954 0.74919 70.088 0.75275 75-3H 0.75876 80.598 0.76727 85.924 Oo 0.71835 64.918 0.71767 70.140 0.71943 75-459 0.72366 80.847 0.73040 86.276 61 0.68935 64.816 0.68692 70.128 0.68700 75.544 0.68955 81.037 0.69463 86.574 62 0.66104 64.644 0.65700 70.047 0.65543 75-565 0.65641 81.167 0.65993 86.8l7 63 0.63346 64.399 0.62786 69.895 0.62475 75.5I9 0.62424 81.235 0.62630 87.004 64 0.60662 64.077 0.59952 69.669 0-59493 75-403 0.59303 81-239 0-59373 87.133 65 0.58054 63.674 0.57193 69.364 0.56596 75.214 0.56272 81.176 0.56217 87.204 66 0.55519 63.185 0.54514 68.978 0.53785 74-948 0-53334 81.044 0.53160 87.214 67 0.53054 62.606 0.51911 68.504 0.51054 74-599 0.50486 80.838 O.5O2O3 87.161 68 0.50658 61.930 0.49386 67.935 0.48404 74.164 0.47724 80.555 0.47340 87.043 69 0.48338 61.151 0.46937 67.266 0.45832 73.634 0.45045 80.190 0.44567 86.858 70 0.46089 60.262 0.44559 66.489 0-43343 73-ooS 0.42448 79.738 0.41883 86.602 0.43911 59.254 0.42256 65.596 0.40928 72.267 0-39934 79-191 0.39283 86.271 72 0.41808 58.117 0.40026 64.576 0.38586 71.411 0-37497 78.541 0.36767 85.860 73 0-39773 56-841 0.37872 63.415 0.36321 70.424 0.35135 77.780 0.34327 85.363 74 0-37814 55-4I4 0.35789 62.101 0.34125 69.292 0.32845 76.894 0.31964 84.772 75 0.35930 53-822 0.33780 60.617 0.32002 67.996 0.30627 75.867 0.29674 84.077 76 0-34123 52.049 0.31846 58.943 0.29951 66.516 0.28478 74.680 0.27452 83.265 77 0.32395 5 -079 0.29987 57.058 0.27971 64.827 0.26397 73.309 0.25296 82.319 78 0.30749 47-893 0.28206 54.935 0.26064 62.897 0.24379 71.721 0.23202 81.216 79 0.29189 45.471 0.26507 52.542 0.24231 60.686 0.22432 69.876 0.21168 79.925 80 0.27718 42-793 0.24894 49-847 0.22476 58.145 0.20551 67.721 0.19191 78.404 81 0.26346 39.837 0.23374 46.812 0.20804 55-217 0.18738 65.183 0.17270 76.592 82 0.25078 36.583 0.21954 43-396 0.19222 51.833 0.16998 62.169 0.15404 74-404 83 0.23922 33.016 0.20644 39.554 0.17741 47.908 0.15343 58.553 0.13596 71-709 84 0.22887 29.126 0.19458 35.252 0.16375 43-351 0.13774 54.i68 0.11849 68.313 85 0.21986 24.910 0.18409 30.458 0.15142 38.066 0.12323 48.795 0.10174 63.903 86 0.21227 20.382 0.17515 25.164 0.14069 31.968 0.11016 42.168 0.085917 57-974 87 0.20623 15.571 0.16794 19.390 0.13184 25.015 0.098934 34-005 0.071443 49-701 88 0.20184 10.524 0.16264 13.194 0.12520 17.244 0.090148 24.110 0.059103 37.843 89 0.19917 5.307 0.15938 6.682 0.12105 8.809 0.084469 12.576 0.050307 21.091 90 0.19827 o.ooo 0.15829 o.ooo 0.11964 o.ooo 0.082845 o.ooo 0.047040 o.ooo Example. sinh (3.0 /86) 0.085917 /57-974 J'.26*. 3-Q /86 [25] TABLE V. CORRECTING FACTOF tanh ti. O.I 0.2 0.3 0.4 o. 5 45 46 47 48 49 I.OOOOO I.OOOIO I.OO02I 1.00033 1.00044 o o.i 88 0.188 0.188 0.188 0.187 0.99985 I.O003O I.OO075 I.OOI2O I.OOI7O 0.765 0.764 0.763 0.76l 0.758 0.99937 1.00040 1.00143 1.00250 1.00357 I.7I8 I.7I9 1.720 I.7I7 I.7IO 0.99803 0.99988 1.00173 1.00358 1.00543 o 3.046 3.055 3.058 3.058 3.053 0.99514 0.99804 1.00094 1.00384 1 .00680 o 4.750 4.771 4.781 4.787 4.787 So 52 53 54 1.00056 1.00068 1.00080 1.00092 1.00103 0.187 0.187 0.186 0.184 0.182 I.OO2I5 I.OO265 I.O03IO 1-00355 I.OO4O5 0-755 0.749 0.744 0.738 0.732 1.00460 1.00563 1.00670 1.00773 1.00877 1-703 1.694 1.685 1.672 1.656 1.00733 1.00925 1.01115 1.01303 1.01490 3.045 3.031 3.014 2.995 2.969 1 .00980 1.01278 1.01578 1.01878 1.02178 4.780 4.767 4-747 4.724 4.692 55 56 57 58 59 1.00114 1.00125 1.00135 1.00146 1.00156 0.180 0.177 o.i75 0.172 0.170 1.00450 1.00495 1-00535 1-00575 I.OO62O 0.724 0.715 0.705 0.694 0.682 1.00980 1.01083 1.01183 1.01283 1.01380 1.638 1.618 1.596 1-574 1.548 1.01675 1.01865 1.02050 1.02230 1.02410 2.942 2.911 2.876 2.836 2.793 1.02478 1.02778 1.03076 1-03374 1.03670 4-656 4.612 4.561 4-506 4-444 6o 61 62 63 64 1.0017 1.0018 1.0019 I.OO20 I.OO2O 0.167 0.163 0.159 0.155 0.151 1.0067 1.0070 1.0074 1.0078 I.OO82 0.668 0.655 0.640 0.625 0.609 1.0148 1.0158 1.0167 1.0176 1.0185 1.521 1.490 1.458 1-425 1.390 1.0259 1.0278 1.0294 1.0311 1.0328 2.745 2.695 2.641 2.583 2.521 1.0397 1.0425 1.0454 1.0482 1.0510 4-375 4-300 4.220 4-133 4.041 65 66 67 68 69 I.OO2I I.OO22 1.0023 1.0024 I.OO25 0.147 0.143 0.138 0.134 0.129 I. OO86 I.OOgO 1.0093 I.OO97 I.OIOO 0.592 0-575 0-557 0-539 0.520 1.0193 I.02O2 I.O2IO 1.0218 I.O226 1-353 I.3I4 1-273 1.231 1.187 1.0344 1.0360 1.0376 1.0391 1.0406 2.456 2.388 2.315 2.240 2.163 1-0538 1.0564 1.0591 1.0617 1.0642 3-942 3-836 3-724 3-609 3-489 70 72 73 74 I.O026 1.0026 1.0027 I.OO28 I.OO28 0.123 0.118 O.II2 O.I07 O.IO2 1.0103 1.0106 1.0109 I.OII2 1.0114 0-499 0.478 0.456 0.434 0.411 1.0234 I.O24I 1.0247 1.0254 I.O260 1.141 1.094 1.045 0-995 0.944 1.0420 1.0434 1.0447 1.0460 1.0472 2.082 1.998 1.911 1.821 1.728 1. 0666 1.0689 1.0712 1-0734 1-0755 3-362 3.229 3.091 2.948 2.800 75 76 77 78 79 I.OO29 I.OO29 1.0030 I.OO3O I.003I O.O96 O.O9O 0.084 0.078 O.072 1.0117 1. 01 2O I.OI22 I.OI23 I.OI25 0.388 0.365 0.341 0.316 0.291 1.0266 I.O272 1.0277 I.O282 I.O287 0.892 0.838 0.783 0.727 0.671 1.0483 1.0494 1.0504 1-0513 1.0522 1.632 1-534 1-434 1-332 1.229 1.0774 1.0793 1.081 1 1.0828 1.0843 2.649 2-493 2-333 2.169 2.OOI 80 81 82 83 84 I.003I 1.0032 1.0032 1.0032 1.0033 O.066 O.O6O 0.054 0.047 O.O4O I.OI27 I.OI29 I.OI30 I.OI32 I-OI33 0.266 0.241 0.215 o.i 88 0.162 I.O29I 1.0295 1.0298 I.030I 1.0304 0.614 0-555 0-494 0-433 0.372 1-0530 1-0538 1.0544 1-0550 1-0555 1.124 1.017 0.908 0.797 0.685 1.0857 1.0870 i. 088 1 1.0892 1.0901 .830 .656 .480 .301 .119 85 86 87 88 89 1-0033 1.0033 1-0033 1.0033 1-0033 0.033 O.O26 0.019 O.OI2 O.OO6 I.OI34 I.OI34 I-OI35 I-OI35 I-OI35 0.136 0.109 0.082 0.054 0.027 1.0307 1.0309 I.03IO I.03II I.03II 0.311 0.249 0.187 0.125 0.063 1.0560 1-0563 1.0566 1.0568 1.0570 0-573 0.460 0.346 0.232 0.117 i .0908 1.0914 1.0919 1.0923 1.0926 0-935 0-749 0.562 0-374 0.187 90 1.0033 O.OOO I.OI36 O.OOO I-03II O.OOO 1.0570 O.OOO 1.0926 0.000 Note. Example. Negative quantities are ii tanh (0.4 /74) i heavy \i.728 type. / = 1.0472 - 1.0472 \i-.43 41*. [26] TABLE V. CORRECTING 0.6 0.7 tanhfl /y. CONTINUED 0.8 0.9 45 46 47 0.99010 0.99417 0.99830 6.817 6.854 6.881 0.98189 0.98727 0.99279 9.214 9.281 9-337 0.96971 0.97646 0-98339 11.902 I2.O2O 12.122 0.95284 0.96089 0.96921 14.839 15.020 15-185 0.93077 0.93999 0.94950 17.956 18.216 18.461 48 1.00248 6.901 0.99840 9-383 0.99050 12.210 0.97779 15-334 0.95938 18.692 49 1.00672 6.912 I 00410 9.419 0.99780 12.285 0.98665 15.469 0.96966 18.908 50 I.OIIOO 6.915 I 00991 9-443 1.00526 12.347 0.99578 15-588 0.98032 19.108 1-01533 6.908 I 01581 9-455 1.01289 12.395 1.00518 15.690 0.99136 19.290 52 1.01970 6.893 I 02181 9-457 1.02069 12.427 1.01486 15-777 1.00282 19-455 53 1.02412 6.870 I 02789 9-445 1.02864 12.444 1.02479 15-845 1.01469 19.600 54 1.02855 6.838 i 03403 9-423 1.03675 12.446 1.03500 15.895 1.02697 19.726 55 1.03300 6.797 I 04024 9-388 1.04501 12.432 1.04549 15-925 1.03970 19.828 56 1.03747 6.746 i 04653 9-338 1-05343 I2.4OI 1.05626 15-934 1.05287 19.909 57 1.04195 6.685 1.05286 9-275 1.06196 12.353 1.06729 15-922 1.06648 19.965 58 1.04642 6.616 I 05921 9.200 1.07061 12.288 1.07854 15-889 1.08054 19.996 59 1.05089 6-537 I 06561 9.112 1.07939 12.203 1.09007 15.833 1.09506 19.998 60 1-05537 6.448 i 07210 9.010 1.08829 I2.IOO 1.10188 15-753 1.11009 19-973 61 1.05980 6.350 i 07856 8.893 1.09726 11.978 1.11390 15-647 1-12555 19.918 62 1.06420 6.242 1.08500 8.762 1.10630 11.835 1.12614 15-514 1.14144 19-832 63 1.06858 6.124 I 09H3 8.617 1.11540 11.673 1.13856 15-355 I-I5777 19.711 64 1.07293 5-998 I 09783 8.458 1.12456 11.491 1.15116 15-169 I.I7454 19.555 65 1.07722 5.862 I 10420 8.285 I.I3373 11.288 1.16392 14.956 1.19173 19-362 66 1.08143 5-7i6 I "053 8.098 1.14291 11.064 1.17684 14.711 1.20935 19.131 67 1.08560 5-561 I 11683 7.895 1.15208 IO.82O 1.18989 14.438 1.22737 18.860 68 1.08967 5-397 i 12299 7.678 1. 16118 10.552 1.20298 14-132 1.24569 18.545 69 1.09363 5-223 I 12904 7-447 1.17020 10.263 1.21608 13-796 1.26429 18.188 70 1.09750 5.041 I 13500 7.202 I.I79I5 9-953 1.22919 13-428 1.28316 17-785 1.10125 4-850 I 14083 6-943 1.18796 9.621 1.24226 13.026 1.30221 17-334 72 1.10490 4-650 I 14651 6.671 1.19659 9.268 1-25524 12.592 1.32140 16.836 73 1.10842 4.441 i 15201 6.386 1.20503 8.894 1.26807 12.126 1.34063 16.290 74 1.11180 4-225 I 15734 6.087 1.21325 8-499 1.28067 11.628 1.35986 I5-693 75 1.11503 4.002 1.16244 5-775 I.22I2I 8.083 1.29300 11.097 1.37894 15-043 76 1. 11810 3-772 1-16733 5-450 1.22889 7.648 1.30500 10.532 1-39775 14.342 77 I.I2I02 3-534 I 17197 5-II4 1.23623 7.192 1-31659 9-936 1.41620 13-591 I-I2375 3.289 1.17636 4.767 I.243I9 6.718 1.32772 9-309 1.43412 12.788 79* I.I2628 3-038 1.18046 4.409 1.24976 6.227 1.33830 8-653 I-45I4I 11.935 80 1.12863 2.781 I 18427 4.042 I.2559I 5.718 1.34827 7.966 1.46790 11.032 81 1.13078 2.519 1.18777 3.666 1.26158 5-195 1-35754 7-254 I-48345 10.081 82 I.I3273 2.251 I 19094 3-281 1.26674 4-656 1.36607 6.517 1.49790 9.087 83 I-I3447 1.979 I 19377 2.888 1.27138 4.104 1.37378 5-757 1.51110 8.052 84 I.I3598 1.704 1.19626 2.488 1-27545 3-540 1.38059 4-975 1.52289 6-977 85 .13727 1.425 I 19839 2.083 1.27895 2.966 1.38646 4-175 I-533I4 5-868 86 .13832 1.144 I 20013 1.672 I.28l85 2.383 1.39132 3-360 1.54170 4-731 87 .13915 0.860 I 20150 1.257 1.28413 1.794 1.39518 2.532 1.54848 3-571 88 13975 0-574 I .20247 0.839 I-28575 1.199 1-39793 1.692 1-55339 2.391 89 .I40IO 0.287 I .20306 0.420 I.2867I 0.600 1-39959 0.847 I-55637 1.198 90 1.14022 o.ooo 1.20327 o.ooo 1.28700 o.ooo 1.40017 o.ooo 1.55740 o.ooo Note. Negative quantities are in heavy type. -49' [27] tanh 6 1ABLE V. I.I ^UK.IVJl,l^ 1 UN U 1.2 r /j. i_or 1.4 JTINUEU 1-5 o o o o 45 0.90354 2I.I67 0.87142 24.384 0.83515 27.536 0.79593 30.516 0.75487 33.288 4 6 0.91359 21.524 0.88183 24.856 0.84561 28.118 0.80600 31.225 0.76427 34.116 47 0.92400 21.869 0.89275 25.315 0.85661 28.699 0.81664 31.936 0.77420 34-952 48 0.93482 22.2OI 0.90425 25.764 0.86815 29.279 0.82786 32.648 0.78467 35-792 49 0.94618 22.52O 0.91625 26.2O2 0.88038 29.852 0.83971 33.360 0-79573 36.638 50 0.95809 22.822 0.92892 26.629 0.89331 30.416 0.85228 34.070 0.80747 37-490 0.97055 23.109 0.94225 27.043 0.90685 30.973 0.86557 34-778 0.81993 38.348 52 0.98345 23.379 0.95617 27.442 0.92123 31.521 0.87964 35.485 0.83320 39-213 53 0.99700 23.630 0.97083 27.826 0.93638 32.059 0.89457 36.189 0.84727 40.086 54 i .01 1 r 23.861 0.98625 28.193 0.95238 32.586 0.91043 36.890 0.86220 40.965 55 1.0258 24.O7O 1.0025 28.540 0.96938 33.100 0.92729 37.587 0.87813 41.849 56 .O4I2 24.256 1.0196 28.867 0.98738 33-599 0.94529 38.279 0.89513 42.738 57 .0572 24.417 1-0375 29.173 1.0065 34-082 0.96443 38.964 0.91333 43.634 58 .0739 24.551 1.0564 29.454 1.0266 34.548 0.98479 39.642 0.93280 44-534 59 .0914 24.657 1.0763 29.708 1.0482 34-993 1. 0066 40.311 0-95367 45-441 6o .1096 24.733 1.0973 29.932 1.0710 35.416 1.0301 40.968 0.97613 46.349 61 .1286 24.774 I.II95 30.124 1-0953 35-8x3 1.0551 41.611 1.0003 47-262 62 .1484 24.778 1.1428 30.280 I.I2I2 36.180 1.0820 42.239 1.0264 48.176 63 1.1689 24.745 1.1674 30-397 I.I488 36.515 .1109 42.850 1.0546 49.092 64 I.I9O4 24.672 I.I934 30.473 1.1784 36.814 .1421 43.438 1.0851 50.009 65 I.2I26 24.554 1. 2208 30.502 1.2099 37.072 .1759 44.001 1.1184 50.924 66 1.2357 24.388 1.2496 30.480 1.2436 37.285 .2124 44.535 1.1546 51-834 67 1.2596 24.173 1.2799 30.403 1.2798 37.446 .2520 45.034 1.1943 52.739 68 1.2845 23.904 1.3119 30.263 i-3 l8 5 37-548 .2951 45-493 1-2377 53.636 69 I.3IOI 23.579 1-3457 30-059 1.3600 37-585 3420 45-904 1.2857 54.520 70 1.3365 23.194 1.3811 29.781 1.4046 37-548 3933 46.260 I-3385 55.388 1.3637 22.744 1.4183 29.425 1-4525 37-430 .4495 46.552 1-3973 56.235 72 I.39I5 22.227 1.4573 28.983 1-5039 37-218 .5113 46.768 1.4628 57.056 73 1.4200 21.639 1.4981 28.447 1.5592 36.899 1.5794 46.894 1-5362 57-841 74 1.4489 20.978 1.5407 27.810 1.6186 36.461 1.6546 46.916 1.6190 58.581 75 1.4783 20.238 1.5849 27.064 1.6822 35.890 1.7380 46.814 1.7129 59.266 76 1.5077 19.419 1.6307 26.202 1-7505 35.170 1.8308 46.567 1.8202 59.880 77 1.5373 I8.5I8 1.6778 25.214 1.8235 34-280 1.9341 46.145 1.9438 60.403 78 1.5665 17.532 1.7261 24.094 1.9012 33.202 2.0498 45.514 2.0875 60.814 79 1.5955 16.462 1.7749 22.835 1.9835 31.912 2.1792 44-633 2.2562 61.076 80 1.6235 15.308 1.8239 21.429 2.0702 30.389 2.3240 43-454 2.4563 61.144 81 1.6505 14.069 1.8726 19.874 2.1604 28.609 2.4862 41-915 2.6973 60.956 82 1.6759 12.748 1.9200 18.168 2.2531 26.553 2.6669 39.944 2.9917 60.422 83 1.6996 11.350 1.9654 16.310 2.3466 24.200 2.8668 37-454 3-3577 59-412 84 I.72I2 9.879 2.0078 14.306 2.4388 21.540 3.0848 34-354 3.8207 57.724 85 1.7402 8.343 2.0463 12.164 2.5267 18.568 3.3164 30.541 4.4198 55-045 86 1.7562 6.748 2.0796 9.898 2.6065 I 5- 2 94 3.5521 25.931 5.2010 50.866 87 1.7691 5.105 2.1068 7.525 2.6744 11.747 3.7765 20.477 6.2275 44-356 88 1.7785 3.425 2.1269 5-067 2.7266 7.970 3.9665 14.214 7.5060 34-274 89 1.7843 I.7I9 2-1393 2.548 2.7596 4.029 4.0952 7.293 8.7880 19.305 90 1.7862 o.ooo 2.1434 o.ooo 2.7709 o.ooo 4.1413 o.ooo 9.4007 o.ooo Note. Negative quantities are in heavy type. tanh (1.3 /45) Example. i-3 /45 0.83515 \27.536 = - 0.83515 \i7.32' .10". [28] TABLE V. CORRECTING ^.^ tanhfl INTJED 1-AC ^ - ry. CONT o 45 46 47 48 49 1.6 o 0.71331 35-799 0.72175 36.733 0.73069 37.682 0.74013 38.642 0.75013 39-6i5 o 0.67224 38.016 0.67959 39-048 0.68741 40.098 0.69565 41.166 0.70435 42.251 1.8 0.63250 39-937 0.63883 41.050 0.64544 42.186 0.65239 43-341 0.65978 44.520 1.9 o 0.59484 41.562 0.60005 42.740 0.60553 43.942 0.61126 45.168 0.61726 46.422 2.0 o 0-55955 42-906 0.56375 44-133 0.56815 45-387 0.57275 46.667 0.57750 47-977 50 52 53 54 0.76069 40.600 0.77194 41.600 0.78381 42.616 0.79650 43.646 0.81000 44.691 0.71359 43-355 0.72329 44.480 o.733 6 5 45-627 0.74465 46.797 0.75629 47-992 0.66756 45.725 0.67572 46.955 0.68433 48.213 0.69350 49.501 0.70317 50.820 0.62358 47.705 0.63026 49.019 0.63721 50.366 0.64458 51.747 0.65232 53-165 0.58250 49-319 0.58770 50.694 o-593 I 5 52-103 0.59885 53-550 0.60480 55-039 55 56 57 58 59 0.82438 45-754 0.83969 46.834 0.85613 47-931 0.87369 49.047 0.89256 50.183 0.76865 49.212 0.78188 50.461 0.79600 51.739 0.81106 53.047 0.82724 54.389 0.71344 52.174 0.72433 53-564 0-73594 54-992 0.74828 56.460 0.76144 57-974 0.66047 54.621 0.66911 56.121 0.67821 57.666 0.68784 59.259 0.69805 60.905 o.6noo 56.571 0.61750 58.149 0.62435 59-777 0.63150 61.458 0.63905 63-195 60 61 62 63 64 0.91288 51.339 0-93475 52.516 0.95834 53.714 0.98394 54-935 1.0117 56.181 0.84459 55-768 0.86324 57.181 0.88335 58.635 0.90506 60.131 0.92853 61.674 0-77550 59-535 0.79056 61.145 0.80672 62.810 0.82406 64.533 0.84272 66.318 0.70895 62.606 0.72047 64.366 0.73274 66.191 0.74579 68.085 0.75974 70.052 0.64695 64.995 0.65530 66.860 0.66405 68.793 0.67330 70.802 0.68310 72.891 65 66 67 68 69 1.0419 57-452 .0749 58.748 .mi 60.072 .1508 61.425 .1948 62.809 0.95400 63.266 0.98176 64.910 I.OI2I 66.6l3 1.0453 68.376 I.o8l7 7O.2O9 0.86283 68.172 0.88461 70.098 0.90817 72.104 0.93372 74-I96 0.96156 76.382 0.77463 72.099 0.79063 74.232 0.80774 76.459 0.82611 78.786 0.84579 8I.22I 0.69340 75.068 0.70430 77-338 0.71585 79-707 0.72810 82.184 0.74105 84.776 70 72 73 74 .2433 64.224 .2974 65.674 .3580 67.160 .4262 68.686 5033 70.256 I.I2I9 72.II6 1.1663 74.IO6 I.2I57 76.187 1.2708 78.371 1.3325 80.670 0.99189 78.671 .0251 81.074 .0614 83.602 .1013 86.266 .1452 89.084 0.86705 83.773 0.88989 86.454 0.91453 89.273 O.94III 92.244 0.96974 95-379 0-75475 87.490 0.76925 90.337 0.78465 93-326 0.80095 96.466 0.81815 99.769 75 76 77 78 79 .5913 71-873 .6924 73-543 .8098 75-274 J-9473 77-078 2.1104 78.966 1.4020 83.IOI 1.4807 85.685 1.5704 88.447 1.6731 91.420 I.79I5 94-644 .1936 92.076 .2470 95.261 1.3061 98.664 1.3714 102.315 1.4436 106.250 1.0005 98.692 1.0337 IO2.2O2 1.0693 105.925 I.I074 IO9.88I I.I478 114.088 0.83625 103.245 0.85525 106.905 0.87515 110.760 0.89585 114.820 0.91720 119.093 80 81 82 83 84 2.3068 80.960 2.5472 83.087 2.8474 85.390 3.2326 87.937 3-7435 90.839 1.9288 98.175 2.0894 IO2.O83 2.2775 106.466 2.4992 111.446 2.7604 II7.I99 1.5231 110.510 1.6104 115.140 1.7056 120.191 1.8078 125.722 1.9155 131-793 I.I906 II8.568 1.2352 123.344 I.28l3 128.428 I-3279 I33.842 1-3739 I39-590 0.93905 123.590 0.96110 128.319 0.98305 133-281 1.0045 138.474 1.0249 143.894 85 86 87 88 89 4.4502 94.289 5.4825 98.668 7.1175 104.791 9.9938 114.760 15.528 135.011 3.0648 123.945 34II4 I3L956 3.7856 I4L527 4.1472 152.868 4.4231 165.896 2.0255 138.448 2.1329 145-720 2.2309 153.606 2.3107 162.043 2.3628 170.905 1.4179 145-673 1.4579 152-075 1.4921 158.768 1.5184 165.700 1.5349 172.805 1.0436 149-528 1.0601 155-359 1.0738 161.359 1.0840 167.494 1.0904 173-723 90 21.395 180.000 4.5275 iSo.OOO 2.3813 180.000 1.5406 180.000 1.0925 180.000 Note. Example. Negative quantiti tanh (2.0 /8o) es are in heavy type. 0.93905 \i23.590 = 0.93905 \ 1 23.35' 24 2.0 /8o [29] TABLE V. 2.1 CORRECTING 2.2 FACTOI 2-. > tanh0 _/- CONTINUED I 2.5 6 j - r/j. 2.i o o o 45 0.52690 43.993 0.49691 44.845 0.46952 45.492 0.44467 45.961 O.422I2 46.283 46 0.53019 45.254 0.49941 46.127 0.47139 46.783 0.44592 47.252 0.42292 47.564 47 0.53362 46.542 O.5O2OO 47.436 0.47322 48.101 0.44717 48.567 0.42368 48.868 48 0.53714 47.859 0.50465 48.774 0.47509 49-446 0.44842 49.908 0.42440 50.195 49 0.54081 49.206 0.50732 50.142 0.47696 50.820 0.44958 51.277 0.42504 5L548 5 0.54462 50.586 O.5IOO9 51.542 0.47883 52.226 0.45075 52.676 0.42560 52.929 51 0.54852 52.001 0.5I29I 52.978 0.48070 53-666 0.45188 54.107 0.42612 54-340 52 0.55257 53.452 0.51578 54.452 0.48256 55-144 0.45292 55.572 0.42652 55.782 53 0.55681 54-943 0.51868 55.965 0.48443 56-659 0.45392 57.073 0.42684 57.256 54 0.56114 56.476 0.52168 57.520 0.48630 58.215 0.45488 58.614 0.42704 58.766 55 0.56562 58.055 0.52473 59-120 0.48813 59.8i5 0.45571 60.196 0.42712 60.314 56 0.57029 59.682 0.52782 60.769 0.48991 61.462 0.45646 61.822 0.42708 61.903 57 0.57514 61.361 0.53095 62.469 0.49170 63.160 0.45713 63.496 0.42688 63-534 58 0.58014 63.094 0.53414 64.226 0-49339 64.912 0.45767 65.220 0.42648 65.211 59 0.58533 64.888 0.53736 66.042 0.49504 66.722 0.45804 66.998 0.42588 66.938 60 0.59071 66.745 0.54064 67.921 0.49661 68.594 0.45829 68.835 0.42512 68.718 61 0.59629 68.671 0-54395 69.869 0.49813 70.531 0.45838 70.735 0.42412 70.556 62 0.60210 70.669 0.54732 71.889 0-49957 72.540 0.45829 72.701 0.42288 72.455 63 0.60814 72.744 0.55073 73-988 0.50087 74.625 0.45800 74.740 O.42I4O 74.419 64 0.61443 74.902 0.55418 76.171 O.5O2O9 76.791 0.45750 76.855 0.41960 76-455 65 0.62090 77.151 0.55764 78.443 0.50317 79-044 0.45675 79.053 0.41752 78.567 66 0.62767 79.496 0.56109 80.810 0.50413 81.390 0-45579 81.341 O.4I5I2 80.761 67 0.63471 81.942 0.56459 83.280 0.50491 83.836 0-45454 83.723 O.4I24O 83.044 68 0.64205 84.499 0.56814 85.858 0.50557 86.389 0.45300 86.208 0.40928 85.424 69 0.64962 87.172 0.57164 88.552 0.50604 89.056 0.45I2I 88.802 0.40580 87.906 70 0.65752 89.968 0.57518 91-370 0.50630 91.843 0.44913 91-514 0.40196 90.5OO 0.66571 92.898 0.57873 94-319 0.50639 94-759 0.44671 94-352 0.39771 93-215 72 0.67424 95-968 0.58223 97.405 0.50626 97.811 0.44396 97-324 0.39306 96.061 73 0.68305 99.185 0.58568 100.636 0.50596 101.006 0.44092 100.439 0.38802 99.048 74 0.69214 102.560 0.58914 104.020 0.50543 104.352 0-43754 103.706 0.38260 IO2.I87 75 0.70152 106.099 0.59255 107.564 0.50470 107.858 0.43388 107.134 0.37681 105.489 76 0.71114 109.811 0.59586 111.274 0.50378 111.531 0.42992 110.730 0.37068 108.966 77 0.72096 113.702 0.59914 115-156 0.50265 "5-374 0.42575 114-505 0.36427 112.633 78 0.73086 117.778 0.60232 119.212 0.50135 II9-392 0.42138 118.467 0.35762 II6.5OI 79 0.74090 122.043 0.60536 123.446 0.49991 123.590 0.41683 122.620 0.35080 120.582 80 0.75086 126.501 0.60832 127.859 0.49835 127.968 O.4I22I 126.968 0.34392 124.887 81 0.76067 131-151 O.6lIO9 132.449 0.49674 132.527 0.40759 131-517 0.33706 129.427 82 0.77019 135-989 0.61364 137-212 0.49509 137-264 0.40305 136.267 0.33033 I34.209 83 0.77924 141.011 O.6l6oO 142.140 0-49343 142.171 0.39869 141.212 0.32391 I39.233 84 0.78767 146.210 0.6l8l4 147-223 0.49183 I47.237 0.39463 146.344 0.31790 144.499 85 0.79524 I5L570 O.62OOO 152.449 0.49039 152.450 0.39094 151.650 0.31248 149.994 86 0.80176 157.070 0.62155 157.800 0.48913 157-793 0.38775 157.113 0.30778 I55-703 87 0.80710 162.690 O.62282 163.254 0.48804 163.245 0.38516 162.710 0.30396 161.598 88 0.81100 168.406 0.62374 168.790 0.48726 168.781 0.38325 168.413 0.30II3 167.645 89 0.81338 174.187 O.62427 174-381 0.48678 I74.376 0.38207 174.188 0.29940 173.796 90 0.81424 180.000 0.62445 180.000 0.48661 180.000 0.38167 180.000 0.29880 iSo.OOO Note Negative quantities are in heavy type. 1 - Atllll^l C 2-3 /9Q " \I50 = 0.48661 Zi_^ [30] 1 TABLE V. CORRECTING FACTOR tanh 6 - I- ^~ NUED JT/V^ J. VJJS-. - r /~y. \^ Cio~* 1 792.03 19.60 522,380 794-05 I.OOO o.oo 1-9*53 u 794.05 19.65 541,220 796.06 I.OOO o.oo 1.8478 " 796.06 19.70 560,650 798.08 I.OOO o.oo 1.7837 " 798.08 19-75 599,830 800.09 I.OOO o.oo 1.6671 " 800.09 19.80 601,730 802. ii I.OOO o.oo 1.6619 802. II 19.85 623,390 804.12 I.OOO o.oo 1.6041 804.12 19.90 645,820 806.14 I.OOO o.oo 1.5484 u 806.14 19-95 669,070 808.15 I.OOO o.oo 1.4946 u 808.15 20.00 693, 1 50 810.17 I.OOO o.oo 1.4426 u 810.17 20.05 718,090 812.18 I.OOO o.oo 1.3926 u 812.18 20. 10 743,930 814.20 I.OOO o.oo 1.3442 u 814.20 20.15 770,710 816.21 I.OOO o.oo 1.2975 u 816.21 20.20 798,440 818.23 I.OOO o.oo 1-2525 u 818.23 20.25 827,160 820.24 I.OOO o.oo 1.2090 u 820.24 20.30 856,940 822.26 I.OOO o.oo 1.1669 u 822.26 20.35 887,770 824.27 I.OOO o.oo 1.1264 a 824.27 20.40 9*9,730 826.29 I.OOO o.oo 1.0873 u 826.29 20.45 952,820 828.30 I.OOO o.oo 1.0496 u 828.30 20.50 987,120 830.32 I.OOO o.oo 1.0130 u 830.32 Note. Negative quantities are in heavy type. Example, sinh (20.0 /45) = cosh (20.0 /45) = 693,i5o/8io.i7 / = 693, 150 , [41] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv x = 0.05 x = 0.15 0.0 0.05 O.I 0.15 O.2 o.oo o.oo o.oo o.oo o.oo o.oo 0.07846 0.15643 0-23345 0.30902 0.05002 0.04987 0.04945 0.04864 0.04757 o.oo 0.07856 0.15663 0.23374 0.30940 0.10017 0.09986 0.09893 0.09740 0.09526 o.oo 0.07885 0.15722 0.23461 0.31056 0.15056 0.15010 0.14871 0.14640 0.14319 o.oo 0.07934 0.15820 0.23608 0.31250 0.20134 0.20072 0.19886 0.19577 0.19148 o.oo 0.08003 0.15957 0.23813 0.31522 0.25 0-3 0-35 0.4 0-45 o.oo o.oo o.oo o.oo o.oo 0.38268 Q-45399 0.52250 0.58778 0.64944 0.04621 0.04457 0.04265 0.04047 0.03804 0.38316 0-45454 0.52313 0.58850 0.65023 0.09254 0.08925 0.08541 0.08104 0.07617 0.38460 0.45626 0.52511 0.59073 0.65270 0.13910 0.13415 0.12838 0.12181 0.11449 0.38700 0.45911 0.52839 0.59441 0.65677 0.18601 0.17939 0.17167 0.16288 0.15310 0.39036 0.46310 0.53298 0.59958 0.66248 0-5 0.55 0.6 0.65 0.7 o.oo o.oo o.oo o.oo o.oo 0.70711 0.76041 0.80902 0.85264 0.89101 0.03537 0.03249 0.02940 0.02614 0.02271 0.70796 0.76133 0.81000 0.85367 0.89208 0.07083 0.06505 0.05888 0.05234 0.04547 0.71065 0.76421 0.81307 0.85691 0.89547 0.10646 0.09778 0.08850 0.07867 0.06835 0.71508 0.76898 0.81814 0.86225 0.90105 0.14237 0.13076 0.11834 0.10520 0.09141 0.72130 0.77567 0.82525 0.86975 0.90889 0-75 0.8 0.85 0.9 0-95 o.oo o.oo o.oo o.oo o.oo 0.92388 0.95106 0.97237 0.98769 0.99692 0.01914 0.01546 0.01168 0.00783 0.00392 0.92503 0.95225 0-97359 0.98892 0.99816 0.03833 0.03095 0.02338 0.01567 0.00786 0.92850 0.95582 0.97724 0.99263 1.00191 0.05762 0.04653 0.03515 0.02355 0.01181 0.93429 0.96178 0.98333 0.99882 1.00815 0.07705 0.06222 0.04700 0.03150 0.01580 0.94242 0.97014 0.99188 1.00751 1.01692 I.O 1.05 i.i I-I5 1.2 o.oo o.oo o.oo o.oo o.oo I.OOOOO 0.99692 0.98769 0.97237 0.95106 o.oo 0.00392 0.00783 0.01168 0.01546 1.00125 0.99816 0.98892 0-97359 0.95225 o.oo 0.00786 0.01567 0.02338 0.03095 1.00500 1.00191 0.99263 0.97724 0.95582 o.oo 0.01181 0.02355 0.03515 0.04653 1.01127 1.00815 0.99882 0.98333 0.96178 o.oo 0.01580 0.03150 0.04700 0.06222 1.02007 1.01692 1.00751 0.99188 0.97014 I.2S i-3 1-35 1.4 1-45 o.oo o.oo o.oo o.oo o.oo 0.92388 0.89101 0.85264 0.80902 0.76041 0.01914 0.02271 0.02614 0.02940 0.03249 0.92503 0.89208 0.85367 0.81000 0.76133 0.03833 0.04547 0.05234 0.05888 0.06505 0.92850 0.89547 0.85691 0.81307 0.76421 0.05762 0.06835 0.07867 0.08850 0.09778 0.93429 0.90105 0.86225 0.81814 0.76898 0.07705 0.09141 0.10520 0.11834 0.13076 0.94242 0.90889 0.86975 0.82525 0.77567 i-5 i-55 1.6 1.65 i-7 o.oo o.oo o.oo o.oo o.oo 0.70711 0.64944 0.58778 0.52250 0-45399 0.03537 0.03804 0.04047 0.04265 0.04457 0.70796 0-65023 0.58850 0.52313 0-45454 0.07083 0.07617 0.08104 0.08541 0.08925 0.71065 0.65270 0-59073 0-52511 0.45626 0.10646 0.11449 o. 12181 0.12838 0.13415 0.71508 0.65677 0.59441 0.52839 0.45911 0.14237 0.15310 0.16288 0.17167 0.17939 0.72130 0.66248 0.59958 0.53298 0.46310 1-75 i. 80 1.85 1.9 1-95 o.oo o.oo o.oo o.oo o.oo 0.38268 0.30902 0.23345 0.15643 0.07845 0.04621 0.04757 0.04864 0.04945 0.04987 0.38316 0.30940 0-23374 0.15663 0.07856 0.09254 0.09526 0.09740 0.09893 0.09986 0.38460 0.31056 0.23461 0.15722 0.07885 0.13910 0.14319 0.14640 0.14871 0.15010 0.38700 0.31250 0.23608 0.15820 0.07934 o. 18601 0.19148 0.19577 0.19886 0.20072 0.39036 0.31522 0.23813 0.15957 0.08003 2.0 o.oo o.oo 0.05002 o.oo 0.10017 o.oo 0.15056 o.oo 0.20134 o.oo Note. Negative quantities are in heavy type. Examples. sinh (o.i + i 0.5) = 0.07083 + * 0.71065. sinh (o.i + i 1.2) = 0.03095 + i 0.95582. [42] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 0.25 x = 0.3 = 0.35 x = 0.4 x = 0.45 o 0.25261 o.oo 0.30452 o.oo 0.35719 o.oo 0.41075 o.oo 0.46534 o.oo 0.05 0.25183 0.08092 0.30358 0.08202 0.35609 0.08331 0.40949 0.08482 0.46391 0.08654 O.I 0.24950 0.16135 0.30077 0.16353 0.35279 0.16611 0.40570 0.16912 0.45961 0.17254 0.15 0.24563 0.24078 0.29611 0.24403 0-34732 0.24789 0-39940 0.25237 0.45249 0.25748 O.2 0.24025 0.31872 0.28962 0.32303 0-33971 0.32814 0.39065 0.33407 0.44257 0.34084 0.25 0.23338 0.39471 0.28134 0.40003 0.33000 0.40636 0-37949 0.41371 0.42992 0.42209 0-3 0.22508 0.46825 0.27133 0-47457 0.31826 0.48208 0.36598 0.49080 0.41462 0.50074 o-35 0.21539 0.53891 0.25965 0.54619 0.30455 0.55483 0.35022 0.56486 0.39677 0.57630 0.4 0.20437 0.60625 0.24636 0.61444 0.28897 0.62416 0.33231 0.63544 0.37647 0.64831 0-45 0.19208 0.66985 0.23156 0.67889 0.27161 0.68964 0.31234 0.70210 0.35385 0.71632 o-5 0.17862 0.72932 0.21533 0.73917 0.25257 0.75086 0.29045 0.76443 0.32905 0.77992 Q-55 o. 1 6406 0.78429 0.19777 0.79488 0.23198 0.80746 0.26676 0.82206 0.30222 0.83871 0.6 0.14848 0.83443 0.17899 0.84570 0.20995 0.85908 0.24143 0.87461 0.27352 0.89232 0.65 0.13199 0.87942 0.15911 0.89130 0.18663 0.90540 0.21462 0.92177 0.24314 0.94044 0.7 0.11468 0.91900 0.13825 0.93140 0.16216 0.94614 0.18648 0.96324 0.21126 0.98275 0-75 0.09667 0.95290 0.11654 0.96577 0.13669 0.98105 0.15719 0.99878 0.17808 1.01901 0.8 0.07801 0.98093 0.09410 0.99418 0.11038 .00991 0.12693 1.02816 0.14380 1.04899 0.85 0.05897 1.00292 0.07109 1.01646 0.08338 03254 0.09589 1.05120 0.10863 1.07250 0.9 0.03952 1.01871 0.04764 1.03247 0.05588 .04880 0.06426 1.06776 0.07280 1.08939 o-95 0.01982 1.02823 0.02389 1.04212 0.02803 .05860 0.03223 1.07774* 0.03651 1.09957 I.O o.oo 1.03141 o.oo 1-04534 0.00 .06188 o.oo 1.08107 o.oo 1.10297 1.05 0.01982 1.02823 0.02389 1.04212 0.02803 .05860 0.03223 1.07774 0.03651 1.09957 i.i 0.03952 1.01871 0.04764 1.03247 0.05588 .04880 0.06426 1.06776 0.07280 1.08939 i-i5 0.05897 1.00292 0.07109 1.01646 0.08338 03254 0.09589 1.05120 0.10863 1.07250 1.2 0.07801 0.98093 0.09410 0.99418 0.11038 .00991 0.12693 1.02816 0.14380 1.04899 1-25 0.09667 0.95290 0.11654 0.96577 0.13669 0.98105 0.15719 0.99878 0.17808 I.OIOOI i-3 0.11468 0.91900 0.13825 0.93140 0.16216 0.94614 0.18648 0.96324 0.21126 0.98275 i-3S 0.13199 0.87942 0.15911 0.89130 0.18663 0.90540 0.21462 0.92177 0.24314 0.94044 i-4 0.14848 0.83443 0.17899 0.84570 0.20995 0.85908 0.24143 0.87461 0.27352 0.89232 1-45 0.16406 0.78429 0.19777 0.79488 0.23198 0.80746 0.26676 0.82206 0.30222 0.83871 i-5 0.17862 0.72932 0.21533 0.73917 0.25257 0.75086 0.29045 0.76443 0.32905 0.77992 1-55 0.19208 0.66985 0.23156 0.67889 0.27161 0.68964 0.31234 0.70210 0.35385 0.71632 1.6 0.20437 0.60625 0.24636 0.61444 0.28897 0.62416 0.33231 0.63544 0.37647 0.64831 1-65 0.21539 0.53891 0.25965 0.54619 0.30455 0.55483 0.35022 0.56486 0.39677 0.57630 i-7 0.22508 0.46825 0.27133 0-47457 0.31826 0.48208 0.36598 0.49080 0.41462 0.50074 i-75 0.23338 0.39471 0.28134 0.40003 0.33000 0.40636 0.37949 0.41371 0.42992 0.42209 1.8 0.24025 0.31872 0.28962 0.32303 0.33071 0.32814 0.39065 0.33407 0.44257 0.34084 1-85 0.24563 0.24078 0.29611 0.24403 0.34732 0.24789 0.39940 0.25237 0.45249 0.25748 1.9 0.24950 0.16135 0.30077 0.16353 0.35279 0.16611 0.40570 0.16912 0.45961 0.17254 i-95 0.25183 0.08092 0.30358 0.08202 0.35609 0.08331 0.40949 0.08482 0.46391 0.08654 2.0 0.25261 o.oo 0.30452 o.oo 0.35719 o.oo 0.41075 o.oo 0.46534 o.oo Note. Negative quantities are in heavy type. Examples. sinh (0.4 + i o) = 0.41075 + to. sinh (0.4 + * i) = o. + i 1.08107. [43] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 0.5 = 0.55 x = 0.6 x = 0.65 x = 0.7 o 0.05 O.I 0.15 O.2 0.52110 0.51949 0.51468 0.50670 0-49559 o.oo 0.08847 0.17640 0.26324 0.34846 0.57815 o.oo 0.57637 0.09063 0.57103 0.18070 0.56218 0.26965 0.54986 0.35695 0.63665 0.63469 0.62882 0.61906 0.60549 o.oo 0.09301 0.18545 0.27674 0.36633 0.69675 0.69460 0.68817 0.67750 0.66265 o.oo 0.09563 0.19066 0.28452 0.37663 0.75858 0.75625 0.74925 0.73763 0.72146 o.oo 0.09848 0.19635 0.29301 0.38787 O.25 0-3 0-35 0.40 0-45 0.48143 0.46430 0.44431 0.42158 0.39624 0.43152 0.51193 0.58918 0.66280 0.73233 0.53414 0.51514 0.49296 0.46773 0.43963 0.44204 0.52441 0.60354 0.67895 0.75018 0.58819 0.56726 0.54284 0.51506 0.48412 0.45366 0.53819 0.61940 0.69680 0.76990 0.64371 0.62081 0.59408 0.56368 0.52981 0.46641 0.55332 0.63682 0.71639 0.79154 0.70084 0.67590 0.64680 0.61371 0.57683 0.48033 0.56984 0.65582 0.73777 0.81517 0-5 0-55 0.6 0.65 0.7 0.36847 0.33842 0.30629 0.27227 0.23657 0-79735 0.85745 0.91227 0.96146 1.00472 0.40882 0.37548 0.33983 0.30208 0.26248 0.81678 0.87835 0.93450 0.98489 1.02920 0.45018 0.41347 0.37422 0.33265 0.28904 0.83825 0.90144 0.95906 1.01078 1.05626 0.49268 0.45250 0.40954 0.36405 0.31632 0.86182 0.92678 0.98602 1.03919 1.08595 0.53640 0.49266 0.44589 0.39636 0.34439 0.88754 0.95444 1.01545 1.07021 1.11836 0-75 0.8 0.85 0.9 0-95 0.19942 0.16103 0.12165 0.08152 0.04088 1.04179 1.07244 1.09647 I-II374 1.12415 0.22125 0.17866 0.13497 0.09044 0.04536 1.06717 1.09857 1.12319 1.14088 1.15154 0.24364 0.19674 0.14862 0.09959 0.04995 1.09523 1.12744 1.15271 1.17087 1. 18181 0.26663 0.21531 0.16265 0.10900 0.05467 1.12602 1.15912 1.18512 1.20379 1.21504 0.29030 0.23442 0.17709 0.11867 0.05952 1.15962 1.19374 1.22049 1.23972 1.25130 .0 05 .1 15 .2 o.oo 0.04088 0.08152 0.12165 0.16103 1.12763 1.12415 I-II374 1.09647 1.07244 o.oo 0.04536 0.09044 0.13497 0.17866 1.15510 1.15154 1.14088 1.12319 1.09857 0.00 0.04995 0.09959 0.14862 0.19674 1.18547 1. 18181 1.17087 1.15271 1.12744 o.oo 0.05467 0.10900 0.16265 0.21531 1.21879 1.21504 1.20379 1.18512 1.15912 o.oo 0.05952 0.11867 0.17709 0.23442 1.25517 1.25130 1.23972 1.22049 1-19374 1-25 i-35 1.4 i -45 0.19942 0.23657 0.27227 0.30629 0.33842 1.04179 1.00472 0.96146 0.91227 0.85745 0.22125 0.26248 0.30208 0.33983 0.37548 1.06717 1.02920 0.98489 0.93450 0.87835 0.24364 0.28904 0.33265 0.37422 0.41347 1.09523 1.05626 1.01078 0.95906 0.90144 0.26663 0.31632 0.36405 0.40954 0.45250 1.12602 1.08595 1.03919 0.98602 0.92678 0.29030 0.34439 0.39636 0.44589 0.49266 1.15962 1.11836 1.07021 1-01545 0-95444 i-SS 1.6 1-65 0.36847 0.39624 0.42158 0.44431 0.46430 0-79735 0.73233 0.66280 0.58918 0.5H93 0.40882 0.43963 0.46773 0.49296 0.51514 0.81678 0.75018 0.67895 0.60354 0.52441 0.45018 0.48412 0.51506 0.54284 0.56726 0.83825 0.76990 0.69680 0.61940 0.53819 0.49268 0.52981 0.56368 0.59408 0.62081 0.86182 0.79154 0.71639 0.63682 0.55332 0.53640 0.57683 0.61371 0.64680 0.67590 0.88754 0.81517 0-73777 0.65582 0.56984 1-75 1.8 1.85 1.9 1-95 0.48143 0.49559 0.50670 0.51468 0.51949 0.43152 0.34846 0.26324 o. 1 7640 6.08847 0.53414 0.54986 0.56218 0.57103 0.57637 0.44204 0.35695 0.26965 0.18070 0.09063 0.58819 0.60549 0.61906 0.62882 0.63469 0.45366 0.36633 0.27674 0.18545 0.09301 0.64371 0.66265 0.67750 0.68817 0.69460 0.46641 0.37663 0.28452 0.19066 0.09563 0.70084 0.72146 0.73763 0.74925 0.75625 0.48033 0.38787 0.29301 0.19635 0.09848 2.0 0.52110 0.00 0.57815 0.00 0.63665 o.oo 0.69675 0.00 0.75858 o.oo Note. Negative quantities are in heavy type. Examples. sinh (0.65 sinh (0.55 f i 0-75) f i 1-40) = 0.26663 + * 1.12602. = -0.33983 + * 0.93450. [44] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 0.75 x = 0.85 x = o.g x = 0.95 o 0.05 O.I 0-15 O.2 0.82232 0.81978 0.8I2I9 0.79960 0.78207 o.oo 0.10158 0.20253 0.30224 0.40079 0.88811 0.88537 0.87717 0.86357 0.84464 o.oo 0.10493 0.20922 0.31222 0.41329 0.95612 0.95317 0-94435 0.92970 0.90932 o.oo 0.10855 0.21643 0.32298 0.42753 1.02652 . 1.02335 1.01388 0.99816 0.97628 o.oo 0.11244 0.22418 0.33455 0.44285 1 .09948 1.09610 1.08595 1.06911 1.04567 o.oo o. 11661 0.23250 0.34695 0.45927 O.25 o-3 0.35 0.4 0.45 0.75972 0.73269 O./OII4 0.66527 0.62529 0.49545 0.58777 0.67647 0.76100 0.84083 0.82050 0.79131 0.75724 0.71849 0.67532 0.51182 0.60718 0.69881 0.78613 0.86859 0.88334 0.85191 0.81522 o.7735i 0.72704 0.52945 0.62811 0.72289 0.81322 0.89853 0.94838 0.91463 0.87525 0.83047 0.78057 0.54842 0.65061 0.74879 0.84235 0.93071 1.01579 0.97965 0.93747 0.88950 0.83605 0.56875 0.67473 0.77655 0.87358 0.96523 -5 0-55 0.6 0.65 0.7 0.58146 0.53405 0.48335 0.42966 0-37332 0.91548 0.98449 1.04742 1.10390 I -iS3S6 0.62799 0.57678 0.52202 0.46403 0.40319 0.94571 1.01700 1.08201 1-14035 1.19166 0.67608 0.62095 0.56199 0-49957 0.43407 0.97830 1.05205 1.11930 1.17965 1.23274 0.72586 0.66667 0.60337 0.53635 0.46603 1-01334 1.08973 i-!5939 1.22191 1.27689 0.77745 0.71406 0.64621 0.57448 0.49916 1.05092 1.13013 1.20238 1.26722 1-32425 0.75 0.8 0.85 0.9 o-95 0.31469 0.254II 0.19197 0.12864 0.06452 1.19613 1.23132 1.25891 1.27874 1.29069 0.33986 0.27444 0.20732 0.13893 0.06968 1-23563 1.27198 1.30048 1.32097 J -3333i 0.36589 0.29546 0.22320 0.14957 0.07502 1.27822 1.31582 1.34530 1.36650 1.37927 0.39283 0.31721 0.23964 0.16058 0.08054 1.32400 1.36294 1-39349 1.41544 1.42867 0.42076 0.33976 0.25667 0.17200 0.08627 1-37309 1.41348 1.44516 1.46793 1.48164 I.O 1.05 i.i 1-15 1.2 o.oo 0.06452 0.12864 0.19197 0.25411 1.29468 1.29069 1.27874 1.25891 1.23132 o.oo 0.06968 0.13893 0.20732 0.27444 1-33743 33331 32097 .30048 .27198 o.oo 0.07502 0.14957 0.22320 0.29546 1.38353 37927 36650 34530 .31582 o.oo 0.08054 0.16058 0.23964 0.31721 I.43309 1.42867 1.41544 1-39349 1.36294 o.oo 0.08627 0.17200 0.25667 0.33976 1.48623 1.48164 1.46793 1.44516 1.41348 1-25 i-3 i-3S 1.4 1-45 0.31469 0.37332 0.42966 0.48335 0.53405 1.19613 I-I5356 1.10390 1.04742 0.98449 0.33986 0.40319 0.46403 0.52202 0.57678 23563 .19166 14035 .08201 .01700 0.36589 0.43407 0.49957 0.56199 0.62095 .27822 .23274 17965 .11930 .05205 0.39283 0.46603 0.53635 0.60337 0.66667 1.32400 1.27689 1.22191 I - I 5939 1.08973 0.42076 0.49916 0.57448 0.64621 0.71406 1.37309 1.32425 1.26722 1.20238 1.13013 i-5 i-SS 1.6 1.65 i-7 0.58146 0.62529 0.66527 0.70114 0.73269 0.91548 0.84083 0.76100 0.67647 0.58777 0.62799 0.67532 0.71849 0.75724 0.79131 0.94571 0.86859 0.78613 0.69881 0.60718 0.67608 0.72704 0.77351 0.81522 0.85191 0.97830 0.89853 0.81322 0.72289 0.62811 0.72586 0.78057 0.83047 0.87525 0.91463 i-oi334 0.93071 0.84235 0.74879 0.65061 0.77745 0.83605 0.88950 0-93747 0.97965 1.05092 0.96523 0.87358 0.77655 0.67473 i-75 1.8 1.85 1.9 i-95 0.75972 0.78207 0.79960 0.81219 0.81978 0-49545 0.40079 0.30224 0.20253 0.10158 0.82050 0.84464 0.86357 0.87717 0.88537 0.51182 0.41329 0.31222 0.20922 0.10493 0.88334 0.90932 0.92970 0.94435 0.95317 0.52945 0.42753 0.32298 0.21643 0.10855 0.94838 0.97628 0.99816 1.01388 1-02335 0.54842 0.44285 0-33455 0.22418 0.11244 1.01579 1.04567 1.06911 1.08595 1.09610 0.56875 0.45927 0-34695 0.23250 0.11661 2.0 0.82232 o.oo 0.88811 o.oo 0.95612 o.oo 1.02652 o.oo 1.09948 o.oo Note. Negative quantities are in heavy type. Examples. sinh (0.8 + * 0.7) = 0.40319 -f- * 1.19166. sinh (0.8 + i 1.7) = 0.79131 + 10.60718. [45] TABLE VII. HYPERBOLIC SINES, sinh (* + iq) = u + iv. CONTINUED = 1.05 = 1.15 0.05 O.I 0.15 0.2 1.17520 o.oo 1.17158 O.I2I07 I.I6073 0.24139 1.14273 0.36023 I.II768 0.47684 1.25386 o.oo 1.24999 0.12583 1.23842 0.25089 1.21921 0.37440 1.19249 0.49560 1-33565 o.oo 1-33153 0.13091 1.31920 0.26101 1.29875 0.38951 1.27028 0.51560 1.42078 o.oo 1.41640 0.13632 1.40329 0.27179 1.38152 0.40559 1.35124 0.53689 1.50946 o.oo 1.50481 0.14206 1.49088 0.28325 1.46776 0.42269 1.43558 0.55952 0.25 o-3 0-35 0.4 0-45 1.08574 0.59051 I.047II 0.70055 I.OO2O2 0.80626 0.95076 0.90700 0.89363 I.002I5 1.15841 0.61375 1.11719 0.72811 1.06909 0.83798 1.01439 0.94269 0.95344 1.04158 1.23398 0.63852 1.19007 0.75749 1.13883 0.87180 1.08056 0.98073 1.01564 1.08362 1.31263 0.66488 1.26592 0.78877 1.21141 0.90780 1.14943 1.02123 1.08037 1.12836 1.39456 0.69291 1.34494 0.82202 1.28703 0.94607 1. 22118 1.06428 1.14781 1.71593 o-5 0-55 0.6 0.65 0.7 0.83099 I.O9II2 0.76323 I.I7337* 0.69077 1.24838 0.61404 1.31569 0-53353 1-3749 0.88661 1.13405 0.81432 1.21953 0.73700 1.29750 0.65514 1.36746 0.56924 1.42899 0.94445 1.17982 0.86743 1.26875 0.78508 1.34986 0.69787 1.42265 0.60637 1.48666 1.00464 1.22854 0.92272 1.32114 0.83511 1.40560 0.74235 1.48139 0.64502 1.54805 1.06735 1-28033 0.98032 1.37684 0.88724 1.46485 0.78869 1.54384 0.68528 1.61331 0-75 0.8 0.85 0.9 0-95 0.44973 1.42562 0.36316 1.46756 0.27435 I-50045 0.18384 1.52408 0.09221 1.53832 0.47983 1.48171 0.38746 1.52530 0.29271 1.55948 0.10615 1.58405 0.09838 1.59885 0.51113 1.54151 0.41274 1.58685 0.31180 1.62242 0.20894 1.64798 0.10479 1.66337 0.54371 1.60517 0.43904 1.65238 0.33167 1.68941 0.22226 1.71602 0.11147 1.73206 0.57765 1.67283 0.46645 1.72204 0.35238 1.76063 0.23613 1.78836 0.11843 1.80507 I.O 1-05 i.i i.iS 1.2 o.oo 1.54308 0.09221 1.53832 0.18384 1.52408 0.27435 1.50045 0.36316 1.46756 o.oo 1.60379 0.09838 1.59885 0.19615 1.58405 0.29271 1.55948 0.38746 1.52530 o.oo 1.66852 0.10479 1.66337 0.20894 1.64798 0.31180 1.62242 0.41274 1.58685 0.00 I.7374I 0.11147 1.73206 0.22226 1.71602 0.33167 1.68941 0.43904 1.65238 o.oo 1.81066 0.11843 1.80507 0.23613 1.78836 0.35238 1.76063 0.46645 1.72204 1.25 i-3 i-35 1.4 i-45 0.44973 1.42562 0.53353 1.37490 0.61404 1.31569 0.69077 1.24838 0.76323 1.17337 0.47983 1.48171 0.56924 1.42899 0.65514 1.36746 0.73700 1.29750 0.81432 1.21953 0.51113 1.54151 0.60637 1.48666 0.69787 1.42265 0.78508 1.34986 0.86743 1.26875 0.54371 1.60517 0.64502 1.54805 0.74235 1.48139 0.83511 .40560 0.92272 .32114 0.57765 1.67283 0.68528 1.61331 0.78869 1.54384 0.88724 1.46485 0.98032 1.37684 i-5 i-SS 1.6 1.65 i-7 0.83099 1.09112 0.89363 1.00215 0.95076 0.90700 I.OO2O2 0.80626 I.047II 0.70055 0.88661 1.13405 0.95344 1.04158 .01439 0.94269 .06909 0.83798 .11719 0.72811 0.94445 1.17982 1.01564 1.08362 1.08056 0.98073 1.13883 0.87180 1.19007 0.75749 1.00464 .22854 1.08037 .12836 1.14943 .02123 1.21141 0.90780 1.26592 0.78877 1.06735 1.28033 1.14781 1.17593 1.22118 1.06428 1.28703 0.94607 1.34494 0.82202 i-7S 1.8 1.85 1.9 i-9S 1.08574 0.59051 I.II768 0.47684 I.I4273 0.36023 I.I6073 0.24139 I.I7I58 O.I2IO7 .15841 0.61375 .19249 0.49560 .21921 0.37440 .23842 0.25089 .24999 0.12583 1.23398 0.63852 1.27028 0.51560 1.29875 0.38951 1.31920 0.26101 1.33153 0.13091 1.31263 0.66488 1.35124 0.53689 1.38152 0.40559 1.40329 0.27179 1.41640 0.13632 1.39456 0.69291 1-43558 0.55952 1.46776 0.42269 1.49088 0.28325 1.50481 0.14206 2.O 1.17520 o.oo 1.25386 o.oo 1.33565 o.oo 1.42078 o.oo 1.50946 o.oo Note. Negative quantities are in heavy type. Examples, sinh (i.o + * i.o) = o + i 1.54308. sinh (i.o + i 1.5) = 0.83099 + i 1.09112. [46] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED = 1.25 x = 1.3 * = i-35 * = 1.4 = 1-45 o 1.60192 o.oo 1.69838 o.oo 1.79909 o.oo 1.90430 o.oo 2.01427 o.oo 0.05 1.59698 0.14816 1.69315 0.15464 1.79354 0.16150 1.89843 0.16876' 2.00806 0.17644 O.I 1.58220 0.29541 1.67747 0.30832 1.77694 0.32199 1.88086 0.33647 1.98947 0.35180 0.15 1.55766 0.44084 1.65146 0.46010 1.74938 0.48051 1.85169 0.50212 1.95862 0.52498 O.2 1.52352 0.58355 1.61526 0.60905 1.71104 0.63606 1.81110 0.66466 1.91569 0.69493 0.25 1.47998 0.72267 1.56910 0.75424 1.66215 0.78769 1.75934 0.82311 1.86094 0.86060 o-3 1.42732 0.85733 1.51327 0.89478 1.60300 0.93446 1.69675 0.97649 1.79473 1.02095 o-35 1.36586 0.98670 1.44811 1.02980 1.53398 1.07548 1.62369 1.12384 1.71745 1.17502 0.4 1.29598 1.10999 1.37402 1.15848 1.45550 1.20986 1.54061 1.26427 1.62958 1.32184 0-45 1. 21811 1.22643 1.29146 1.28001 1.36804 1.33678 1.44804 1.39690 1.53166 1.46051 -5 1.13273 1.33532 1.20094 1.39365 1.27215 1.45546 1.34655 1.52092 1.42431 1.59017 Q-55 1.04036 1.43597 1.10301 1.49870 1.16842 1.56517 1.23674 1.63556 1.30817 1.71007 0.6 0.94158 1.52777 0.99829 1.59451 1.05748 1.66523 1.11932 1.74012 1.18396 1.81935 0.65 0.83700 1.61014 0.88740 1.68048 0.94002 1.75502 0.99500 1.83394 1.05245 1.91745 0.7 0.72726 1.68260 0.77105 1.75610 0.81677 1.83399 0.86454 1.91646 0.91446 2.00373 0-75 0.61303 1.74467 0.64994 1.82089 0.68848 1.90165 0.72875 1.98717 0.77083 2.07766 0.8 0.49502 1.79600 0.52483 1.87445 0-55595 1-95759 0.58846 2.04562 0.62244 2.13878 0.85 0.37396 1.83624 0.39648 1.91646 0.41999 2.00146 0.44455 2.09147 0.47022 2.18671 0.9 0.25060 1.86517 0.26569 1.94665 0.28144 2.03299 0.29790 2.12442 0.31510 2.22115 o-9S 0.12569 1.88260 0.13325 1.96484 0.14116 2.05199 0.14427 2.14427 0.15804 2.24191 I.O o.oo 1.88842 o.oo 1.97091 o.oo 2.05833 o.oo 2.15090 o.oo 2.24884 1.05 0.12569 1.88260 0.13325 1.96484 0.14116 2.05199 0.14427 2.14427 0.15804 2.24191 i.i 0.25060 1.86517 0.26569 1.94665 0.28144 2.03299 0.29790 2.12442 0.31510 2.22115 i- J 5 0.37396 1.83624 0.39648 1.91646 0.41999 2.00146 0.44455 2.09147 0.47022 2.18671 1.2 0.49502 1.79600 0.52483 1.87445 0-55595 1-95759 0.58846 2.04562 0.62244 2.13878 I- 2 5 0.61303 1.74467 0.64994 1-82089 0.68848 1.90165 0.72875 1-98717 0.77083 2.07766 i-3 0.72726 1.68260 0.77105 1.75610 0.81677 1-83399 0.86454 1.91646 0.91446 2.00373 i-35 0.83700 1.61014 0.88740 1.68048 0.94002 1.75502 0.99500 1.83394 1.05245 1.91745 1.4 0.94158 1.52777 0.99829 1.59451 1.05748 1.66523 1.11932 1.74012 1.18396 1.81935 i-45 1.04036 1.43597 1.10301 1.49870 1.16842 1.56517 1.23674 1.63556 1.30817 1.71007 i-5 1.13273 1.33532 .20094 1.39365 1.27215 1.45546 1-34655 1.52092 1.42431 1.59017 i-55 1. 21811 1.22643 .29146 1.28001 1.36804 1.33678 1.44804 1.39690 1.53166 1.46051 1.6 1.29598 1.10999 .37402 1.15848 I -45550 1.20986 1.54061 1.26427 1.62958 1.32184 1.65 1.36586 0.98670 .44811 1.02980 I-53398 1.07548 1.62369 1.12384 1.71745 1.17502 i-7 1.42732 0.85733 .51327 0.89478 1.60300 0.93446 1.69675 0.97649 1-79473 1-02095 i-75 1.47998 0.72267 .56910 0.75424 1.66215 0.78769 1-75934 0.82311 1.86094 0.86060 1.8 1.52352 0.58355 .61526 0.60905 1.71104 0.63606 1.81110 0.66466 1.91569 0.69493 1.85 1-55766 0.44084 .65146 0.46010 1.74938 0.48051 1.85169 0.50212 1.95862 0.52498 1.9 1.58220 0.29541 .67747 0.30832 1.77694 0.32199 1.88086 0.33647 1.98947 0.35180 i-95 1.59698 0.14816 .69315 0.15464 1-79354 0.16150 1.89843 0.16876 2.00806 0.17644 2.O I.6OI92 O.OO 1.69838 o.oo 1.79909 o.oo 1.90430 o.oo 2.01427 o.oo Note. Negative quantities are in heavy type. Examples, sinh (1.35 + to) = 1.70909 + io. sinh (1.4 + i 1.15) = -0.44455 + i 2.09147. [47] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED * = 1.5 = i-SS 1.6 * = 1.65 x = 1.7 0.05 O.I 0.15 0.2 2.12928 o.oo 2.12272 0.18457 2.10307 0.36800 2.07045 0.54916 2.02507 0.72693 2.24961 o.oo 2.24268 0.19316 2.22191 0.38512 2.18745 0.57471 2.13951 0.76076 2-37557 0.00 2.36824 0.20223 2.34632 0.40320 2.30993 0.60170 2.25930 0.79648 2.50746 o.oo 2.49973 0.21180 2.47659 0.42230 2.43818 0.63019 2.38474 0.83420 2.64563 o.oo 2.63747 0.22191 2.61306 0.44245 2.57253 0.66026 2.51614 0.87400 0.25 o-3 o-3S 0.4 o-4S 1.96720 0.90023 1.89720 1.06797 1.81551 1.22913 1.72263 1.38271 1.61912 1.52777 2.07837 0.94211 2.00442 1.11766 1.91811 1.28632 1.81997 1.44704 1.71062 1.59885 2.19473 0.98636 2.11664 1.17015 2.02550 1.34672 1.92187 1.51500 1.80640 1.67393 2.31660 1.03306 2.23417 1.22556 2.13797 1.41050 2.02858 1.58674 1.90669 1.75320 2.44424 1.08235 2.35727 1.28403 2.25577 1-47779 2.14036 1.66244 2.01175 1-83684 o-S 0-55 0.6 0.65 0.7 1 -55 6 3 1-66341 1.38286 1.78879 1.25156 1.90314 1.11255 2.00576 0.96667 2.09601 1.59071 1.74080 1.46101 1.87201 1.32229 1.99169 1.17542 2.09908 1.02130 2.19353 1.67978 1.82254 1.54281 1.95992 1.39632 2.08522 1.24123 2.19765 1.07848 2.29654 1-77305 1.90885 1.62847 2.05274 1.47385 2.18396 1.31015 2.30173 1.13837 2.40529 1.87074 1.99992 1.71820 2.15067 1.55506 2.28816 1.38234 2.41154 1.20109 2.52005 0-75 0.8 0.85 0.9 o.QS 0.81484 2.17334 0.65798 2.23727 0.49707 2.28742 0.33309 2.32345 0.16706 2.34516 0.86089 2.27446 0.69517 2.34137 0.52516 2.39384 0.35192 2.43155 0.17650 2.45427 0.90909 2.38127 0.73409 2.45131 0.55456 2.50625 0.37163 2.54573 0.18639 2.56952 0.95957 2.49404 0.77485 2.56740 0.58536 2.62494 0.39225 2.66629 0.19673 2.69121 1.01244 2.61302 0.81754 2.68989 0.61761 2.75017 0.41387 2.79350 0.20757 2.81960 I.O 1.05 i.i i-iS 1.2 o.oo 2.35241 0.16706 2.34516 0.33309 2.32345 0.49707 2.28742 0.65798 2.23727 o.oo 2.46186 0.17650 2.45427 0.35192 2.43155 0.52516 2.39384 0.69517 2.34137 o.oo 2.57746 0.18639 2.56952 0.37163 2.54573 0.55456 2.50625 0.73409 2.45131 o.oo 2.69951 0.19673 2.69121 0.39225 2.66629 0.58536 2.62494 0.77485 2.56740 o.oo 2.82832 0.20757 2.81960 0.41387 2.79350 0.61761 2.75017 0.81754 2.68989 1.25 i-3 i-35 1.4 1-45 0.81484 2.17334 0.96667 2.09601 1. 11255 2.00576 1.25156 1.90314 1.38286 1.78879 0.86089 2.27446 1.02130 2.19353 1.17542 2.09908 1.32229 1.99169 1.46101 1.87201 0.90909 2.38127 1.07848 2.29654 1.24123 2.19765 1.39632 2.08522 1.54281 1.95992 0.95957 2.49404 1.13837 2.40529 1.31015 2.30173 1.47385 2.18396 1.62847 2.05274 1.01244 2.61302 1.20109 2.52005 1.38234 2.41154 1.55506 2.28816 1.71820 2.15067 i-5 i-5S 1.6 1.65 i-7 1.50563 1.66341 1.61912 1.52777 1.72263 1.38271 1.81551 1.22913 1.89720 1.06797 1.59071 1.74080 1.71062 1.59885 1.81997 1.44704 1.91811 1.28632 2.00442 1.11766 1.67978 1.82254 1.80640 1.67393 1.92187 1.51500 2.02550 1.34672 2.11664 1.17015 1.77305 1.90885 1.90669 1.75320 2.02858 1.58674 2.13797 1.41050 2.23417 1.22556 1.87074 1.99992 2.01175 1.83684 2.14036 1.66244 2.25577 1-47779 2.35727 1-28403 1-75 1.8 i.8 S 1.9 i-95 1.96720 0.90023 2.02507 0.72693 2.07045 0.54916 2.10307 0.36800 2.12272 0.18457 2.07837 0.94211 2.13951 0.76076 2.18745 0.57471 2.22191 0.38512 2.24268 0.19316 2.19473 0.98636 2.25930 0.79648 2.30993 0.60170 2.34632 0.40320 2.36824 0.20223 2.31660 1.03306 2.38474 0.83420 2.43818 0.63019 2.47659 0.42230 2.49973 0.21 180 2.44424 1.08235 2.51614 0.87400 2.57253 0.66026 2.61306 0.44245 2.63747 0.22191 2.0 2.12928 o.oo 2.24961 o.oo 2-37557 o.oo 2.50746 o.oo 2.64563 o.oo Note. Negative quantities are in heavy type. Examples, sinh (1.7 -f 1*0.7) = 1.20109 4 / 2.52005. sinh (1.7 + 11-2) = -2.35727 + * 1.28403. [48] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv, CONTINUED = i-7S x = 1.8 = 1.85 * = 1.9 x = 1.95 o 2.79041 o.oo 2.94217 o.oo 3.10129 o.oo 3.26816 o.oo 3.44321 o.oo 0.05 2.78181 0.23257 2.93310 0.24381 3-09173 0.25566 3.25809 0.26815 3.43259 0.28131 O.I 2.75606 0.46370 2-90595 0.48612 3.06311 0.50975 3.22793 0.53465 3.40081 0.56089 0.15 2-71331 0.69198 2.86088 0.72542 3.01560 0.76069 3.17787 0.79785 3.34807 0.83701 O.2 2.65384 0.91599 2.79817 0.96026 2.94950 1.00694 3.10821 1.05614 3.27468 1.10797 0.25 2.57800 1-13435 2.71821 1.18918 2.86522 1.24698 3-01939 1.30791 3.18111 1.37210 0-3 2.48627 1-34571 2.62149 1.41076 2.76327 1-47934 2.91196 1.55162 3.06792 1.62777 -3S 2.37922 1.54878 2.50862 1.62365 2.64429 1.70258 2.78657 1.78576 2.93582 1.87341 0.4 2.25749 1.74231 2.38027 1.82653 2.50900 I-9I53I 2.64400 2.00889 2.78561 2.10749 0-45 2.12185 1.92509 2.23725 2.01814 2-35824 2.11624 2.48513 2.21964 2.61823 2.32858 o-5 1.97312 2.09600 2.08043 2.19731 2.19294 2.30413 2.31094 2.41670 2.43471 2.53532 o-5S 1.81223 2.25399 1.91079 2.36294 2.01413 2.47780 2.12250 2.59887 2.23618 2.72642 0.6 1.64016 2.39808 1.72937 2.51400 1.82289 2.63620 1.92098 2.76501 2.02387 2.90071 0.65 1-45799 2-52739 1-53728 2.64956 1.62042 2.77835 1.70761 2.91409 1.79907 3-05713 0.7 1.26682 2.64111 1-33572 2.76878 1.40796 2.90337 1.48372 3.04522 1.56318 3.19469 0-75 1.06784 2.73855 1.12592 2.87093 1. 18681 3.01049 1.25067 3.15757 1.31766 3.31255 0.8 0.86229 2.81911 0.90918 2.95538 0-95835 3.09904 1.00992 3-25045 1.06401 3.41000 0.85 0.65141 2.88229 0.68684 3.02161 0.72398 3.16850 0.76294 3.32330 0.80380 3.48641 o.g 0.43652 2.92769 0.46026 3.06921 0.48627 3.21841 0.51125 3.37565 0.53864 3-54134 o-95 0.21893 2-9555 0.23084 3.09789 0.24332 3.24848 0.25642 3.40719 0.27015 3-57443 I.O o.oo 2.96419 o.oo 3.10747 o.oo 3-25853 o.oo 3.41773 o.oo 3-58548 1.05 0.21893 2.95505 0.23084 3.09789 0.24332 3.24848 0.25642 3.40719 0.27015 3-57443 i.i 0.43652 2.92769 0.46026 3.06921 0.48627 3.21841 0.51125 3.37565 0.53864 3-54I34 i-i5 0.65141 2.88229 0.68684 3.02161 0.72398 3.16850 0.76294 3-32330 0.80380 3-48641 1.2 0.86229 2.81911 0.90918 2.95538 0.95835 3.09904 1.00992 3-25045 1.06401 3.41000 1-25 1.06784 2-73855 1.12592 2.87093 1. 18681 3.01049 1.25067 3.15757 1.31766 3-3i 2 55 i-3 1.26682 2.64111 1-33572 2.76878 1.40796 2.90337 1.48372 3.04522 1.56318 3.19469 i-35 1-45799 2.52739 1.53728 2.64956 1.62042 2.77835 1.70761 2.91409 1.79907 3-05713 1.4 1.64016 2.39808 1.72937 2.51400 1.82289 2.63620 1.92098 2.76501 2.02387 2.90071 1-45 1.81223 2-25399 1.91079 2.36294 2.01413 2.47780 2.12250 2.59887 2.23618 2.72642 i-5 1.97312 2.09600 2.08043 2.19731 2.19294 2.30413 2.31094 2.41670 2.43471 2.53532 i-SS 2.12185 1.92509 2.23725 2.01814 2.35824 2.11624 2.48513 2.21964 2.61823 2.32858 1.6 2.25749 1-74231 2.38027 1.82653 2.50900 I-9I53I 2.64400 2.00889 2.78561 2.10749 1.65 2.37922 1.54878 2.50862 1.62365 2.64429 1.70258 2.78657 1.78576 2.93582 1.87341 i-7 2.48627" I-3457I 2.62149 1.41076 2.76327 1-47934 2.91196 1.55162 3.06792 1.62777 i-TS 2.57800 I-I3435 2.71821 1.18918 2.86522 1.24698 3.01939 1.30791 3.18111 1.37210 1.8 2.65384 0.91599 2.79817 0.96026 2.94950 i .00694 3.10821 1.05614 3.27468 1.10797 1.85 2-71331 0.69198 2.86088 0.72542 3.01560 0.76069 3.17787 0.79785 3.34807 0.83701 1.9 2.75606 0.46370 2.90595 0.48612 3.06311 0.50975 3-22793 0.53465 3.40081 0.56089 i-95 2.78181 0.23257 2.93310 0.24381 3-09173 0.25566 3.25809 0.26815 3.43259 0.28131 2.O 2.79041 o.oo 2.94217 o.oo 3.10129 o.oo 3.26816 o.oo 3.44321 0.00 Note. Negative quantities are in heavy type. Examples. sinh (1.85 + *' Q-75) s= I. l868l + 3.01049. sinh (1.85 + i 1.35) = -1.62042 + * 2.77835. [49] TABLE VII. HYPERBOLIC SINES, sinh (* + ig) = u + iv. CONTINUED x = 2.05 x = 2.15 o 0.05 O.I 0.15 0.2 3.62686 o.oo 3.61568 0.29518 3.58221 0.58854 3.52666 0.87827 3.44935 1.16258 3.81958 o.oo 3.80781 0.30978 3.77256 0.61765 3.71404 0.92172 3.63264 I.220IO 4.02186 o.oo 4.00946 0.32516 3.97235 0.64831 3.91074 0.96747 3.82501 1.28066 4.23419 o.oo 4.22113 0.34135 4.18206 0.68060 4.11720 1.01564 4.02695 1.34443 4.45711 o.oo 4-44337 0.35839 4.40223 0.71458 4-33396 1.06636 4.23896 1.41156 0.25 0-3 o-3S 0.4 0-45 3.35078 1.43973 3.23156 1.70800 3.09241 1.96574 2.93420 2.21136 2.75789 2.44335 3.52883 1.51096 3.40327 1.79250 3.23673 2.06299 3.09OII 2.32076 2.90443 2.56423 3.71571 1.58596 3.58351 1.88148 3.42920 2.16540 3.25376 2.43597 3.05825 2.69152 3.91188 1.66493 3.77269 1.97516 3.61024 2.27322 3.42553 2.55726 3.21970 2.82553 4.11783 1.74806 3.97131 2.07379 3.80031 2.38672 3.60588 2.68495 3.38921 2.96661 o-S o-SS 0.6 0.65 0.7 2.56458 2.66027 2.35546 2.86080 2.13182 3.04368 1-89503 3-20780 1.64656 3-35214 2.70085 2.79188 2.48062 3.00232 2.24509 3.19426 1-99573 3-36649 I.73405 3-5I798 2.84389 2.93048 2.61199 3-15137 2.36399 3-35283 2.10142 3.53361 1.82589 3.69261 2.99402 3-07639 2.74988 3.30828 2.48879 3-51977 2.21236 3.70956 1.92228 3.87647 3.15165 3.23000 2.89466 3-47347 2.61982 3.69552 2.32883 3-89478 2.02349 4.07003 0-75 0.8 0.85 0.9 0-95 1-38794 3.4758i 1.12076 3.57806 0.84667 3.65825 0.56737 3-7I587 0.28456 3-75059 1.46169 3.64777 1.18032 3.75507 0.89166 3.83923 0.59751 3-8997I 0.29968 3.93615 1.53910 3.82885 1.24282 3.94148 0.93888 4.02981 0.62916 4.09329 0.31555 4-13154 1.62035 4-01950 1.30844 4-13773 0.98845 4.23046 0.66237 4-29711 0.33221 4.33726 1.70566 4.22019 1-37732 4-34433 1.04049 4.44170 0.69724 4.51167 0.34970 4-55382 I.O 1.05 i.i i.iS 1.2 o.oo 3.76220 0.28456 3.75059 0.56737 3-71587 0.84667 3.65825 1.12076 3.57806 o.oo 3-94832 0.29968 3.93615 0-59751 3-8997I 0.89166 3.83923 1.18032 3.75507 o.oo 4.14431 0.31555 4-13154 0.62916 4.09329 0.93888 4.02981 1.24282 3.94148 o.oo 4-35067 0.33221 4.33726 0.66237 4-29711 0.98845 4.23046 1.30844 4.13773 o.oo 4.56791 0.34970 4.55382 0.69724 4.51167 1.04049 4.44170 1-37732 4-34433 I-2S i-3 i -35 1.4 1-45 1.38794 3-47581 1.64656 3.35214 1.89503 3.20780 2.13182 3.04368 2.35546 2.86080 1.46169 3.64777 1-73405 3-5I798 1-99573 3-36649 2.24509 3.19426 2.48062 3.00232 1.53910 3.82885 1.82589 3.69261 2.10142 3.53361 2.36399 3-35283 2.61199 3.15137 1.62035 4.01950 1.92228 3.87647 2.21236 3-70956 2.48879 3-51977 2.74988 3.30828 1.70566 4.22019 2.02349 4-07003 2.32883 3-89478 2.61982 3.69552 2.89466 3-47347 i-5 i-SS 1.6 i.6 S i-7 2.56458 2.66027 2.75789 2.44335 2.93420 2.21136 3.09241 1.96574 3.23156 1.70800 2.70085 2.79188 2.90443 2.56423 3.09011 2.32076 3-23673 2.06299 3.40327 1.79250 2.84389 2.93048 3.05825 2.69152 3.25376 2.43597 3.42920 2.16540 3.58351 1.88148 2.99402 3.07639 3.21970 2.82553 3-42553 2.55726 3.61024 2.27322 3.77269 1.97516 3.15165 3.23000 3.38921 2.96661 3.60588 2.68495 3.80031 2.38672 3.97131 2.07379 1-75 1.8 1.85 1.9 i-9S 3.35078 1.43973 3-44935 1.16258 3.52666 0.87827 3.58221 0.58854 3.61568 0.29518 3.52883 1.51096 3.63264 I.22OIO 3.71404 0.92172 3.77256 0.61765 3.80781 0.30978 3.71571 1.58596 3.82501 1.28066 3.91074 0.96747 3.97235 0.64831 4.00946 0.32516 3.91188 1.66493 4.02695 1.34443 4.11720 1.01564 4.18206 0.68060 4.22113 0.34135 4.11783 1.74806 4.23896 1.41156 4.33396 1.06636 4.40223 0.71458 4-44337 0.35839 2.0 3.62686 o.oo 3.81958 o.oo 4.02186 o.oo 4.23419 o.oo 4.45711 o.oo Note. Negative quantities are in heavy type. Examples, sinh (2.2 + *' i-o) =0 + * 4.56791. sinh (2.2 + i 1.5) = -3.15165 + i 3.23000. [So] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED q X = 2.2$ x = 2.3 a: = 2.35 x = 24 x = 245 0.05 O.I 0.15 O.2 4.69117 o.oo 4.67671 0.37633 4.63341 0.75035 4.56155 I.II974 4.46157 1.48222 4.93696 o.oo 4.92174 0.39522 4.87618 0.78799 4.80056 1.17592 4.69533 1.55659 5.19510 o.oo 5.17909 0.41509 5.13114 0.82761 5.05156 1.23504 4.94083 1.63485 5.46623 o.oo 5.44938 0.43599 5.39893 0.86930 5.31521 1.29724 5.19869 1.71719 5.75103 o.oo 5.73330 0.45799 5.68022 0.91316 5.59213 1.36269 5.46955 1.80383 0.25 0.03 o-3S 0.4 o-45 4.33407 1.83557 4.17986 2.I776O 3.99987 2.50620 3.79523 2.81935 3.56719 3.II5I2 4.56116 1.92766 4.39887 2.28685 4.20946 2.63194 3.99409 2.96081 3.75410 3-27141 4.79965 2.02457 4.62887 2.40182 4.42955 2.76426 4.20292 3.10966 3-95038 343588 5.05014 2.12655 4.87045 2.52280 4.66073 2.90349 4.42228 3.26629 4.15656 3.60894 5.31325 2.23385 5.12420 2.65009 4-90356 3-04999 4.65268 3-43 I0 9 4-373" 3-79I04 o-5 0-55 0.6 0.65 0.7 3.31716 3.39168 3.04677 3.64734 2-7574 3.88050 2.45II3 4.08975 2.12975 4.27377 3.49096 3-56185 3.20630 3.83034 2.90188 4.07520 2.57956 4-29494 2.24134 4.48820 3.6735I 3-74093 3-37395 4-02290 3.05360 4.28008 2.71443 4.51087 2.35853 4-71384 3.86521 3.92935 3.55003 4.22554 3.21297 4-49567 2.85610 4.73808 2.48162 4.95127 4.06659 4.12761 3-73499 4-43873 3.38036 4-72249 3.00490 4.97713 2.61091 5.20109 0-75 0.8 0.85 0.9 0-9S 1.79523 4.43145 1.44965 4.56181 1.09513 4.66404 0.73386 4-7375 1 0.36807 4.78178 1.88930 4-65378 1.52560 4-79058 1.15251 4.89805 0.77231 4-97521 0.38735 5-02169 1.98808 4.88776 1-60537 5-03I53 1.21277 5.14429 0.81269 5.22533 0.40760 5.27416 2.09184 5-13394 1.68916 5.28496 1.27607 5.40341 0.85511 5.48853 0.42888 5-53981 2.20082 5.39298 i.777i6 5-55162 I-34255 5-67603 0.89966 5.76545 0.45122 5-8I933 I.O 1.05 i.i i.iS 1.2 o.oo 4.79657 0.36807 4.78178 0.73386 4.73751 1.09513 4.66404 1.44965 4.56181 0.00 5-03722 0.38735 5-02169 0.77231 4.97521 1.15251 4.89805 1.52560 4-79058 o.oo 5.29047 0.40760 5.27416 0.81269 5.22533 1.21277 5-14429 1-60537 5.03153 o.oo 5-55695 0.42888 5.53981 0.85511 5.48853 1.27607 5.40341 1.68916 5.28496 o.oo 5-83732 0.45122 5.81933 0.89966 5.76545 I.34255 5-67603 1.77716 5.55162 1.25 i-3 i-3S 1.4 1-45 1.79523 443145 2.12975 4.27377 2.45113 4.08975 2-75740 3-88050 3.04677 3.64734 1.88930 4-65378 2.24134 4.48820 2.57956 4-29494 2.90188 4.07520 3.20630 3.83034 1.98808 4.88776 2.35853 4-71384 2.71443 4.51087 3.05360 4.28008 3-37395 4-02290 2.09184 5.13394 2.48162 4,95127 2.85610 4.73808 3.21297 4.49567 3.55003 4.22554 2.20082 5.39298 2.61091 5.20109 3.00490 4-977I3 3.38036 4.72249 3-73499 443873 i-5 i-SS 1.6 1.65 i-7 3.31716 3.39168 3-56719 3- I][ 5i2 3.79523 2.81935 3.99987 2.50620 4.17986 2.17760 3.49096 3.56185 3.75410 3.27141 3.99409 2.96081 4.20946 2.63194 4.39887 2.28685 3-6735I 3.74093 3.95038 3.43588 4.20292 3.10966 4-42955 2.76426 4.62887 2.40182 3.86521 3.92935 4.15656 3-60894 4.42228 3.26629 4.66073 2.90349 4.87045 2.52280 4x16659 4.12761 4-373II 3.79104 4.65268 3.43109 4.90356 3-04999 5.12420 2.65009 1-75 1.8 1.85 1.9 1-95 4.33407 1.83557 4.46157 1.48222 4-56i55 I-II974 4.63341 0.75035 4.67671 0.37633 4.56116 1.92766 4-69533 1-55659 4.80056 1.17592 4.87618 0.78799 4.92174 0.39522 4.79965 2.02457 4.94083 1.63485 5.05156 1.23504 5.13114 0.82761 5.17909 0.41509 5.05014 2.12655 5.19869 1.71719 5.31521 1.29724 5.39893 0.86930 5.44938 0.43599 5.31325 2.23385 5.46955 1-80383 5.59213 1.36269 5.68022 0.91316 5-73330 0.45799 2.0 4.69117 o.oo 4.93696 o.oo 5.19510 o.oo 5.46623 o.oo 5.75103 o.oo Note. Negative quantities are in heavy type. Examples, sinh (2.4 + i 0.05) = 5.4493 s + i 0.43599. sinh (2.4 + i 1.95) = -5.44938 + * 0.43599. CsxJ TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 2.5 = 2.55 x = 2.6 x = 2.65 = 2.7 o 0.05 O.I 0.15 O.2 6.05020 o.oo 6.03155 0.48113 5.97572 0.95930 5.88304 1.43155 5.75408 1.89498 6.36451 o.oo 6.34489 0.50548 6.28615 1.00784 6.18866 1.50399 6.05301 1.99087 6.69473 o.oo 6.67409 0.53109 6.61231 1.05891 6.50976 1.58019 6.36706 2.09174 7.04169 o.oo 7.01998 0.55803 6.95500 1.11262 6.84713 1.66034 6.69705 2.19784 7.40626 o.oo 7-38343 0.58636 7.31508 1.16911 7.20163 1.74465 7.04377 2.30943 0.25 o-3 o-35 0.4 o-45 5.58966 2.34673 5.39077 2.78401 5-15865 3.20411 4.89472 3.60448 4.60062 3.98260 5.88004 2.46547 5.67082 2.92488 5.42664 3.36624 5.14900 3.78686 4.83961 4.18413 6.18512 2.59039 5-96505 3-07306 5.70819 3.53680 5.41615 3-97872 5.09071 4-39612 6.50567 2.72178 6.27419 3.22894 6.00403 3.71619 5.69685 4-18053 5-35454 4-61910 6.84249 2.85997 6.59903 3-39288 6.31488 3.90488 5.99179 4-39279 5.63177 4.85363 o-S o-55 0.6 0.65 0.7 4.27814 4-33619 3.92929 4.66304 3.55622 4-96113 3.16122 5.22864 2.74674 5.46392 4.50039 4.55560 4.13342 4.89898 3.74097 5.21217 3-32545 5-49321 2.88943 5-74039 4-73389 4-78641 4.34788 5-14719 3-93506 5.47624 3-49799 5-77I53 3.03934 6.03123 4.97923 5.02919 4.57321 5.40827 4.13900 5-75401 3.67927 6.06427 3.19686 6.33714 5.23702 5-28454 4.80998 5.68287 4-35329 6.04606 3.86976 6.37218 3.36237 6.65891 0-75 0.8 0.85 0.9 o-9S 2.31531 5.66550 1.86961 5.83215 1.41239 5.96287 0.94646 6.05680 0.47469 6.11339 2-43559 5-952i8 1.96674 6.12727 1.48577 6.26458 0.99563 6.36327 0.49935 6.42273 2.56196 6.25374 2.06878 6.43770 1.56285 6.58199 1.04729 6.68567 0.52526 6.74814 2.69474 6.57095 2.17600 6.76424 1.64385 6.91583 1.10156 7.02478 0.55249 7.09042 2.83425 6.90458 2.28866 7.10769 1.72896 7.26697 1-15859 7.38146 0.58109 7-45043 I.O 1.05 .1 15 .2 o.oo 6.13229 0.47469 6.11339 0.94646 6.05680 1.41239 5.96287 1.86961 5.83215 o.oo 6.44259 0.49035 6.42273 0.99563 6.36327 1.48577 6.26458 1.96674 6.12727 o.oo 6.76901 0.52526 6.74814 1.04729 6.68567 1.56285 6.58199 2.06878 6.43770 o.oo 7.11234 0.55249 7-09042 1.10156 7.02478 1.64385 6.91583 2.17600 6.76424 o.oo 7-47347 0.58109 7.45043 1-15859 7-38146 1.72896 7.26697 2.28866 7.10769 25 3 35 4 45 2.31531 5-66550 2.74674 5.46392 3.16122 5.22864 3.55622 4.96113 3.92929 4.66304 2.43559 5-95218 2.88943 5.74039 3.32545 5-49321 3.74097 5-21217 4.13342 4.89898 2.56196 6.25374 3.03934 6.03123 3.49799 5-77153 3.93506 5-47624 4.34788 5.14719 2.69474 6.57095 3.19686 6.33714 3.67927 6.06427 4.13900 5.75401 4.57321 5-40827 2.83425 6.90458 3.36237 6.65891 3.86976 6.37218 4-35329 6.04606 4.80998 5.68287 i-5 i-SS 1.6 1.65 i-7 4.27814 4-33619 4.60062 3.98260 4.89472 3.60448 5-15865 3-20411 5.39077 2.78401 4-50039 4.55560 4.83961 4-18413 5.14900 3.78686 5.42664 3.36624 5.67082 2.92488 '4-73389 4-78641 5.09071 4-39612 5.41615 3.97872 5.70819 3.53680 5-96505 3-07306 4.97923 5-02919 5-35454 4-61910 5.69685 4-18053 6.00403 3.71619 6.27419 3.22894 5.23702 5.28454 5.63177 4.85363 5.99179 4-39279 6.31488 3.90488 6.59903 3-39288 i-75 1.8 1.85 1.9 1-95 5.58966 2.34673 5.75408 1.89498 5.88304 1.43155 5.97572 0.95930 6.03155 0.48113 5.88004 2.46547 6.05301 1.99087 6.18866 J. 50399 6.28615 1.00784 6.34489 0.50548 6.18512 2.59039 6.36706 2.09174 6.50976 1.58019 6.61231 1.05891 6.67409 0.53109 6.50567 2.72178 6.69705 2.19784 6.84713 1.66034 6.95500 1.11262 7.01998 0.55803 6.84249 2.85997 7.04377 2.30943 7.20163 1.74465 7.31508 1.16911 7-38343 0.58636 2.O 6.05020 o.oo 6.36451 o.oo 6.69473 o.oo 7.04169 o.oo 7.40626 o.oo Note. Negative quantities are in heavy type. Examples, sinh (2.7 + *Q-7) = 3.36237 + i 6.65891. sinh (2.5 + i 1.25) = 2.31531 + i 5.66550. [52] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 2.75 x = 2.85 x 2.9 x = 2.95 o 0.05 O.I 0.15 0.2 7-78935 o.oo 7.76534 0.61616 7.69345 1.22852 7.57413 1.83331 7.40811 2.42680 8.19192 o.oo 8.16666 0.64750 8.09106 1.29101 7.96557 1.92656 7.79098 2.55023 8.61497 o.oo 8.58841 0.68046 8.50891 1.35673 8.37694 2.02463 8.19332 2.68005 9.05956 o.oo 9.03163 0.71512 8.94802 1.42583 8.80924 2.12776 8.61616 2.81656 9.52681 o.oo 9.49744 0.75157 9.40952 1.49851 9.26358 2.23621 9.06053 2.96012 0.25 o-3 0.35 0.4 0-45 7.19642 3.00532 6.94036 3-56531 6.64151 4-10333 6.30172 4.61604 5.92307 5.10030 7.56834 3.15818 7.29905 3.74666 6.98476 4.31204 6.62740 4.85083 6.22918 5.35972 7-959 I 9 3-31894 7.67599 3-93738 7-34547 4-53I53 6.96966 5.09775 6.55088 5-63254 8.36994 3.48800 8.07213 4-13793 7.72455 4-76236 7.32934 5-35742 6.88894 6.91945 8.80162 3.66578 8.48845 4-34884 8.12294 5.00509 7-70735 5.63049 7.24424 6.22116 o-S 0-55 0.6 0.65 o-7 5.50790 5-553" 5.05878 5.97168 4.57847 6.35344 4.06993 6.69602 3.53629 6.99732 5.79256 5-83556 5.32022 6.27542 4.81509 6.67660 4.28026 7.03661 3.71905 7.35323 6.09170 6.13261 5.59498 6.59486 5-06375 7.01646 4.50131 7-39479 3.91112 7.72754 6.40608 6.44498 5.88371 6.93078 5.32508 7-37385 4.73361 7.77146 4.11295 8.12115 6.73647 6.77348 6.18717 7.28404 5.63048 7.74969 4-97774 8.16757 4.32508 8.53508 0.75 0.8 0.85 0.9 o-95 2.98086 7.25548 2.40704 7.46891 1.81839 7.63629 1.21852 7-75659 0.61115 7.82907 3.13491 7-62452 2.53144 7.84881 1.91237 8.02470 1.28150 8.15112 0.64273 8.22728 3.29681 8.01263 2.66217 8.24834 2.OIII2 8.43318 1.34768 8.56604 0.67592 8.64608 3.46694 8.42078 2.79956 8.66850 2.11491 8.86275 1.41723 9.00237 0.71081 9.08649 3.64575 8.84998 2-94395 9.11031 2.22399 9-3I447 1.49032 9.46121 0-74747 9-54962 I.O 1.05 i.i 1.2 o.oo 7.85328 0.61115 7.82907 1.21852 7.75659 1.81839 7.63629 2.40704 7.46891 o.oo 8.25273 0.64273 8.22728 1.28150 8.15112 1.91237 8.02470 2.53144 7.84881 o.oo 8.67281 O.67592 8.64608 1.34768 8.56604 2.0III2 8.43318 2.66217 8.24834 o.oo 9.11458 0.71081 9.08649 1.41723 9.00237 2.11491 8.86275 2.79956 8.66850 0.00 9-579I5 0.74747 9-54962 1.49032 9.46121 2.22399 9-3I447 2.94395 9.11031 1-25 1-35 i-4 i-45 2.98086 7-25548 3.53629 6.99732 4.06993 6.69602 4.57847 6.35344 5.05878 5.97168 3.13491 7-62452 3-71905 7-35323 4.28026 7.03661 4.81509 6.67660 5.32022 6.27542 3.29681 8.01263 3.9III2 7.72754 4.50131 7-39479 5-06375 7.01646 5.59498 6.59486 3.46694 8.42078 4.11295 8.12115 4.73361 7.77146 5.32508 7-37385 5.88371 6.93078 3.64575 8.84998 4.32508 8.53508 4-97774 8.16757 5.63048 7.74969 6.18717 7.28404 i-55 1.6 1.65 5-50790 5-553" 5.92307 5.10030 6.30172 4.61604 6.64151 4-10333 6.94036 3.56531 5.79256 5-83556 6.22918 5-35972 6.62740 4.85083 6.98476 4.31204 7.29905 3.74666 6.09170 6.13261 6.55088 5.63254 6.96966 5.09775 7-34547 4-53I53 7.67599 3-93738 6.40608 6.44498 6.88894 5.91945 7.32934 5-35742 7.72455 4.76236 8.07213 4.13793 6.73647 6.77348 7.24424 6.22116 7-70735 5-63049 8.12294 5.00509 8.48845 4.34884 1-75 1.8 1.85 1.9 1-95 7.19642 3.00532 7.40811 2.42680 7.57413 1.83331 7-69345 1.22852 7-76534 0.61616 7.56834 3-15818 7.79098 2.55023 7-96557 1.92656 8.09106 1.29101 8.16666 0.64750 7.959I9 3-31894 8.19332 2.68005 8.37694 2.02463 8.50891 1.35673 8.58841 0.68046 8.36994 3.48800 8.61616 2.81656 8.80924 2.12776 8.94802 1.42583 9.03163 0.71512 8.80162 3.66578 9.06053 2.96012 9.26358 2.23621 9.40952 1.49851 9.49744 0.75157 2.0 7.78935 o.oo 8.19192 o.oo 8.61497 o.oo 9.05956 o.oo 9.52681 o.oo Note. Negative quantities are in heavy type. Examples, sinh (2.9 + i Q-9) = 1-41723 + i 9.00237. sinh (2.8 + i 1.4) = 4.81509 + * 6.67660. [53] TABLE VII. HYPERBOLIC SINES, sinh (* + iq) = u + iv. CONTINUED x = 3.0 x - 3.05 x = 3.10 = 3-iS x = 3.20 o.o 10.01787 o.oo 10.53399 o.oo 11.07645 o.oo 11.64661 o.oo 12.24588 o.oo 0.05 9.98699 0.78990 10.50150 0.83020 11.04230 0.87258 11.61070 0.91714 12.20810 0.96399 O.I 9.89454 1-57493 10.40430 1.65529 10.94010 1-73979 11.50320 1.82863 12.09510 1.92205 0.15 9.74108 2.35025 10.24290 2.47017 10.77040 2.59626 11.32480 2.72885 11.90750 2.86826 0.2 9-52757 3.11108 10.01840 3.26982 10.53430 3.43673 11.07660 3.61224 11.64650 3.79678 0.25 9-2553I 3-85273 9.73216 4.04931 10.23330 4.25602 10.76010 4-47336 11.31370 4.70190 o-3 8.92599 4-57062 9.38586 4.80383 9.86919 5.04906 10.37720 5.30690 10.91120 5-57802 0-35 8.54164 5.26034 8.98171 5-52874 9-44423 5.81097 9.93036 6.10773 10.44130 6.41976 0.4 8.10463 5.91762 8.52218 6.21956 8.96104 6-537Q6 9.42230 6.87089 9.90712 7.22191 0-45 7.61765 6.53842 8.01011 6.87204 8.42260 7.22284 8.85615 7.59170 9.31184 7-97954 o-5 7.08371 7.11891 7.44866 7.48215 7.83223 7.86409 8.23539 8.26570 8.65915 8.68797 o.SS 6.50609 7-6555I 6.84128 8.04612 7-19358 8.45686 7-56387 8.88873 7-95306 9.34284 0.6 5.88836 8.14491 6.19173 8.56049 6.51058 8.99749 6.84570 9-45697 7-19795 9.94011 0.65 5-23433 8.58409 5.50400 9.02209 5.78743 9.48264 6.08533 9.96690 6.39846 10.47610 0.7 4.54806 8.97035 4-78233 9.42805 5.02860 9-90933 5-28745 10.41540 5-55952 10.94750 0-75 3-83368 9.30131 4.03119 9-77589 4.23878 10.27490 4.45696 10.79970 4.68630 11.35140 0.8 3-09569 9-57492 3-255i8 10.06350 3.42281 10.57720 3-59900 11.11730 3.78419 11.68530 0.85 2 -33 86 3 9.78949 2.45911 10.28900 2-58575 10.81420 2.71885 11.36650 2.85874 11.94720 o.g 1.56714 9-94371 1.64788 10.45110 1.73274 10.98460 1.82193 11-54550 1.91568 12.13540 o-95 0.78599 10.03660 0.82649 10.54870 0.86905 11.08720 0.91378 11.65340 0.96080 12.24880 I.O o.oo 10.06766 o.oo 10.58135 o.oo 11.12150 0.00 11.68946 o.oo 12.28665 1.05 0.78599 10.03660 0.82649 10.54870 0.86905 11.08720 0.91378 11.65340 , 0.96080 12.24880 i.i 1.56714 9-94371 1.64788 10.45110 1.73274 10.98460 1.82193 11-54550 1.91568 12.13540 IS 2.33863 9.78949 2.45911 10.28900 2.58575 10.81420 2.71885 11.36650 2.85874 11.94720 .2 3-09569 9-57492 3.25518 10.06350 3.42281 10.57720 3-59900 11.11730 3.78419 11.68530 25 3.83368 9.30131 4.03119 9.77589 4.23878 10.27490 4.45696 10.79970 4.68630 11.35140 3 4.54806 8.97035 4.78233 9.42805 5.02860 9-90933 5.28745 10.41540 5.55952 10.94750 35 5-23433 8.58409 5.50400 9.02209 5.78743 9.48264 6.08533 9.96690 6.39846 10.47610 4 5.88836 8.14491 6.19173 8.56049 6.51058 8.99749 6.84570 9-45697 7.19795 9.94011 45 6.50609 7-6555I 6.84128 8.04612 7.19358 8.45686 7-56387 8.88873 7.95306 9.34284 5 7.08371 7.11891 7.44866 7.48215 7.83223 7.86409 8.23539 8.26570 8.65915 8.68797 55 7.61765 6.53842 8.OIOH 6.87204 8.42260 7.22284 8.85615 7.59170 9.31184 7-97954 .6 8.10463 5.91762 8.52218 6.21956 8.96104 6.53705 9.42230 6.87089 9.90712 7.22191 .65 8.54164 5.26034 8.98171 5-52874 9.44423 5.81097 9.93036 6.10773 10.44130 6.41976 7 8.92599 4-57062 9.38586 4.80383 9.86919 5.04906 10.37720 5.30690 10.91120 5.57802 i-75 9-25531 3-85273 9-73216 4.04931 10.23330 4.25602 10.76010 4-47336 11.31370 4.70190 1.8 9.52757 3.11108 10.01840 3.26982 10.53430 3-43673 11.07660 3.61224 11.64650 3.79678 i.8 S 9.74108 2.35025 10.24290 2.47017 10.77040 2.59626 11.32480 2.72885 11.90750 2.86826 1.9 9.89454 1 -5 7493 10.40430 1.65529 10.94010 1-73979 11.50320 1.82863 12.09510 1.92205 i-95 9.98699 0.78990 10.50150 0.83020 11.04230 0.87258 11.61070 0.91714 12.20810 0.96399 2.0 10.01787 o.oo 10.53399 o.oo 11.07645 o.oo 11.64661 o.oo 12.24588 0.00 Note. Negative quantities are in heavy type. Examples. sinh (3.0 + i 0.95) = 0.78599 + i 10.03660. sinh (3.0 + * 1.05) = 0.78599 + i 10.03660. [54] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 3- 2 5 x = 3.30 = 3-35 x = 3-4 x = 3-45 o.o 12.87578 O.OO 13-53788 o.oo 14.23382 o.oo 14.96536 o.oo I5-73432 o.oo 0.05 12.83610 1.01326 13.49615 1.06507 14.18950 1-11953 14.91923 1.17679 15.68581 1.23699 O.I 12.71726 2.02028 13.37120 2.12356 14.05851 2.23215 14.78111 2.34632 15.54061 2.46636 0.15 12.52002 3.01484 13.16388 3.16897 13.84054 3-33101 14.55188 3-50139 I5-29959 3.68053 O.2 12.24560 3.99082 12.87530 4.19483 I3-537I7 4-40933 14.23291 4.63487 14.96423 4.87198 0.25 11.89566 4.94218 12.50736 5-19485 i3- I 5Q33 5-46075 13.82620 5-73977 14.53662 6.03341 o-3 11.47240 5.86309 12.06234 6.16381 12.68244 6-47795 13-33423 6.80928 14.01940 7-15765 0-35 10.97841 6.74784 11.54294 7.09279 12.13633 7-45549 12.76006 7.83682 I3-4I572 8.23775 0.4 10.41673 7-59099 10.95235 7-97905 H-5I54I 8-38705 12.10723 8.81604 12.72933 9.26706 0-45 9.79082 8.38733 10.29428 8.81610 10.82349 9.26691 n-37975 9.74090 11.96447 10.23924 o-5 9-10455 9.13197 9-57273 9.59881 10.06483 10.08964 10.58212 10.60570 11.12585 11.14830 0-55 8.36217 9.82031 8.79215 10.32233 9.24413 10.85016 9.71923 11.40512 10.21863 11.98861 0.6 7.56820 10.44810 7-95737 10.98222 8.36643 n-54379 8.79642 12.13423 9.24840 12.75501 0.65 6.72758 11.01147 7-07352 11.57440 7-437I5 12.16623 7-81938 12.78852 8.22116 13.44278 0.7 S-84548 11.50695 6.14670 12.09520 6.46202 12.71369 6.79414 I3-36397 7.14324 14.04764 0-75 4-9 2 735 11.93150 5.18072 12.54145 5-44705 13.18275 5.72700 13.85702 6.02127 14.56595 0.8 3-97883 12.28247 4-18343 12.91035 4-39850 I3-57054 4-62455 14.26465 4.86217 14.99442 0.85 3-00579 12.55773 3.16036 13.19970 3.32282 13.87465 3-49359 14.58432 3.67311 15-33045 0.9 2.01422 12.75556 2.11780 13.40764 '2.22666 14.09323 2.34110 14.81410 2.46139 15-57196 o-95 I.OIO22 12.87474 1.06217 13-5329 1.11677 14.22498 1.17417 14.95250 1.23450 15.71746 I.O o.oo 12.91456 o.oo I3-57476 0.00 14.26891 o.oo 14.99874 0.00 15.76607 1.05 I.OIO22 12.87474 1.06217 13-53290 1.11677 14.22498 1.17417 14.95250 1.23450 15.71746 i.i 2.01422 12.75556 2.11780 13.40764 2.22666 14.09323 2.34110 14.81410 2.46139 15-57196 i-iS 3-00579 12.55773 3.16036 13.19970 3-32282 13.87465 3-49359 14.58432 3.67311 15-33045 1.2 3-97883 12.28247 4.18343 12.91035 4-39850 I3-57054 4-62455 14.26465 4.86217 14.99442 1-25 4-92735 11.93150 5.18072 12.54145 5.44705 13.18275 5.72700 13.85702 6.02127 14.56595 !-3 5.84548 11.50695 6.14670 12.09520 6.46202 12.71369 6.79414 I3-36397 7.14324 14.04764 i-3S 6.72758 11.01147 7.07352 11.57440 7-43715 12.16623 7.81938 12.78852 8.22116 13.44278 1.4 7.56820 10.44810 7-95737 10.98222 8.36643 n-54379 8.79642 12.13423 9.24840 12.75501 i -45 8.36217 9.82031 8.79215 10.32233 9.24413 10.85016 9.71923 11.40512 10.21863 11.98861 i-S 9-10455 9.13197 9-57273 9.59881 10.06483 10.08964 10.58212 10.60570 11.12585 11.14830 i-SS 9.79082 8.38733 10.29428 8.81610 10.82349 9.26691 n-37975 9.74090 11.96447 10.23924 1.6 10.41673 7-59099 10.95235 7-97905 11-51541 8.38705 12.10723 8.81604 12.72933 9.26706 1.65 10.9784! 6.74784 11.54294 7.09279 12.13633 7-45549 12.76006 7.83682 13.41572 8.23775 i-7 11.47240 5.86309 12.06234 6.16381 12.68244 6-47795 I3-33423 6.80928 14.01940 7-15765 i-75 11.89566 4.94218 12.50736 5- I 9485 13-15033 5-46075 13.82620 5-73977 14.53662 6.03341 1.8 I2.2456O 3.99082 12.87530 4.19483 13-53717 4.40933 14.23291 4-63487 14.96423 4.87198 1.85 I2.52O02 3.01484 13.16388 3.16897 13.84054 3-33 101 14.55188 3-50I39 15.29959 3-68053 1.9 12.71726 2.02028 13.37120 2.12356 14.05851 2.23215 14.78111 2.34632 15.54061 2.46636 i-95 12.83610 1.01326 13.49615 1.06507 14.18950 I.II953 14.91923 1.17679 15.68581 1.23699 2.O 12.87578 O.OO 13.53788 o.oo 14.23382 o.oo 14.96536 o.oo 15.73432 o.oo Note. Examples, sinh (3.40 + * o) = 14.96536 + i o. sinh (3.45 + i 1.45) = 10.21863 + i 11.98861. -/ TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED x = 3-50 = 3-55 x = 3.60 = 3-65 x = 3.70 0.0 16.54263 o.oo 17.39230 o.oo 18.28546 o.oo 19.22434 0.00 20.21129 o.oo 0.05 16.49163 1.30029 17-33870 1.36684 18.22900 1.43680 19.16506 1.51036 20.14900 1.58770 O.I 16.33896 2.59256 17.17817 2.72525 18.06033 2.86475 18.98765 3.01142 19.96246 3.16561 0.15 16.08555 3-86885 16.91175 4.06686 17.78022 4-27503 18.69316 4.49390 19.65290 4.72401 0.2 15-733 5.12129 16.54105 5-38339 17.39050 5.65896 18.28342 5.94868 19.22208 6.25327 0.2S 15.28340 6.34215 16.06840 6.66674 16.89356 7.00800 17.76096 7.36678 18.67280 7-74399 o-3 14.73960 7-5239I 15.49665 7.90898 16.29246 8.31383 17.12900 8.73946 18.00840 9.18696 t>-35 14.10490 8.65928 14.82933 9.10246 15.59090 9.56840 16.39143 10.05827 17.23295 10.57330 0.4 13-3833 9.74126 14.07066 10.23982 14.79326 10.76400 15-55281 11.31505 16.35127 11.89444 o-4S 12.57910 10.76320 13.22520 11.31405 13.90436 11.89320 14.61830 12.50208 15.36878 13.14223 o-5 11.69740 11.71820 12.29820 12.31852 12.92978 12.94910 13-59365 13.61203 14.29155 14.30902 0-55 JO-74357 12.60210 11.29540 13.24705 11-87545 13-92515 12.48520 14.63806 13.12620 15.38760 0.6 9.72361 13.40770 10.22293 14.09390 10.74789 14.81535 11.29972 !5-57383 11.87990 16.37127 0.65 8.64360 14.13065 9.08745 14.85386 9-55412 15.61421 10.04469 16.41360 10.56037 17.25404 0.7 7.51020 14.76650 7-89594 15.52225 8.30143 16.31680 8.72766 17.15216 9-17573 18.03040 0-75 6.3359 i5-3"3o 6.65574 16.09492 6.99754 16.91880 7.35683 17.78497 7-73453 18.69565 0.8 S-n^S 15.76170 5-37451 16.56840 5-65052 17.41650 5.94065 18.30814 6.24563 19.24560 0.85 3.86180 16.11491 4.06015 16.93970 4.26865 17.80680 4.48783 18.71843 4.71823 19.67690 0.9 2.58783 16.36878 2.72075 17.20653 2.86047 18.08732 3-00735 19.01331 3.16174 19.98688 o.QS 1.29792 16.52173 1.36458 17.36731 1.43466 18.25632 1.50832 19.19100 I-58576 20.17362 I.O 0.00 16.57282 0.00 17.42102 o.oo 18.31278 o.oo 19-25033 o.oo 20.23601 1.05 1.29792 16.52173 1.36458 17.36731 1.43466 18.25632 1.50832 19.19100 1.58576 20.17362 i.i 2.58783 16.36878 2.72075 17.20653 2.86047 18.08732 3-00735 19.01331 3.16174 19.98688 i-iS 3.86180 16.11491 4.06015 16.93970 4.26865 17.80680 4.48783 18.71843 4.71823 19.67690 1.2 5.11195 15.76170 5.37451 16.56840 5.65052 17.41650 5.94065 18.30814 6.24563 19.24560 1-25 6.33059 15-3113 6.65574 16.09492 6.99754 16.91880 7.35683 17.78497 7-73453 18.69565 i-3 7.51020 14.76650 7.89594 15.52225 8.30143 16.31680 8.72766 17.15216 9-17573 18.03040 i-3S 8.64360 14.13065 9.08745 14.85386 9-55412 15.61421 10.04469 16.41360 10.56037 17.25404 1.4 9.72361 13.40770 10.22293 14.09390 10.74789 14.81535 11.29972 I5-57383 11.87990 16.37127 i. 45 10.74357 12.60210 11.29540 13.24705 11.87545 13.92515 12.48520 14.63806 13.12620 15.38760 i-5 11.69740 11.71820 12.29820 12.31852 12.92978 12.94910 13.59365 13.61203 14-29155 14.30902 i-55 12.57910 10.76320 13.22520 11.31405 13-90436 11.89320 14.61830 12.50208 15.36878 13.14223 1.6 13-38330 9.74126 14.07066 10.23982 14.79326 10.76400 15.55281 11-31505 16.35127 11.89444 i.6 S 14.10490 8.65928 14.82933 9.10246 15-59090 9.56840 16.39143 10.05827 17.23295 ^10.57330 i-7 14.73960 7-52391 15-49665 7.90898 16.29246 8.31383 17.12900 8.73946 18.00840 9.18696 i-75 15.28340 6-34215 16.06840 6.66674 16.89356 7.00800 17.76096 7.36678 18.67280 7-74399 1.8 15.73300 5.12129 16.54105 5-38339 17.39050 5.65896 18.28342 5.94868 19.22208 6.25327 1.85 16.08555 3-86885 16.91175 4.06686 17.78022 4.27503 18.69316 4.49390 19.65290 4.72401 1.9 16.33896 2.59256 17.17817 2.72525 18.06033 2.86475 18.98765 3.01142 19.96246 3.16561 i-95 16.49163 1.30029 17.33870 1.36684 18.22900 1.43680 19.16506 1.51036 20.14900 1.58770 2.O 16.54263 0.00 17.39230 o.oo 18.28546 0.00 19.22434 o.oo 20.21129 0.00 Note. Negative quantities are in heavy type. Examples, sinh (3.70 + i 0.5) = 14.29155 -4- i 14.30902 . - sinh (3.70 + i 1.5) = 14.29155 + * 14.30902. [56] TABLE VII. HYPERBOLIC SINES, sinh (x + iq) = u + iv. CONTINUED = 3-75 x = 3.80 * = 3.85 x = 3.90 = 3-95 o.o 21.24878 o.oo 22.33941 o.oo 23.48589 o.oo 24.69110 o.oo 25.95806 o.oo 0.05 21.18327 1.66900 22.27052 1.75448 23.41348 1.84435 24.61500 1.93883 25.87805 2.03816 O.I 20.98716 3.32772 22.06437 3-49815 23-19673 3.67733 24.38710 3-86571 25.63849 4-06375 0.15 20.66167 4.96592 21.72216 5.22025 22.83696 5.48764 24.00888 5-76875 25.24084 6.06429 O.2 20.20879 6.57350 21.24603 6.91017 22.33640 7.26411 23.48262 7.63623 24.68760 8.02744 0.25 19.63131 8.14055 20.63891 8.55748 21.69813 8.99580 22.81160 9.45662 23.98212 9.94109 o-3 18.93280 9.65741 19-90455 10.15203 20.92608 10.67203 21-99993 11.21871 23.12881 11.79366 -3S 18.11756 11.11473 19.04746 11.68400 20.02501 12.28246 21.05272 12.91164 22.13290 13-57315 0.4 17.19062 12.50353 18.07296 13-14392 19.00048 13.81716 I9-97556 14.52497 21.00052 15.26910 o.4S 16.15770 13.81524 16.98701 14.52281 17.85880 15.26668 18.77526 16.04874 19.73867 16.87098 o-5 15.02516 15.04177 15-79634 15.81216 16.60702 16.62208 17.45924 17-47355 18.35512 18.36873 0-55 13.79998 16.17556 14.50828 17.00402 15.25286 17.87500 16.03558 18.79065 16.85842 19.75331 0.6 12.48971 17.20963 13.13076 18.09105 13.80465 19.01770 14-51307 19.99190 15.25776 21.01610 0.65 11.10246 18.13760 11.67230 19.06655 12.27134 20.04315 12.90106 21.06988 13.56305 22.14931 0.7 9.64674 18.95373 10.14188 19.92448 10.66233 20.94503 11.20952 22.01797 11.78474 23.14597 0.75 8.13156 19.65301 8.54892 20.65958 8.98766 21.71778 9.44887 22.83030 9-93373 23.99991 0.8 6.56624 20.23113 6.90325 21.26731 7-25754 22.35664 7.62997 23.50188 8.02149 24.70590 0.85 4.96043 20.68452 5-21503 21.74391 5.48267 22.85766 .5.76402 24.02856 6.05979 25.25957 0.9 3-32404 21.01038 3-49465 22.08646 3.67400 23.21775 3.86254 24.40710 4.06074 25.65749 o-9S 1.66716 21.20670 1-75273 22.29283 1.84268 23.43470 1.93724 24.63516 2.03665 25.89724 I.O o.oo 21.27230 o.oo 22.36178 o.oo 23.50717 o.oo 24-71135 o.oo 25-97731 1.05 1.66716 21.20670 1.75273 22.29283 1.84268 23.43470 1.93724 24.63516 2.03665 25.89724 i.i 3-32404 21.01038 3.49465 22.08646 3.67400 23.21775 3.86254 24.40710 4.06074 25-65749 i-iS 4.96043 20.68452 5-21503 21.74391 5.48267 22.85766 5.76402 24.02856 6.05979 25.25957 1.2 6.56624 20.23113 6.90325 21.26731 7.25754 22.35664 7.62997 23.50188 8.02149 24.70590 1-25 8.13156 19.65301 8.54892 20.65958 8.98766 21.71778 9.44887 22.83030 9-93373 23.99991 i-3 9.64674 18.95373 10.14188 19.92448 10.66233 20.94503 11.20952 22.01797 11.78474 23.14597 1-35 11.10246 18.13760 11.67230 19.06655 12.27134 20.04315 12.90106 21.06988 13-56305 22.14931 1.4 12.48971 17.20963 13.13076 18.09105 13.80465 19.01770 14.51307 19.99190 15-25776 21.01610 1-45 13.79998 16.17556 14.50828 17.00402 15.25286 17.87500 16.03558 18.79065 16.85842 19.75331 i-5 15.02516 15.04177 15-79634 15.81216 16.60702 16.62208 17.45924 17-47355 18.35512 18.36873 1-55 16.15770 13.81524 16.98701 14.52281 17.85880 15.26668 18.77526 16.04874 19.73867 16.87098 1.6 17.19062 12.50353 18.07296 13-14392 19.00048 13.81716 19.97556 14.52497 21.00052 15.26910 1-65 18.11756 11.11473 19.04746 11.68400 20.02501 12.28246 21.05272 12.91164 22.13290 13.57315 i-7 18.93280 9.65741 19.90455 10.15203 20.92608 10.67203 21.99993 11.21871 23.12881 11.79366 1-75. 19.63131 8.14055 20.63891 8.55748 21.69813 8.99580 22.81160 9.45662 23.98212 9.94109 1.8 20.20879 6.57350 21.24603 6.91017 22.33640 7.26411 23.48262 7.63623 24.68760 8.02744 1.85 20.66167 4.96592 21.72216 5.22025 22.83696 5.48764 24.00888 5.76875 25.24084 6.06429 1.9 20.98716 3.32772 22.06437 3.49815 23-19673 3.67733 24.38710 3-86571 25.63849 4.06375 i -95 21.18327 1.66900 22.27052 1.75448 23.41348 1.84435 24.61500 1.93883 25.87805 2.03816 2.0 21.24878 o.oo 22.33941 o.oo 23.48589 0.00 24.69110 o.oo 25.95806 o.oo Note. Negative quantities are in heavy type. Examples, sinh (3.90 + fr'o-75) = 9.44887 + i 22.83030. sinh (3.95 + i 1.95) = -25.87805 + i 2.03816. [57] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv x = 0.05 x = 0.15 I.OOOO O.OO 1.00125 o.oo 1.00500 o.oo 1.01127 o.oo 1.02007 o.oo 0.05 0.99692 o.oo 0.99816 0.00392 1.00191 0.00786 1.00815 0.01181 1.01692 0.01580 O.I 0.98769 o.oo 0.98892 0.00783 0.99263 0.01567 0.99882 0.02355 1.00751 0.03150 0.15 0.97237 o.oo 0.97385 0.01168 0.97724 0.02338 0.98333 0.03515 0.99188 0.04700 O.2 0.95106 o.oo 0.95225 0.01546 0.95582 0.03095 0.96178 0.04653 0.97014 0.06222 0.2S 0.92388 o.oo 0.92503 0.01914 0.92850 0.03833 0.93429 0.05762 0.94242 0.07705 0-3 0.89101 o.oo 0.89208 0.02271 0.89547 0.04547 0.90105 0.06835 0.90889 0.09141 o-3S 0.85264 o.oo 0.85367 0.02614 0.85691 0.05234 0.86225 0.07867 0.86975 0.10520 0.4 0.80902 o.oo 0.81000 0.02940 0.81307 0.05888 0.81814 0.08850 0.82525 0.11834 0-45 0.76041 o.oo 0.76133 0.03249 0.76421 0.06505 0.76898 0.09778 0.77567 0.13076 o.S 0.70711 o.oo 0.70796 0.03537 0.71065 0.07083 0.71508 0.10646 0.72130 0.14237 o-SS 0.64945 o.oo 0.65023 0.03804 0.65270 0.07617 0.65677 0.11449 0.66248 0.15310 0.6 0.58779 o.oo 0.58850 0.04047 0.59027 0.08104 0.59441 0.12181 0.59958 0.16288 0.65 0.52250 o.oo 0.52313 0.04265 0.52511 0.08541 0.52839 0.12838 0.53298 0.17167 0.7 0.45399 o.oo 0.45439 0.04457 0.45626 0.08925 0.45911 0.13415 0.46310 0.17939 o.7S 0.38268 o.oo 0.38316 0.04621 0.38460 0.09254 0.38700 0.13910 0.39036 o.i 8601 0.8 0.30902 o.oo 0.30940 0.04757 0.31056 0.09526 0.31250 0.14319 0.31522 0.19148 0.85 0.23345 o.oo 0.23374 0.04864 0.23461 0.09740 0.23608 0.14640 0.23813 0.19577 0.9 0.15643 o.oo 0.15663 0.04941 0.15722 0.09893 0.15820 0.14871 0.15957 0.19886 o-9S 0.07846 o.oo 0.07856 0.04987 0.07885 0.09986 0.07934 0.15010 0.08003 0.20072 I.O o.oo o.oo o.oo 0.05002 o.oo 0.10017 o.oo 0.15056 o.oo 0.20134 1-05 0.07846 o.oo 0.07856 0.04987 0.07885 0.09986 0.07934 0.15010 0.08003 0.20072 i.i 0.15643 o.oo 0.15663 0.04941 0.15722 0.09893 0.15820 0.14871 0.15957 0.19886 i-iS 0.23345 o.oo 0.23374 0.04864 0.23461 0.09740 0.23608 0.14640 0.23813 0.19577 1.2 0.30902 o.oo 0.30940 0.04757 0.31056 0.09526 0.31250 0.14319 0.31522 0.19148 1-25 0.38268 o.oo 0.38316 0.04621 0.38460 0.09254 0.38700 0.13910 0.39036 0.18601 i-3 0.45399 o.oo 0.45439 0.04457 0.45626 0.08925 0.45911 0.13415 0.46310 0.17939 35 0.52250 o.oo 0.52313 0.04265 0.52511 0.08541 0.52839 0.12838 0.53298 0.17167 4 0.58779 o.oo 0.58850 0.04047 0.59027 0.08104 0.59441 0.12181 0.59958 0.16288 45 0.64945 o.oo 0.65023 0.03804 0.65270 0.07617 0.65677 0.11449 0.66248 0.15310 5 0.70711 o.oo 0.70796 0.03537 0.71065 0.07083 0.71508 0.10646 0.72130 0.14237 55 0.76041 o.oo 0.76133 0.03249 0.76421 0.06505 0.76898 0.09778 0.77567 0.13076 .6 0.80902 o.oo 0.81000 0.02940 0.81307 0.05888 0.81814 0.08850 0.82525 0.11834 65 0.85264 o.oo 0.85367 0.02614 0.85691 0.05234 0.86225 0.07867 0.86975 0.10520 1-7 0.89101 o.oo 0.89208 0.02271 0.89547 O-O4547 0.90105 0.06835 0.90889 0.09141 1-75 0.92388 o.oo 0.92503 0.01914 0.92850 0.03833 0.93429 0.05762 0.94242 0.07705 1.8 0.95106 o.oo 0.95225 0.01546 0.95582 0.03095 0.96178 0.04653 0.97014 0.06222 1.85 0.97237 o.oo 0.97385 0.01168 0.97724 0.02338 0.98333 0.03515 0.99188 0.04700 1.9 0.98769 o.oo 0.98892 0.00783 0.99263 0.01567 0.99882 0.02355 1.00751 0.03150 1-95 0.99692 o.oo 0.99816 0.00392 1.00191 0.00786 1.00815 0.01181 1.01692 0.01580 I.OOOO O.OO I.OOI25 O.OO 1.00500 o.oo 1.01127 o.oo 1.02007 o.oo Note. Negative quantities are in heavy type. Examples, cosh (o + i 0.75) = 0.38268 + o. cosh (0.2 + i 1.5) = 0.72130 + * 0.14237 [58] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED x = 0.25 x = 0.3 x = 0.35 x = 0.4 x = 0.45 o.os O.I 0.15 0.2 1.03141 o.oo 1.02823 0.01982 1.01871 0.03952 1.00292 0.05897 0.98093 0.07806 1.04534 o.oo 1.04212 0.02389 1.03247 0.04764 1.01646 0.07109 0.99418 0.09410 i. 06 1 88 o.oo 1.05860 0.02803 1.04880 0.05588 1.03254 0.08338 1.00991 0.11038 1.08107 o.oo 1.07774 0.03223 1.06776 0.06426 1.05120 0.09589 1.02816 0.12693 1.10297 o.oo 1.09957 0.03651 1.08939 0.07280 1.07250 0.10863 1.04899 0.14380 0.25 0-3 0-35 0.4 0-4S 0.95290 0.09667 0.91900 0.11468 0.87942 0.13199 0.83443 0.14848 0.78429 0.16406 0.96577 0.11654 0.93140 0.13825 0.89130 0.15911 0.84570 0.17899 0.79488 0.19777 0.98105 0.13669 0.94614 0.16216 0.90540 0.18663 0.85908 0.20995 0.80746 0.23198 0.99878 0.15719 0.96324 0.18648 0.92177 0.21462 0.87461 0.24143 0.82206 0.26676 1.01901 0.17808 0.98275 0.21126 0.94044 0.24314 0.89232 0.27352 0.83871 0.30222 0-5 0-55 0.6 0.65 0.7 0.72932 0.17862 0.66985 0.19208 0.60625 . 20 43 7 0.53891 0.21539 0.46825 0.22508 0.73917 0.21533 0.67889 0.23156 0.61444 0.24636 0.54619 0.25965 0.47457 0.27133 0.75086 0.25257 0.68964 0.27161 0.62416 0.28897 0.55483 0.30455 0.48208 0.31826 0.76443 0.29045 0.70210 0.31234 0.63544 0.33231 0.56486 0.35022 0.49080 0.36598 0.77992 0.32905 0.71632 0.35385 0.64831 0.37647 0.57630 0.39677 0.50074 0.41462 o.7S 0.8 0.85 0.9 o-9S 0.39471 0.23338 0.31872 0.24025 0.24078 0.24563 0.16135 0.24950 0.08092 0.25183 0.40003 0.28134 0.32303 0.28962 0.24403 0.29611 0.16353 0.30077 0.08202 0.30358 0.40636 0.33000 0.32814 0.33971 0.24789 0.34732 0.16611 0.35279 0.08331 0.35609 0.41371 0.37949 0.33407 0.39065 0.25237 0.39940 0.16912 0.40570 0.08482 0.40949 0.42209 0.42992 0.34084 0.44257 0.25748 0.45249 0.17254 0.45961 0.08654 0.46391 I.O 1.05 i.i 1.2 o.oo 0.25261 0.08092 0.25183 0.16135 0.24950 0.24078 0.24563 0.31872 0.24025 o.oo 0.30452 0.08202 0.30358 0.16353 0.30077 0.24403 0.29611 0.32303 0.28962 o.oo 0.35719 0.08331 0.35609 o. 16611 0.35279 0.24789 0.34732 0.32814 0.33971 o.oo 0.41075 0.08482 0.40949 0.16912 0.40570 0.25237 0.39940 0.33407 0.39065 o.oo 0.46534 0.08654 0.46391 0.17254 0.45961 0.25748 0.45249 0.34084 0.44257 1-25 i-35 1.4 1-45 0.39471 0.23338 0.46825 0.22508 0.53891 0.21539 0.60625 0.20437 0.66985 0.19208 0.40003 0.28134 0.47457 0.27133 0.54619 0.25965 0.61444 0.24636 0.67889 0.23156 0.40636 0.33000 0.48208 0.31826 0.55483 0.30455 0.62416 0.28897 0.68964 0.27161 0.41371 0.37949 0.49080 0.36598 0.56486 0.35022 0.63544 0.33231 0.70210 0.31234 0.42209 0.42992 0.50074 0.41462 0.57630 0.39677 0.64831 0.37647 0.71632 0.35385 i-SS 1.6 1.65 0.72932 0.17862 0.78429 0.16406 0.83443 0.14848 0.87942 0.13199 0.91900 0.11468 0.73917 0.21533 0.79488 0.19777 0.84570 0.17899 0.89130 0.15911 0.93140 0.13825 0.75086 0.25257 0.80746 0.23198 0.85908 0.20995 0.90540 0.18663 0.94614 0.16216 0.76443 0.20945 0.82206 0.26676 0.87461 0.24143 0.92177 0.21462 0.96324 0.18648 0.77992 0.32905 0.83871 0.30222 0.89232 0.27352 0.94044 0.24314 0.98275 0.21126 i-75 1.8 1.85 1.9 I.9S 0.95290 0.09667 0.98093 0.07806 1.00292 0.05897 1.01871 0.03952 1.02823 0.01982 0.96577 0.11654 0.99418 0.09410 1.01646 0.07109 1.03247 0.04764 1.04212 0.02389 0.98105 0.13669 1.00991 0.11038 1.03254 0.08338 1.04880 0.05588 1.05860 0.02803 0.99878 0.15719 1.02816 0.12693 1.05120 0.09589 1.06776 0.06426 1.07774 0.03223 1.01901 0.17808 1.04899 0.14380 1.07250 0.10863 1.08939 0.07280 1.09957 0.03651 2.0 1.03141 o.oo 1.04534 o.oo i. 06 1 88 o.oo 1.08107 o.oo 1.10297 o/oo Note. Negative quantities are in heavy type. Examples, cosh (0.3 + i 0.9) = 0.16353 + i 0.30077. cosh (0.45 + i 1.7) = 0.98275 + i 0.21126. [59] TABLE VTII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED 0.5 x = 0.55 x = 0.6 x = 0.65 x 0.7 o 0.05 O.I 0.15 0.2 1.12763 o.oo 1.12415 0.04088 1.11374 0.08152 1.09647 0.12165 1.07244 0.16103 1.15510 o.oo 1.15154 0.04536 1.14088 0.09044 1.12319 0.13497 1.09857 0.17866 1.18547 o.oo 1. 18181 0.04995 1.17087 0.09959 1.15271 0.14862 1.12744 0.19674 1.21879 o.oo 1.21504 0.05467 1.20379 0.10900 1.18512 0.16265 1.15912 0.21531 1.25517 o.oo 1.25130 0.05952 1.23972 0.11867 1.22049 0.17709 1.19374 0.23442 0.25 0-3 0-35 0.4 0.45 1.04179 0.19942 1.00472 0.23657 0.96146 0.27227 0.91227 0.30629 0.85745 0.33842 1.06717 0.22125 1.02920 0.26248 0.98489 0.30208 0.93450 0.33983 0.87835 0.37548 1.09523 0.24364 1.05626 0.28904 1.01078 0.33265 0.95906 0.37422 0.90144 0.41347 1.12602 0.26663 1.08595 0.31632 1.03919 0.36405 0.98602 0.40954 0.92678 0.45250 1.15962 0.29030 1.11836 0.34439 1.07021 0.39636 1.01545 0.44589 0.95444 0.49266 0-5 0-55 0.6 0.65 0.7 0.79735 0.36847 0.73233 0.39624 0.66280 0.42158 0.58918 0.44431 0.51193 0.46430 0.81678 0.40882 0.75018 0.43963 0.67895 0.46773 0.60354 0.49296 0.52441 0.51514 0.83825 0.45018 0.76990 0.48412 0.69680 0.51506 0.61940 0.54284 0.53819 0.56726 0.86182 0.49268 0.79154 0.52981 0.71639 0.56368 0.63682 0.59408 0.55332 0.62081 0.88754 0.53640 0.81517 0.57683 0.73777 0.61371 0.65582 0.64680 0.56984 0.67590 0-75 0.8 0.85 0.9 0-9.5 0.43152 0.48143 0.34846 0.49559 0.26324 0.50670 0.17640 0.51468 0.08847 0.51949 0.44204 0.53414 0.35695 0.54986 0.26965 0.56218 0.18070 0.57103 0.09063 0.57637 0.45366 0.58819 0.36633 0.60549 0.27674 0.61906 0.18545 0.62882 0.09301 0.63469 0.46641 0.64371 0.37663 0.66265 0.28452 0.67750 0.19066 0.68817 0.09563 0.69460 0.48033 0.70084 0.38787 0.72146 0.29301 0.73763 0.19635 0.74925 0.09848 0.75625 1.0 1.05 i.i 1.2 o.oo 0.52110 0.08847 0.51949 0.17640 0.51468 0.26324 0.50670 0.34846 0.49559 o.oo 0.57815 0.09063 0.57637 0.18070 0.57103 0.26965 0.56218 0.35695 0.54986 o.oo 0.63665 0.09301 0.63469 0.18545 0.62882 0.27674 0.61906 0.36633 0.60549 o.oo 0.69675 0.09563 0.69460 0.19066 0.68817 0.28452 0.67750 0.37663 0.66265 o.oo 0.75858 0.09848 0.75625 0.19635 0.74925 0.29301 0.73763 0.38787 0.72146 1.25 1.3 1-35 1.4 i-45 0.43152 0.48143 0.51193 0.46430 0.58918 0.44431 0.66280 0.42158 0.73233 0.39624 0.44204 0.53414 0.52441 0.51514 0.60354 0.49296 0.67895 0.46773 0.75018 0.43963 0.45366 0.58819 0.53819 0.56726 0.61940 0.54284 0.69680 0.51506 0.76990 0.48412 0.46641 0.64371 0.55332 0.62081 0.63682 0.59408 0.71639 0.56368 0.79154 0.52981 0.48033 0.70084 0.56984 0.67590 0.65582 0.64680 0.73777 0.61371 0.81517 0.57683 i-SS 1.6 1.65 1-7 0.79735 0.36847 0.85745 0.33842 0.91227 0.30629 0.96146 0.27227 1.00472 0.23657 0.81678 0.40882 0.87835 0.37548 0.93450 0.33983 0.98489 0.30208 1.02920 0.26248 0.83825 0.45618 0.90144 0.41347 0.95906 0.37422 1.01078 0.33265 1.05626 0.28904 0.86182 0.49268 0.92678 0.45250 0.98602 0.40954 1.03919 0.36405 1.08595 0.31632 0.88754 0.53640 0.95444 0.49266 1.01545 0.44589 1.07021 0.39636 1.11836 0.34439 i!s 1.85 1.9 1-95 1.04179 0.19942 1.07244 0.16103 1.09647 0.12165 1.11374 0.08152 1.12415 0.04088 1.06717 0.22125 1.09857 0.17866 1.12319 0.13497 1.14088 0.09044 1.15154 0.04536 1.09523 0.24364 1.12744 0.19674 1.15271 0.14862 1.17087 0.09959 1. 18181 0.04995 1.12602 0.26663 1.15912 0.21531 1.18512 0.16265 1.20379 0.10900 1.21504 0.05467 1.15962 0.29030 1.19374 0.23442 1.22049 0.17709 1.23972 0.11867 1.25130 0.05952 2.0 1.12763 o.oo 1.15510 o.oo 1.18547 o.oo 1.21879 o.oo 1.25517 o.oo Note. Negative quantities are in heavy type. Examples, cosh (0.6 + i 0.95) = 0.09301 + i 0.63469. cosh (0.6 + i 1.05) = 0.09301 + i 0.63469. [60] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED 0.75 0.8 x = 0.85 x = o.g x = 0.95 o 0.05 O.I 0.15 O.2 1.29468 o.oo 1.29069 0.06452 1.27874 0.12864 1.25891 0.19197 1.23132 0.25411 1.33743 o.oo I -3333 I 0.06968 1.32097 0.13893 1.30048 0.20732 1.27198 0.27444 1.38353 o.oo 1.37927 0.07502 1.36650 0.14957 1.34530 0.22320 1.31582 0.29546 1.43309 o.oo 1.42867 0.08054 1.41544 0.16058 1.39349 0.23964 1.36294 0.31721 1.48623 o.oo 1.48164 0.08627 1.46793 0.17200 1.44516 0/25667 1.41348 0.33976 0.25 -3 0-35 0.4 0-45 1.19613 0.31469 1.15356 0.37332 1.10390 0.42966 1.04742 0.48335 0.98449 0.53405 1.23563 0.33986 1.19166 0.40319 1.14035 0.46403 1.08201 0.52202 1.01700 0.57678 1.27822 0.36589 1.23274 0.43407 1.17965 0.49957 1.11930 0.56199 1.05205 0.62095 1.32400 0.39283 1.27689 0.46603 1.22194 0.53635 1.15939 0.60337 1.08973 0.66667 1.37309 0.42076 1.32425 0.49916 1.26722 0.57448 1.20238 0.64626 1.13013 0.71406 o-5 0-55 0.6 0.65 0.7 0.91548 0.58147 0.84083 0.62529 0.76100 0.66527 0.67647 0.70114 0.58777 0.73269 0.94571 0.62799 0.86859 0.67532 0.78613 0.71849 0.69881 0.75724 0.60718 0.79131 0.97830 0.67608 0.89853 0.72704 0.81322 0.77351 0.72290 0.81522 0.62811 0.85191 1.01334 0.72586 0.93071 0.78057 0.84235 0.83047 0.74879 0.87525 0.65061 0.91463 1.05092 0.77745 0.96523 0.83605 0.87358 0.88950 0.77655 0.93747 0.67473 0.97965 0.75 0.8 0.85 0.9 o.95 0.49545 0.75972 0.40008 0.78207 0.30224 0.79960 0.20253 0.81219 0.10158 0.81978 0.51182 0.82050 0.41329 0.84464 0.31222 0.86357 0.20922 0.87717 0.10493 0.88537 0.52945 0.88334 0.42753 0.90932 0.32298 0.92970 0.21643 0.94435 0.10855 0.95317 0.54842 0.94838 0.44285 0.97628 0-33455 0.99816 0.22418 1.01388 0.11244 1-02335 0.56875 1.01579 0.45927 1.04567 0.34695 1.06911 0.23250 1.08595 0.11661 1.09610 .0 05 .1 IS .2 o.oo 0.82232 0.10158 0.81978 0.20253 0.81219 0.30224 0.79960 0.40008 0.78207 o.oo 0.88811 0.10493 0.88537 0.20922 0.87717 0.31222 0.86357 0.41329 0.84464 o.oo 0.95612 0.10855 0.95317 0.21643 0.94435 0.32298 0.92970 0.42753 0.90932 o.oo 1.02652 0.11244 1-02335 0.22418 1.01388 0.33455 0.99816 0.44285 0.97628 o.oo 1.09948 o. 11661 1.09610 0.23250 1.08595 0.34695 1.06911 0.45927 1.04567 25 3 i-35 1.4 1-45 0.49545 0.75972 0.58777 0.73269 0.67647 0.70114 0.76100 0.66527 0.84083 0.62529 0.51182 0.82050 0.60718 0.79131 0.69881 0.75724 0.78613 0.71849 0.86859 0.67532 0.52945 0.88334 0.62811 0.85191 0.72290 0.81522 0.81322 0.77351 0.89853 0.72704 0.54842 0.94838 0.65061 0.91463 0.74879 0.87525 0.84235 0.83047 0.93071 0.78057 0.56875 1.01579 0.67473 0.97965 0.77655 0.93747 0.87358 0.88950 0.96523 0.83605 i-5 1-55 1.6 1.65 i-7 0.91548 0.58147 0.98449 0.53405 1.04742 0.48335 1.10390 0.42966 I.I5356 0.37332 0.94571 0.62799 1.01700 0.57678 1.08201 0.52202 1.14035 0.46403 1.19166 0.40319 0.97830 0.67608 1.05205 0.62095 1.11930 0.56199 1.17965 0.49957 1.23274 0.43407 1.01334 0.72586 1.08973 0.66667 I-I5939 0.60337 1.22194 0.53635 1.27689 0.46603 1.05092 0.77745 1.13013 0.71406 1.20238 0.64626 1.26722 0.57448 1.32425 0.49916 1-75 1.8 1.85 1.9 1-95 1.19613 0.31469 1.23132 0.25411 1.25891 0.19197 1.27874 0.12864 1.29069 0.06452 1.23563 0.33986 1.27198 0.27444 1.30048 0.20732 1.32097 0.13893 1.33331 0.06968 1.27822 0.36589 1.31582 0.29546 1.34530 0.22320 1.36650 0.14957 1.37927 0.07502 1.32400 0.39283 1.36294 0.31721 1-39349 0.23964 1.41544 0.16058 1.42867 0.08054 1.37309 0.42076 1.41348 0.33976 1.44516 0.25667 1.46793 0.17200 1.48164 0.08627 2.0 1.29468 o.oo 1.33743 0.00 1.38353 o.oo 1.43309 o.oo 1.48623 o.oo Note. Negative quantities are in heavy type. Examples, cosh (0.9 + i i.o) = o + * 1.02652. cosh (0.9 + i i.io) = 0.22418 + i 1.01388. [61] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED x = 1.05 0.05 O.I 0.15 O.2 1.54308 o.oo 1.53832 0.09221 1.52408 0.18384 1.50045 0.27435 1.46756 0.36316 1.60379 o.oo 1.59885 0.09838 1.58405 0.19615 1.55948 0.29271 1.52530 0.38746 1.66852 o.oo 1.66337 0.10479 1.64798 0.20894 1.62242 0.31180 1.58685 0.41274 1.73741 o.oo 1.73206 0.11147 1.71602 0.22226 1.68941 0.33167 1.65238 0.43904 i. 81066 o.oo 1.80507 0.11843 1.78836 0.23613 1.76063 0.35238 1.72204 0.46645 0.25 0.3 o-3S 0.4 o-4S 1.42562 0.44973 1.37490 0.53353 1.31569 0.61404 1.24838 0.69077 1.17337 0.76323 1.48171 0.47983 1.42899 0.56924 1.36746 0.65514 1.29750 0.73700 1.21953 0.81432 1.54151 0.51113 1.48666 0.60637 1.42265 0.69787 1.34986 0.78508 1.26875 0.86743 1.60517 0.54371 1.54805 0.64502 1.48139 0.74235 1.40560 0.83511 1.32114 0.92272 1.67283 0.57765 1.61331 0.68528 1.54384 0.78869 1.46485 0.88724 1.37684 0.98032 o-S o.SS 0.6 0.65 0.7 1.09112 0.83099 1.00215 0.89363 0.90700 0.95076 0.80626 I.OO2O2 0.70055 1-04711 1.13405 0.88661 1.04158 0.95344 0.94269 1.01439 0.83798 1.06909 0.72811 1.11719 1.17982 0.94445 1.08362 1.01564 0.98073 1.08056 0.87180 1.13883 0.75749 1.19007 1.22854 1.00464 1.12836 1.08037 1.02123 I-I4943 0.90780 1.21141 0.78877 1.26592 1.28033 1-06735 1.17593 1.14781 1.06428 I. 22118 0.94607 1.28703 0.82202 1.34494 0-75 0.8 0.85 0.9 o-9S 0.59051 1.08574 0.47684 1.11768 0.36023 1.14273 0.24139 1.16073 0.12107 1.17158 0.61375 1.15841 0.49560 1.19249 0.37440 1.21921 0.25089 1.23842 0.12583 1.24999 0.63852 1.23398 0.51560 1.27028 0.38951 1.29875 0.26101 1.31920 0.13091 1.33153 0.66488 1.31263 0.53689 1.35124 0.40559 1.38152 0.27179 1.40329 0.13632 1.41640 0.69291 1.39456 0.55952 1.43558 0.42269 1.46776 0.28325 1.49088 0.14206 1.50481 I.O 1.05 i.i i-iS 1.2 o.oo 1.17520 0.12107 1.17158 0.24139 1.16073 0.36023 1.14273 0.47684 1.11768 o.oo 1.25386 0.12583 1.24999 0.25089 1.23842 0.37440 1.21921 0.49560 1.19249 o.oo 1.33565 0.13091 1.33153 0.26101 1.31920 0.38951 1.29875 0.51560 1.27028 o.oo 1.42078 0.13632 1.41640 0.27179 1.40329 0.40559 1.38152 0.53689 1.35124 o.oo 1.50946 0.14206 1.50481 0.28325 1.49088 0.42269 1.46776 0.55952 1.43558 1.25 i-3 1-35 1.4 i.4S 0.59051 1.08574 0.70055 1.04711 O.8O626 1.00202 O.9O7OO 0.95076 I.002I5 0.89363 0.61375 1.15841 0.72811 1.11719 0.83798 1.06909 0.94269 1.01439 1.04158 0.95344 0.63852 1.23398 0.75749 1.19007 0.87180 1.13883 0.98073 1.08056 1.08362 1.01564 0.66488 1.31263 0.78877 1.26592 0.90780 1.21141 I.02I23 I.I4943 1.12836 1.08037 0.69291 1.39456 0.82202 1.34494 0.94607 1.28703 1.06428 I. 22118 1.17593 1.14781 i.S i-SS 1.6 1.65 1.7 .09112 0.83099 .17337 0.76323 .24838 0.69077 .31569 0.61404 .37490 0.53353 1.13405 0.88661 1.21953 0.81432 1.29750 0.73700 1.36746 0.65514 1.42899 0.56924 1.17982 0.94445 1.26875 0.86743 1.34986 0.78508 1.42265 0.69787 1.48666 0.60637 1.22854 1.00464 I.32II4 0.92272 1.40560 0.835II I.48I39 0.74235 1.54805 0.64502 1.28033 1.06735 1.37684 0.98032 1.46485 0.88724 1.54384 0.78869 1.61331 0.68528 1-75 1.8 1.85 1.9 1.95 1.42562 0.44973 X.46756 0.36316 1.50045 0.27435 1.52408 0.18384 1.53832 O.O922I 1.48171 0.47983 1.52530 0.38746 1.55948 0.29271 1.58405 0.19615 1.59885 0.09838 1.54151 0.51113 1.58685 0.41274 1.62242 0.31180 1.64798 0.20894 1.66337 0.10479 I.605I7 0.54371 1.65238 0.43904 I.6894I 0.33167 I.7I6O2 O.22226 I.732O6 O.III47 1.67283 0.57765 1.72204 0.46645 1.76063 0.35238 1.78836 0.23613 1.80507 0.11843 2.O 1.54308 o.oo 1.60379 o.oo 1.66852 o.oo I 7374i o.oo i. 81066 o.oo Note. Negative qua n tities are in heavy type. Examples, cosh (1.2 4- i o) = i. 81066 + i o. cosh (E.I + i.i) = 0*26101 + i 1.31920. [62] TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED 9 x = 1.25 *= 1.3 x = i-35 x = 1.4 x = 1.45 o 0.05 O.I o-iS O.2 1.88842 o.oo 1.88260 0.12569 1.86517 0.25060 .83624 0.37396 .79600 0.49502 1.97091 o.oo 1.96484 1.13325 1.94665 0.26569 1.91646 0.39648 1.87445 0.52483 2.05833 o.oo 2.05200 0.14116 2.03299 0.28144 2.00146 0.41999 1-95759 0.55595 2.15090 o.oo 2.14427 0.14941 2.12442 0.29790 2.09147 0.44455 2.04562 0.58846 2.24884 o.oo 2.24191 0.15804 2.22115 0.31510 2.18671 0.47022 2.13878 0.62244 0.25 0-3 0-35 0.4 0-45 .74467 0.61303 .68260 0.72726 .61014 0.83700 1.52777 0.94158 1.43597 1.04036 1.82089 0.64994 1.75610 0.77105 1.68048 0.88740 1.59451 0.99829 1.49870 1.10301 1.90165 0.68848 1.83399 0.81677 1.75502 0.94002 1.66523 1.05748 1.56517 1.16842 1.98717 0.72875 1.91646 0.86454 1.83394 0.99500 1.74012 1.11932 1.63556 1.23674 2.07766 0.77083 2.00373 0.91446 1.91745 1.05245 1.81935 1.18396 1.71007 1.30817 o-5 o.SS 0.6 0.65 0.7 1.33532 1.13273 1.22643 1.2x811 1.10999 1.29598 0.98670 1.36586 0.85733 1-42732 1.39365 1.20094 1.28001 1.29146 1.15848 1.37402 1.02980 1.44811 0.89478 1.51327 1.45546 1.27215 1.33678 1.36804 1.20986 1.45550 1.07548 1.53398 0.93446 1.60300 1.52092 1.34655 1.39690 1.44804 1.26427 1.54061 1.12384 1.62369 0.97649 1.69675 1.59017 1.42431 1.46051 1.53166 1.32184 1.62958 1.17502 1.71745 1.02095 1-79473 0.75 0.8 0.85 0.9 o-95 0.72267 1.47998 0.58355 1-52352 0.44084 1.55766 0.29541 1.58220 0.14816 1.59698 0.75424 1.56910 0.60905 1.61526 0.46010 1.65146 0.30832 1.67747 0.15464 1.69315 0.78769 1.66215 0.63606 1.71104 0.48051 1.74938 0.32199 1.77694 0.16150 1.79354 0.82311 1.75934 0.66466 1.81110 0.50212 1.85169 0.33647 1.88086 0.16876 1.89843 0.86060 1.86094 0.69493 1.91569 0.52498 1.95862 0.35180 1.98947 0.17644 2.00806 I.O 1.05 i.i i-iS 1.2 o.oo 1.60192 0.14816 1.59698 0.29541 1.58220 0.44084 1.55766 0.58355 1-52352 o.oo 1.69838 0.15464 1.69315 0.30832 1.67747 0.46010 1.65146 0.60905 1.61526 o.oo 1.79909 0.16150 1.79354 0.32199 1.77694 0.48051 1.74938 0.63606 1.71104 o.oo 1.90430 0.16876 1.89843 0.33647 1.88086 0.50212 1.85169 0.66466 1.81110 o.oo 2.01427 0.17644 2.00806 0.35180 1.98947 0.52498 1.95862 0.69493 1.91569 I.2S i-3 i-35 1.4 i-4S 0.72267 1.47998 0.85733 1.42732 0.98670 1.36586 .10999 1-29598 .22643 1. 21811 0.75424 1.56910 0.89478 1.51327 1.02980 1.44811 1.15848 1.37402 1.28001 1.29146 0.78769 1.66215 0.93446 1.60300 1.07548 1.53398 1.20986 1.45550 1.33678 1.36804 0.82311 1.75934 0.97649 1.69675 1.12384 1.62369 1.26427 1.54061 1.39690 1.44804 0.86060 1.86094 1.02095 1.79473 1.17502 1.71745 1.32184 1.62958 1.46051 1.53166 i-5 55 .6 .65 7 .33532 1.13273 43597 1.04036 52777 0.94158 .61014 0.83700 .68260 0.72726 1-39365 1.20094 1.49870 1.10301 1.59451 0.99829 1.68048 0.88740 1.75610 0.77105 1.45546 1.27215 1.56517 1.16842 1.66523 1.05748 1.75502 0.94002 1.83399 0.81677 1.52092 1.34655 1.63556 1.23674 1.74012 1.11932 1.83394 0.99500 1.91646 0.86454 1.59017 1.42431 1.71007 1.30817 1.81935 1.18396 1.91745 1.05245 2.00373 0.91446 75 .8 85 1.9 1-95 .74467 0.61303 .79600 0.49502 .83624 0.37396 .86517 0.25060 .88260 0.12569 1.82089 0.64994 1.87445 0.52483 1.91646 0.39648 1.94665 0.26569 1.06484 0.13325 1.90165 0.68848 X-95759 0.55595 2.00146 0.41999 2.03299 0.28144 2.05200 0.14116 1.98717 0.72875 2.04562 0.58846 2.09147 0.44455 2.12442 0.29790 2.14427 0.14941 2.07766 0.77083 2.13878 0.62244 2.18671 0.47022 2.22115 0.31510 2.24191 0.15804 2.O 1.88842 O.OO 1.97091 o.oo 2.05833 o.oo 2.15090 o.oo 2.24884 o.oo Note. Negative quantities are in heavy type. Examples, cosh (1.4 + i 1.9) = 2.12442 + * 0.29790. cosh (1.4 + i 1.4) = 1.26427 + * 1.54061. [63] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED *= 1.5 = i-55 x = 1.6 x = 1.65 x = 1.7 o 2.35241 o.oo 2.46186 o.oo 2.57746 o.oo 2.69952 o.oo 2.82832 o.oo 0.05 2.34516 0.16706 2.45427 0.17650 2.56952 0.18639 2.69121 0.19673 2.81960 0.20757 O.I 2.32345 0.33309 2.43155 0.35192 2-54573 0.37162 2.66629 -39 22 5 2.79350 0.41387 0.15 2.28742 0.49707 2.39384 0.52516 2.50625 0.55456 2.62494 0.58536 2.75017 0.61761 O.2 2.23727 0.65798 2.34137 0.69517 2.45131 0.73409 2.56740 0.77485 2.68989 0.81754 0.25 2.17334 0.81484 2.27446 0.86089 2.38127 0.90909 2.49404 0.95957 2.61302 1.01244 -3 2.09601 0.96667 2.19353 1.02130 2.29654 1.07848 2.40529 1.13837 2.52005 1.20109 0.35 2.00576 1.11255 2.09908 1.17542 2.19765 1.24123 2.30173 1.31015 2.41154 1.38234 0.4 1.90314 1.25156 1.99169 1.32229 2.08522 1.39632 2.18396 1.47385 2.28816 1.55506 o-4S 1.78879 1.38286 1.87201 1.46101 1.95992 1.54281 2.05274 1.62847 2.15067 1.71820 o-S 1.66341 1.50563 1.74080 1.59071 1.82254 1.67978 1.00885 1.77305 1.99992 1.87074 0-55 1.52777 1.61912 1.59885 1.71062 1.67393 1.80640 1.75320 1.90669 1.83684 2.01175 0.6 1.38271 1.72263 1.44704 1.81997 1.51500 1.92187 1.58674 2.02858 1.66244 2.14036 0.65 1.22913 1.81551 1.28632 1.91811 1.34672 2.02550 1.41050 2.13797 1.47779 2.25577 0.7 1.06797 1.89720 1.11766 2.00442 1.17015 2.11664 1.22556 2.23417 1.28403 2.35727 0-75 0.90023 1.96720 0.94211 2.07837 0.98636 2.19473 1.03306 2.31660 1.08235 2.44424 0.8 0.72693 2.02507 0.76076 2.13951 0.79648 2.25930 0.83420 2.38474 0.87400 2.51614 0.85 0.54916 2.07045 0.57471 2.18745 0.60170 2.30993 0.63019 2.43818 0.66026 2.57253 0.9 0.36800 2.10307 0.38512 2.22191 0.40320 2.34632 0.42230 2.47659 0.44245 2.61306 0-95 0.18457 2.12272 0.19316 2.24268 0.20223 2.36824 0.21180 2.49973 0.22191 2.63747 I.O o.oo 2.12928 o.oo 2.24961 o.oo 2.37557 0,00 2.50747 o.oo 2.64563 i. 05 0.18457 2.12272 0.19316 2.24268 0.20223 2.36824 o. 21180 2.49973 0.22191 2.63747 i.i 0.36800 2.10307 0.38512 2.22191 0.40320 2.34632 0.42230 2.47659 0.44245 2.61306 i-iS 0.54916 2.07045 0.57471 2.18745 0.60170 2.30993 0.63019 2.43818 0.66026 2.57253 1.2 0.72693 2.02507 0.76076 2.13951 0.79648 2.25930 0.83420 2.38474 0.87400 2.51614 I.2S 0.90023 1.96720 0.94211 2.07837 0.98636 2.19473 1.03306 2.31660 1.08235 2.44424 i-3 1.06797 1.89720 1.11766 2.00442 1.17015 2.11664 1.22556 2.23417 1.28403 2.35727 i-35 1.22913 1.81551 1.28632 1.91811 1.34672 2.02550 1.41050 2.13797 1.47779 2.25577 1.4 1.38271 1.72263 1.44704 1.81997 1.51500 1.92187 1.58674 2.02858 1.66244 2.14036 i-4S 1-52777 1.61912 1.59885 1.71062 1-67393 1.80640 1-75320 1.90669 1.83684 2.01175 i-5 1.66341 1.50563 1.74080 1.59071 1.82254 1.67978 1.90885 1.77305 1.99992 1.87074 i-SS 1.78879 1.38286 1.87201 1.46101 1.95992 1.54281 2.05274 1.62847 2.15067 1.71820 1.6 1.90314 1.25156 1.99169 1.32229 2.08522 1.39632 2.18396 1.47385 2.28816 1.55506 1.65 2.00576 1.11255 2.09908 1.17542 2.19765 1.24123 2.30173 1.31015 2.41154 1.38234 i-7 2.09601 0.96667 2.19353 1.02130 2.29654 1.07848 2.40529 1.13837 2.52005 1.20109 1-75 2-17334 0.81484 2.27446 0.86089 2.38127 0.90909 2.49404 0.95957 2.61302 1.01244 1.8 2.23727 0.65798 2.34137 0.69517 2.45131 0.73409 2.56740 0.77485 2.68989 0.81754 1.85 2.28742 0.49707 2.39384 0.52516 2.50625 0.55456 2.62494 0.58536 2.75017 0.61761 1.9 2.32345 0-33309 2.43155 0.35192 2-54573 0.37162 2.66629 0.39225 2.79350 0.41387 i. 95 2.34516 0.16706 2.45427 0.17650 2.56952 0.18639 2.69121 0.19673 2.81960 0.20757 2.35241 o.oo 2.46186 o.oo 2.57746 o.oo 2.69952 o.oo 2.82832 o.oo Note. Negative quantities are in heavy type. Examples, cosh (1.6 + i 0.4) = 2.08522 -f- i 1.39632. cosh (1.7 + * 1.2) = 0.87400 + i 2.51614. [64] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + n>. CONTINUED ? x = i-75 x = 1.8 X = 1.85 X = 1.9 x = i-95 o 2.96419 o.oo 3.10747 o.oo 3-25853 o.oo 3-41773 o.oo 3-58548 o.oo 0.05 2.95505 0.21893 3.09789 0.23084 3.24848 0.24332 3.40719 0.25642 3-57443 0.27015 O.I 2.92769 0.43652 3.06921 0.46026 3.21841 0.48627 3-37565 0.51125 3-54I34 0.53864 0.15 2.88229 0.65141 3.02161 0.68684 3.16850 0.72398 3-32330 0.76294 3.48641 0.80380 O.2 2.81911 0.86229 2.95538 0.90918 3.09904 0.95835 3-25045 1.00992 3.41000 1.06401 0.25 2.73855 1.06784 2.87093 1.12592 3.01049 1. 18681 3- I 57S7 1.25067 3-31255 1.31766 o-3 2.64111 1.26682 2.76878 1-33572 2.90337 1.40796 3.04522 1.48372 3.19469 1.56318 0-35 2-52/39 1.45799 2.64956 1.53728 2.77835 1.62042 2.91409 1.90761 3-057I3 1.79907 0.4 2.39808 1.64016 2.51400 1.72937 2.63620 1.82289 2.76501 1.92098 2.90071 2.02387 0-45 2-25399 1.81223 2.36294 1.91079 2.47780 2.01413 2.59887 2.12250 2.72642 2.23618 o-5 2.09600 1.97312 2.19731 2.08043 2.30413 2.19294 2.41670 2.31094 2-53532 2.43471 Q-55 1.92509 2.12185 2.01814 2.23725 2.11624 2.35824 2.21964 2.48513 2.32858 2.61823 0.6 1.74231 2.25749 1.82653 2.38027 i-9 I 53i 2.50900 2.00889 2.64400 2.10749 2.78561 0.65 1.54^78 2.37922 1.62365 2.50862 1.70258 2.64429 1.78576 2.78657 r.87341 2.93582 0.7 I-3457I 2.48627 1.41076 2.62149 1-47934 2.76327 1-55162 2.91196 1.62777 3.06792 o-75 I-I3435 2.57800 1.18918 2.71821 1.24698 2.86522 1.30791 3.01939 1.37210 3.18111 0.8 0.91599 2-65384 0.96026 2.79817 1.00694 2.94950 1.05614 3.10821 1.10797 3.27468 0.85 0.69198 2.71331 0.72542 2.86088 0.76069 3.01560 0.79785 3.17787 0.83701 3-34807 0.9 0.46370 2.75606 0.48612 2.90595 0.50975 3.06311 0.53465 3-22793 0.56089 3.40081 o-95 0.23257 2.78181 0.24381 2.93310 0.25566 3-09173 0.26815 3.25809 0.28131 3.43259 I.O o.oo 2.79041 o.oo 2.94217 o.oo 3.10129 o.oo 3.26816 0.00 3-44321 1.05 0.23257 2.78181 0.24381 2.93310 0.25566 2.09173 0.26815 3.25809 0.28131 3-43259 i.i 0.46370 2.75606 0.48612 2-90595 0.50975 3.06311 0.53465 3-22793 0.56089 3.40081 i-i5 0.69198 2.71331 0.72542 2.86088 0.76069 3.01560 0.79785 3.17787 0.83701 3.34807 1.2 0.91599 2.65384 0.96026 2.79817 1.00694 2.94950 1.05614 3.10821 1.10797 3-27468 1.25 .13435 2.57800 1.18918 2.71821 1.24698 2.86522 1.30791 3- OI 939 1.37210 3.18111 i-3 .34571 2.48627 1.41076 2.62149 1.47934 2.76327 1.55162 2.91196 1.62777 3.06792 i-35 .54878 2.37922 1.62365 2.50862 1.70258 2.64429 1.78576 2.78657 1.87341 2.93582 1.4 .74231 2.25749 1.82653 2.38027 1-91531 2.50900 2.00889 2.64400 2.10749 2.78561 i-4S .92509 2.12185 2.01814 2.23725 2.11624 2.35824 2.21964 2.48513 2.32858 2.61823 i-5 2.09600 1.97312 2.19731 2.08043 2.30413 2.19294 2.41670 2.31094 2.53532 2.43471 i-SS 2.25399 1.81223 2.36294 1.91079 2.47780 2.01413 2.59887 2.12250 2.72642 2.23618 1.6 2.39808 1.64016 2.51400 1.72937 2.63620 1.82289 2.76501 1.92098 2.90071 2.02387 1.65 2.52739 1-45799 2.64956 1-53728 2.77835 1.62042 2.91409 1.90761 3.05713 1.79907 1-7 2.64111 1.26682 2.76878 1-33572 2.90337 1.40796 3.04522 1-48372 3.19469 1.56318 i-75 2.73855 1.06784 2.87093 1.12592 3.01049 1. 18681 3.15757 1.25067 3.31255 1.31766 1.8 2.81911 0.86229 2.95538 0.90918 3.09904 0.95835 3-25045 1.00992 3.41000 1.06401 1.85 2.88229 0.65141 3.02161 0.68684 3.16850 0.72398 3-32330 0.76294 3.48641 0.80380 1.9 2.92769 0.43652 3.06921 0.46026 3.21841 0.48627 3.37565 0.51125 3.54134 0.53864 i-95 2.95505 0.21893 3.09789 0.23084 3.24848 0.24332 3.40719 0.25642 3.57443 0.27015 2.O 2.96419 O.OO 3.10747 o.oo 3.25853 o.oo 3.41773 o.oo 3.58548 o.oo Note. Negative quantities are in heavy type. Examples, cosh (1.8 + o. 2) = 2.95538 + i 0.90918. cosh (1.8 + i 2.0) = 3.10747 + o. [65] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED 9 x = 2.0 x = 2.05 X = 2.1 x = 2.15 X = 2.2 o 0.05 O.I 0.15 0.2 3.76220 o.oo 3.75059 0.28456 3.71587 0.56737 3.65825 0.84667 3.57806 1.12076 3.94832 o.oo 3.93615 0.29968 3.89971 0.59751 3.83923 0.89166 3.75507 1.18032 4.14431 o.oo 4.13154 0.31555 4.09329 0.62916 4.02981 0.93888 3.94148 1.24282 4.35067 o.oo 4.33726 0.33221 4.29711 0.66237 4.23046 0.98845 4-13773 1-30844 4.56791 o.oo 4.55382 0.34970 4.51167 0.69724 4.44170 1.04049 4-34433 1-37732 0.25 0-35 0.4 0-45 3.47581 1.38794 3.35214 1.64656 3.20780 1.89503 3.04368 2.13182 2.86080 2.35546 3.64777 1.46169 3.51798 1.73405 3.36649 1.99573 3.19426 2.24509 3.00232 2.48062 3.82885 1.53910 3.69261 1.82589 3.53361 2.10142 3.35283 2.36399 3.15137 2.61199 4.01950 1.62035 3.87647 1.92228 3.70956 2.21236 3.51977 2.48879 3.30828 2.74988 4.22019 1.70566 4.07003 2.02349 3.89478 2.32883 3.69552 2.61982 3-47347 2.89466 0-5 0-55 0.6 0.65 0-7 2.66027 2.56458 2-44335 2.75789 2.21136 2.93420 I-96574 3-09241 1.70800 3.23156 2.79188 2.70085 2.56423 2.90443 2.32076 3.09011 2.06299 3.23673 1.79250 3.40327 2.93048 2.84389 2.69152 3.05825 2-43597 3-25376 2.16540 3.42920 1.88148 3-5835I 3.07639 2.99402 2.82553 3.21970 2.55726 3-42553 2.27322 3.61024 1.97516 3-77269 3.23000 3.15165 2.96661 3.38921 2.68495 3.60588 2.38672 3.80031 2.07379 3.97I3I 0.75 0.8 0.85 0.9 0.95 1-43973 3-35078 1.16258 3-44935 0.87827 3.52666 0-58854 3.58221 0.29518 3.61568 1.51096 3.52883 I.22OIO 3.63264 0.92172 3.71404 0.61765 3.77256 0.30978 3.80781 1-58596 3-7I57I 1.28066 3.82501 0.96747 3-9 I0 74 0.64831 3-97235 0.32516 4.00946 1.66493 3-91188 1-34443 4-02695 1.01564 4.11720 0.68059 4.18206 0.34135 4-22113 1.74806 4.11783 1.41156 4.23896 1.06636 4-33396 0.71458 4.40223 0.35839 4-44337 I.O 1.05 i.i 1.2 o.oo 3.62686 0.29518 3.61568 0.58854 3.58221 0.87827 3.52666 1.16258 3-44935 o.oo 3.81958 0.30978 3.80781 0.61765 3.77256 0.92172 3.71404 1. 22010 3.63264 o.oo 4.02186 0.32516 4.00946 0.64831 3-97235 0.96747 3-91074 1.28066 3.82501 o.oo 4.23419 0.34135 4-22113 0.68059 4.18206 1.01564 4.11720 1.34443 4-02695 0.00 4-457II 0.35839 4-44337 0.71458 4.40223 1.06636 4-33396 1.41156 4.23896 1.25 i-3S 1.4 MS 1-43973 3-35078 1.70800 3.23156 I-96574 3-09241 2.21136 2.93420 2.44335 2.75789 I.5I096 3.52883 1.79250 3-40327 2.06299 3.23673 2.32O76 3.OOOII 2.56423 2.90443 1.58596 3-71571 1.88148 3.58351 2.16540 3.42920 2.43597 3-25376 2.69152 3.05825 1.66493 3.91188 i.975i6 3.77269 2.27322 3.61024 2.55726 3.42553 2.82553 3-2I970 1.74806 4.11783 2.07379 3-97I3I 2.38672 3.80031 2.68495 3-60588 2.96661 3.38921 i-SS .6 .65 7 2.66027 2.56458 2.86080 2.35546 3.04368 2.13182 3.20780 1.89503 3.35214 1.64656 2.79188 2.70085 3.OO232 2.48062 3.19426 2.24509 3.36649 1.99573 3.51798 L73405 2.93048 2.84389 3.15137 2.61199 3.35283 2.36399 3.53361 2.10142 3.69261 1.82589 3.07639 2.99402 3.30828 2.74988 3.51977 2.48879 3.70956 2.21236 3.87647 1.92228 3.23000 3.15165 3-47347 2.89466 3.69552 2.61982 3.89478 2.32883 4.07003 2.02349 75 .8 -85 1.9 3.47581 1.38794 3.57806 1.12076 3.65825 0.84667 3.71587 0.56737 3.75059 0.28456 3.64777 1.46169 3-75507 I.I8032 3.83923 0.89166 3.89971 0.59751 3.93615 0.29968 3.82885 1.53910 3.94148 1.24282 4.02981 0.93888 4.09329 0.62916 4.13154 0.31555 4.01950 1.62035 4-13773 1-30844 4.23046 0.98845 4.29711 0.66237 4.33726 0.33221 4.22019 1.70566 4-34433 1-37732 4.44170 1.04049 4.51167 0.69724 4.55382 0.34970 2.O 3.76220 o.oo 3.94832 o.oo 4.14431 o.oo 4.35067 o.oo 4.56791 o.oo Note. Negative quantities are in heavy type. Examples, cosh (2.1 + * 0.8) = 1.28066 + i 3.82501. cosh (2.2 + i 1.25) = 1.74806 + i 4.11783. [66] TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED 2.25 x = 2.3 x = 2.35 2.4 x = 2.45 o 0.05 O.I 0.15 O.2 4.79657 o.oo 4.78178 0.36807 4-737SI 0.73386 4.66404 1.09513 4.56l8l 1.44965 5.03722 o.oo 5.02169 0.38735 4.97521 0.77231 4.89805 1.15251 4.79058 1.52560 5.29047 o.oo 5.27416 0.40760 5.22533 0.81269 5.14429 1.21277 5.03153 1.60537 5.55695 o.oo 5.53981 0.42888 5.48853 0.85511 5.40341 1.27607 5.28496 1.68916 5.83732 o.oo 5.81933 0.45122 5.76545 0.89966 5.67603 1.34255 5.55162 1.77716 0.25 o-3 o-3S 0.4 0-45 443145 1.79523 4.27377 2.12975 4.08975 2.45II3 3.88050 2.75740 3-64734 3-04677 4.65378 1.88930 4.48820 2.24134 4.29494 2.57956 4.07520 2.90188 3.83034 3.20630 4.88776 1.98808 4.71384 2.35853 4.51087 2.71443 4.28008 3.05360 4.02200 3.37395 5.13394 2.09184 4.95127 2.48162 4.73808 2.85610 4.49567 3.21297 4.22554 3.55003 5.39298 2.20082 5.20109 2.61091 4-97713 3-00490 4.72249 3.38036 443873 3-73499 o-5 0-55 0.6 0.65 0.7 3.39168 3.3I7I6 3.II5I2 3.567I-9 2.81935 3.79523 2.50620 3.99987 2.17760 4.17986 3.56185 3.49096 3.27141 3.75410 2.96081 3.99409 2.63194 4.20946 2.28685 4.39887 3-74093 3.67351 3.43588 3.95038 3.10966 4.20292 2.76426 4.42955 2.40182 4.62887 3-92935 3-86521 3.60894 4-15656 3.26629 4.42228 2.90349 4.66073 2.52280 4.87045 4.12761 4.06659 3.79104 4-373" 3.43109 4.65268 3.04999 4.90356 2.65009 5.12420 0-75 0.8 0.85 0.9 o-95 I-83557 4-33407 1.48222 4.46157 I.II974 4-56I5S 0.75035 4-6334I 0.37633 4.6767I 1.92766 4.56116 1.55659 4-69533 1.17592 4.80056 0.78799 4.87618 0.39522 4-92174 2.02457 4.79965 1.63485 4.94083 1.23504 5.05156 0.82761 5.13114 0.41509 5.17909 2.12655 5.05014 1.71719 5.19869 1.29724 5.31521 0.86930 5.39893 0.43599 5-44938 2-23385 5-3I325 1.80383 5-46955 1.36269 5.59213 0.91316 5.68022 0-45799 5-73330 I.O 1-05 i.i "5 1.2 o.oo 4.69117 0.37633 4-67671 0.75035 4-63341 1.11974 4.56155 1.48222 4.46157 o.oo 4.93696 0.39522 4.92174 0.78799 4.87618 1.17592 4.80056 1.55659 4.69533 o.oo 5- I 95 I o 0.41509 5.17909 0.82761 5.13114 1.23504 5.05156 1.63485 4-94083 o.oo 5.46623 0.43509 5.44938 0.86930 5.39893 1.29724 5.31521 1.71719 5.19869 o.oo 5-75103 0.45799 5-73330 0.91316 5.68022 1.36269 5.59213 1.80383 5-46955 1.25 i-3 1-35 1.4 1-45 1.83557 4.33407 2.17760 4.17986 2.50620 3.99987 2.81935 3.79523 3.11512 3.56719 1.92766 4.56116 2.28685 4.39887 2.63194 4.20946 2.96081 3.99409 3.27141 3.75410 2.02457 4.79965 2.40182 4.62887 2.76426 4-42955 3.10966 4.20292 3.43588 3.95038 2.12655 5.05014 2.52280 4.87045 2.90349 4.66073 3.26629 4.42228 3.60894 4.15656 2.23385 5-3 r 325 2.65009 5.12420 3.04999 4-90356 3.43109 4.65268 3.79104 4.37311 i-5 i-55 1.6 1.65 i-7 3.39168 3.31716 3.64734 3-04677 3.88050 2.75740 4.08975 2.45113 4.27377 2.12975 3.56185 3.49096 3.83034 3.20630 4.07520 2.90188 4.29494 2.57956 4.48820 2.24134 3.74093 3-6735I 4.02290 3-37395 4.28008 3.05360 4.51087 2.71443 4.71384 2.35853 3.92935 3-86521 4.22554 3-55003 4.49567 3-21297 4.73808 2.85610 4.95127 2.48162 4.12761 4.06659 4.43873 3-73499 4.72249 3.38036 4-97713 3-00490 5.20109 2.61091 i-75 1.8 i.5 1.9 i-95 4.43145 1.79523 4.56181 1.44965 4.66404 1.09513 4.73751 0.73386 4.78178 0.36807 4.65378 1.88930 4.79058 1.52560 4.89805 1.15251 4.97521 0.77231 5.02169 0.38735 4.88776 1.98808 5.03153 1-60537 5.14429 1.21277 5-22533 0.81269 5.27416 0.40760 5.13394 2.09184 5.28496 1.68916 5.40341 1.27607 5.48853 0.85511 5.53981 0.42888 5.39298 2.20082 5.55162 1.77716 5.67603 1.34255 5.76545 0.89966 5.81933 0.45122 2.O 4.79657 o.oo 5.03722 o.oo 5.29047 o.oo 5.55<595 o.oo 5.83732 o.oo Note. Negative quantities are in heavy type. Examples, cosh (2.4 + 10.4) = 4-4956? + * 3-21297- cosh (2.4 + i 1.5) = 3-92935 + * 3-86521 [67] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED q x = 2.5 x = 2.55 X = 2.6 x = 2.65 x = 2.7 0.05 O.I 0.15 0.2 6.13229 o.oo 6.II339 0.47469 6.05680 0.94646 5.96287 I.4I239 5.83215 1.86961 6.44259 o.oo 6.42273 0.49935 6.36327 0.99563 6.26458 1.48577 6.12727 1.96674 6.76901 o.oo 6.74814 0.52526 6.68567 1.04729 6.58199 1.56285 6.43770 2.06878 7.11234 o.oo 7.09042 0.55249 7.02478 1.10156 6.91583 1.64385 6.76424 2.17600 7-47347 o.oo 7.45043 0.58109 7.38146 1.15859 7.26697 1.72896 7.10769 2.28866 0.25 0-3 0-35 0.4 0-45 5.66550 2.3I53I 5.46392 2.74674 5.22864 3.l6l22 4.96113 3.55622 4.66304 3.92929 5.95218 2.43559 5.74039 2.88943 5.49321 3.32545 5.21217 3.74097 4.89898 4-13342 6.25374 2.56196 6.03123 3.03934 5-77153 3-49799 5.47624 3.93506 5.14719 4-34788 6.57095 2.69474 6.33714 3- I 9686 6.06427 3.67927 5.75401 4-13900 5.40827 4-57321 6.90458 2.83425 6.65891 3.36237 6.37218 3.86976 6.04606 4-35329 5.68287 4.80998 0-5 0-55 0.6 0.65 0.7 4.33619 4.27814 3.98260 4.60062 3.60448 4-89472 3.204II 5.15865 2.78401 5.39077 4.55560 4.50039 4.18413 4.83961 3.78686 5.14900 3.36624 5.42664 2.92488 5.67082 4.78641 4-73389 4.39612 5.09071 3.97872 5-41615 3.53680 5.70819 3.07306 5.96505 5.02919 4.97923 4.61910 5.35454 4.18053 5.69685 3.71619 6.00403 3.22894 6.27419 5.28454 5.23702 4-85363 5-63177 4.39279 5.99I79 3.90488 6.31488 3.39288 6.59903 0-75 0.8 0.85 0.9 0-95 2.34673 5.S8966 1.89498 5.75408 I.43I5S 5.88304 0-9593 5-97572 0.48113 6.03155 2.46547 5.88004 1.99087 6.05301 1.50399 6.18866 1.00784 6.28615 0.50548 6.34489 2.59039 6.18512 2.09174 6.36706 1.58019 6.50976 1.05891 6.61231 0.53109 6.67409 2.72178 6.50567 2.19784 6.69705 1.66034 6.84713 1.11262 6.95500 0-55803 7.01998 2.85997 6.84249 2-30943 7-04377 1.74465 7.20163 1.16911 7.31508 0.58636 7.38343 I.O 1.05 i.i i.iS 1.2 o.oo 6.05020 0.48113 6.03155 0.95930 5-97572 1.43155 5.88304 1.89498 5.75408 o.oo 6.36451 0.50548 6.34489 1.00784 6.28615 1.50399 6.18866 1.99087 6.05301 o.oo 6.69473 0.53109 6.67409 1.05891 6.61231 1.58019 6.50976 2.09174 6.36706 o.oo 7.04169 0.55803 7.01998 1.11262 6.95500 1.66034 6.84713 2.19784 6.69705 o.oo 7.40626 0.58636 7.38343 1.16911 7.31508 1.74465 7.20163 2.30943 7-04377 1.25 i-3 i-35 1.4 1-45 2.34673 5.58966 2.78401 5.39077 3.20411 5.15865 3.60448 4.89472 3.98260 4.60062 2.46547 5.88004 2.92488 5.67082 3.36624 5.42664 3.78686 5.14900 4.18413 4.83961 2.59039 6.18512 3.07306 5.96505 3.53680 5.70819 3.97872 5.41615 4.39612 5.09071 2.72178 6.50567 3.22894 6.27419 3.71619 6.00403 4.18053 5.69685 4.61910 5.35454 2.85997 6.84249 3.39288 6.59903 3.90488 6.31488 4-39279 5-99179 4-85363 5-63177 i-5 i-55 1.6 1.65 i-7 4.33619 4.27814 4.66304 3.92929 4.96113 3.55622 5.22864 3.16122 5.46392 2.74674 4.55560 4.50039 4.89898 4.13342 5.21217 3.74097 5.49321 3.32545 5.74039 2.88943 4.78641 4.73389 5.14719 4.34788 5.47624 3.93506 5-77153 3-49799 6.03123 3.03934 5.02919 4.97923 5.40827 4-573 21 5.75401 4.13900 6.06427 3.67927 6.33714 3-19686 5.28454 5.23702 5.68287 4.80998 6.04606 4-35329 6.37218 3.86976 6.65891 3.36237 i-75 1.8 1.85 1.9 1-95 5.66550 2.31531 5.83215 1.86961 5.96287 1.41239 6.05680 0.94646 6.1 1339 0.47469 5.95218 2.43559 6.12727 1.96674 6.26458 1.48577 6.36327 0.99563 6.42273 0.49935 6.25374 2.56196 6.43770 2.06878 6.58199 1.56285 6.68567 1.04729 6.74814 0.52526 6.57095 2.69474 6.76424 2.17600 6.91583 1.64385 7.02478 1.10156 7.09042 0.55249 6.90458 2.83425 7.10769 2.28866 7.26697 1.72896 7.38146 1.15859 7.45043 0.58109 2.0 6.13229 0.00 6.44259 o.oo 6.76901 o.oo 7.11234 o.oo 7-47347 o.oo Note. Negative quantities are in heavy type. Examples, cosh (2.7 + * i.oo) = o + i 7.40626. cosh (2.6 + j 1.2) = 2.09174 + i 6.36706. [68] TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED x = 2.75 x = 2.80 x = 2.85 x = 2.90 x = 2.95 o 0.05 O.I 0.15 O.2 7.85328 o.oo 7.82907 0.61115 7.75659 1.21852 7.63629 1.81839 7.46891 2.40704 8.25273 o.oo 8.22728 0.64273 8.15112 1.28150 8.02470 1.91237 7.84881 2.53144 8.67281 o.oo 8.64608 0.67592 8.56604 1.34768 8.43318 2.OIII2 8.24834 2.66217 9.11458 o.oo 9.08649 0.71081 9.00237 1.41723 8.86275 2.11491 8.66850 2.79956 9.57915 o.oo 9.54962 0.74747 9.46121 1.49032 9.31447 2.22399 9.11031 2.94395 0.25 0-3 o-35 0.4 0-45 7.25548 2.98086 6.99732 3.53629 6.69602 4.06993 6.35344 4-57847 5.97168 5.05878 7.62452 3.13491 7-35323 3-71905 7.03661 4.28026 6.67660 4.81509 6.27542 5.32022 8.01263 3.29681 7.72754 3-9HI2 7-39479 4-50I3I 7.01646 5.06375 6.59486 5.59498 8.42078 3.46694 8.12115 4.11295 7.77146 4-73361 7-37385 5-32508 6.93078 5.88371 8.84998 3.64575 8.53508 4.32508 8.16757 4-97774 7.74969 5.63048 7.28404 6.18717 o-S 0-55 0.6 0.65 0.7 5-55311 5-50790 5.10030 5.92307 4.61604 6.30172 4.10333 6.64151 3.56531 6.94036 5-83556 5-79256 5.35972 6.22918 4.85083 6.62740 4.31204 6.98476 3.74666 7.29905 6.13261 6.09170 5.63254 6.55088 5.09775 6.96966 4.53I53 7-34547 3.93738 7.67599 6.44498 6.40608 5.91945 6.88894 5-35742 7-32934 4.76236 7.72455 4.13793 8.07213 6.77348 6.73647 6.22116 7.24424 5.63048 7.70735 5.00509 8.12294 4.34884 8.48845 0-75 0.8 0.85 0.9 o-95 3.00532 7.19642 2.42680 7.40811 1-83331 7-57413 1.22852 7.69345 0.61616 7.76534 3.15818 7.56834 2.55023 7-79098 1.92656 7-96557 1.29101 8.09106 0.64750 8.16666 3.31894 7-959I9 2.68005 8.19332 2.02463 8.37694 I-35673 8.50891 0.68046 8.58841 3.48800 8.36994 2.81656 8.61616 2.12776 8.80924 1.42583 8.94802 0.71512 9.03163 3.66578 8.80162 2.96012 9.06053 2.23621 9.26358 1.49851 9.40952 0-75*57 9-49744 I.O 1.05 i.i i-iS 1.2 o.oo 7-78935 0.61616 7-76534 1.22852 7.69345 1.83331 7-574I3 2.42680 7.40811 o.oo 8.19192 0.64750 8.16666 1.29101 8.09106 1.92656 7-96557 2.55023 7-79098 o.oo 8.61497 0.68046 8.58841 1-35673 8.50891 2.02463 8.37694 2.68005 8.19332 o.oo 9-05956 0.71512 9.03163 1.42583 8.94802 2.12776 8.80924 2.81656 8.61616 o.oo 9.52681 0.75157 9-49744 1.49851 9.40952 2.23621 9.26358 2.96012 9.06053 1-25 i-3 1-35 1.4 1-45 3.00532 7.19642 3-5653I 6.94036 4.10333 6.64151 4.61604 6.30172 5.10030 5.92307 3.15818 7-56834 3.74666 7.29905 4.31204 6.98476 4.85083 6.62740 5-35972 6.22918 3.31894 7.95919 3.93738 7-67599 4.53I53 7-34547 5.09775 6.96966 5.63254 6.55088 3.48800 8.36994 4-13793 8.07213 4.76236 7.72455 5-35742 7-32934 5.91945 6.88894 3.66578 8.80162 4.34884 8.48845 5.00509 8.12294 5.63048 7.70735 6.22116 7.24424 i-5 1-55 1.6 1.65 1-7 5-553II 5.50790 5.97168 5.05878 6-35344 4-57847 6.69602 4.06993 6-99732 3-53629 5.83556 5-79256 6.27542 5.32022 6.67660 4.81509 7.03661 4.28026 7.35323 3-71905 6.13261 6.09170 6.59486 5.59498 7.01646 5.06375 7.39479 4-50131 7.72754 3-9 1112 6.44498 6.40608 6.93078 5.88371 7-37385 5-32508 7.77146 4.73361 8.12115 4.11295 6.77348 6.73647 7.28404 6.18717 7.74969 5-63048 8.16757 4-97774 8.53508 4.32508 i-75 1.8 1.85 1.9 1-95 7.25548 2.98086 7.46891 2.40704 7.63629 1.81839 7.75659 1-21852 7.82907 0.61115 7.62452 3.1349! 7.84881 2.53144 8.02470 1.91237 8.15112 1.28150 8.22728 0.64273 8.01263 3.29681 8.24834 2.66217 8.43318 2.0III2 8.56604 1.34768 8.64608 0.67592 8.42078 3.46694 8.66850 2.79956 8.86275 2.11491 9.00237 1.41723 9.08649 0.71081 8.84998 3.64575 9.11031 2.94395 9.31447 2.22399 9.46121 1.49032 9.54962 0.74747 2.0 7.85328 o.oo 8.25273 o.oo 8.67281 o.oo 9.11458 o.oo 9.579I5 0.00 Note. Negative quantities are in heavy type. Examples, cosh (2.95 + to) = 9.57915 + *'o. cosh (2.8 + i 1.2) = 2.55023 + * 7-79098. [69] TABLE VIII. HYPERBOLIC COSINES, cosh (* + ig) = u + iv. CONTINUED x = 3-0 3-S x = 3.10 = 3-iS = 3.20 o 10.06766 o.oo 10.58135 o.oo 11.12150 o.oo 11.68946 o.oo 12.28665 0.00 0.05 10.03660 0.78599 10.54870 0.82649 11.08720 0.86905 11.65340 0.91378 12.24880 0.96080 O.I 9-94371 1.56714 10.45110 1.64788 10.98460 1.73274 11.54550 1.82193 12.13540 1.91568 0.15 9.78949 2.33863 10.28900 2.45911 10.81420 2.58575 11.36650 2.71885 11.94720 2.85874 O.2 9-57492 3-09569 10.06350 3.25518 10.57720 3.42281 11.11730 3.59900 11.68530 3.78419 0.25 9.30131 3.83368 9-77589 4.03120 10.27490 4.23878 10.79970 4-45696 11.35140 4.68630 o-3 8.97035 4.54806 9-42805 4-78233 9-90933 5.02860 10.41540 5-28745 10.94750 5-55952 o.3S 8.58409 5-23433 9.02209 5.50400 9.48264 5-78743 9.96690 6.08533 10.47610 6.39846 0.4 8.14491 5.88836 8.56049 6.19173 8.99749 6.51058 9-45697 6.84570 9.94011 .7-19795 0-45 7-655SI 6.50609 8.04612 6.84128 8.45686 7.19358 8.88873 7-56387 9.34284 7-95306 o-S 7.11891 7.08371 7.48215 7.44866 7.86409 7-83223 8.26570 8.23539 8.68797 8.65915 o-SS 6.53842 7.61765 6.87204 8.OIOH 7.22284 8.42260 7.59170 8.85615 7.97954 9.31184 0.6 5.91762 8.10463 6.21956 8.52218 6.53705 8.96104 6.87089 9.42230 7.22191 9.90712 0.65 5-26034 8.54164 5-52874 8.98171 5.81097 9.44423 6.10773 9-93036 6.41976 10.44130 0.7 4.57062 8.92599 4.80383 9.38586 5.04906 9.86919 5.30690 10.37720 5.57802 10.91120 0-75 3-85273 9-25531 4.04931 9.73216 4.25602 10.23330 4-47336 10.76010 4.70190 11.31370 0.8 3.11108 9-52757 3.26982 10.01840 3-43673 10.53430 3.61224 11.07660 3.79678 11.64650 0.85 2.35025 9.74108 2.47017 10.24290 2.59626 10.77040 2.72885 11.32480 2.86826 11.90750 0.9 1-57493 9-89454 1.65529 10.40430 1-73979 10.94010 1.82863 11.50320 1.92205 12.09510 Q-9S o. 78990 9.98699 0.83020 10.50150 0.87258 11.04230 0.91714 11.61070 0.96400 12.20810 I.O 0.00 10.01787 o.oo 10.53399 0.00 11.07645 o.oo 11.64661 o.oo 12.24588 1.05 0.78990 9.98699 0.83020 10.50150 0.87258 11.04230 0.91714 11.61070 0.96400 12.20810 i.i i -57493 9-89454 1.65529 10.40430 1.73979 10.94010 1.82863 11.50320 1.92205 12.09510 i-i5 2.35025 9.74108 2.47017 10.24290 2.59626 10.77040 2.72885 11.32480 2.86826 11.90750 1.2 3.11108 9-52757 3.26982 10.01840 3.43673 10.53430 3.61224 11.07660 3.79678 11.64650 1-25 3-85273 9-25531 4.04931 9.73216 4.25602 10.23330 4-47336 10.76010 4.70190 11.31370 i-3 4-57062 8.92599 4.80383 9.38586 5.04906 9.86919 5.30690 10.37720 5.57802 10.91120 1.35 5-26034 8.54164 5.52874 8.98171 5.81097 9-44423 6.10773 9.93036 6.41976 10.44130 1.4 5-91762 8.10463 6.21956 8.52218 6.53705 8.96104 6.87089 9.42230 7.22191 9.90712 i-45 6.53842 7.61765 6.87204 8.01011 7.22284 8.42260 7.59170 8.85615 7-97954 9.31184 i-S 7.11891 7.08371 7.48215 7.44866 7.86409 7.83223 8.26570 8.23539 8.68797 8.65915 i-5S 7.65551 6.50609 8.04612 6.84128 8.45686 7.19358 8.88873 7.56387 9.34284 7-95306 1.6 8.14491 5.88836 8.56049 6.19173 8.99749 6.51058 9.45697 6.84570 9.94011 7-19795 1.65 8.58409 5-23433 9.02209 5.50400 9.48264 5-78743 9.96690 6.08533 10.47610 6.39846 i-7 8.97035 4.54806 9.42805 4-78233 9.90933 5.02860 10.41540 5.28745 10.94750 5-55952 1-75 9-30131 3.83368 9.77589 4.03120 10.27490 4.23878 10.79970 4-45696 11.35140 4.68630 1.8 9-57492 3-09569 10.06350 3-255i8 10.57720 3.42281 11.11730 3-59900 11.68530 3.78419 1.85 9.78949 2.33863 10.28900 2.45911 10.81420 2.58575 11.36650 2.71885 11.94720 2.85874 1.9 9-94371 1.56714 10.45110 1.64788 10.98460 1.73274 11.54550 1.82193 12.13540 1.91568 1-95 10.03660 0.78599 10.54870 0.82649 11.08720 0.86905 11.65340 0.91378 12.24880 0.96080 2.0 10.06766 0.00 10.58135 o.oo 11.12150 o.oo 11.68946 o.oo 12.28665 o.oo \ Note. Negative quantities are in heavy type. Examples, cosh (3.10 + i 0.5) = 7.86409 + * 7.83223. cosh (3.10 + i 1.55) = -8.45686 + i 7.19358. [7o] TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED = 3-25 x = 3.30 = 3-35 = 3-40 3-45 12.91456 o.oo I3-57476 o.oo 14.26891 o.oo 14.99874 o.oo 15-76607 o.oo 0.05 12.87474 I.OIO22 I3-53290 1.06217 14.22498 1.11677 14.95250 1.17417 15.71746 1.23450 O.I 12.75555 2.OI422 13.40764 2.11780 14.09323 2.22666 14.81410 2.34110 I5-57I96 2.46139 0.15 12-55773 3-00579 13.19970 3.16036 13.87465 3.32282 14.58432 3-49359 I5-33045 3.673" 0.2 12.28247 3.97883 12.91035 4.18343 I3-57054 4.39850 14.26465 4.62455 14.99442 4.86217 0.25 11.93150 4-92735 I2.54U5 5.18072 13.18275 5-44705 13.85702 5.72700 14.56595 6.02127 o-3 11.50695 5.84548 I2.O952O 6.14670 12.71369* 6.46202 13-36397 6.79414 14.04764 7.14324 o-35 11.01147 6.72758 11.57440 7.07352 12.16623 7.43715 12.78852 7-81938 13.44278 8.22116 0.4 10.44810 7.56820 IO.98222 7.95737 "54379 8.36643 12.13423 8.79642 12.75501 9.24840 o.45 9.82031 8.36217 10.32233 8.79215 10.85016 9.24413 11.40512 9.71923 11.98861 10.21863 0-5 9.13197 9.10456 9.59881 9.57273 10.08964 10.06483 10.60570 10.58212 11.14830 11.12585 o-SS 8.38733 9.79082 8.81610 10.29428 9.26691 10.82349 9.74090 "-37975 10.23924 11.96447 0.6 7.59099 10.41673 7-97905 10.95235 8.38705 11.51541 8.81604 12.10723 9.26706 12.72933 0.65 6.74784 10.9/841 7.09279 11.54294 7-45549 12.13633 7.83682 12.76006 8.23775 13.41572 0.7 5.86309 II.4724O 6.16381 12.06234 6-47795 12.68244 6.80928 I 3-33423 7-I5765 14.01940 0-75 4.94218 11.89566 5-I9485 12.50736 5-46075 13-15033 5-73977 13.82620 6.03341 14.53662 0.8 3.99082 I2.2456O 4.19483 12.87530 4.40933 13-53717 4.63487 14.23291 4.87198 14.96423 0.85 3.01484 I2.52OO2 3.16897 13.16388 3-33IOI 13.84054 3-50I39 14.55188 3-68053 15.29960 0.9 2.02028 12.71726 2.21356 13.37120 2.23215 14.05851 2.34632 14.78111 2.46636 15.54061 0-95 1.01326 12.83610 1.06507 13.49615 I-II953 14.18950 1.17679 14.91923 1.23699 15.68581 I.O 0.00 12.87578 o.oo 13.53788 o.oo 14.23382 0.00 14.96536 0.00 15.73432 1.05 1.01326 12.83610 1.06507 13.49615 1.11953 14.18950 1.17679 14.91923 1.23699 15.68581 i.i 2.02028 12.71726 2.12356 13.37120 2.23215 14.05851 2.34632 14.78111 2.46636 15.54061 i-iS 3.01484 I2.52OO2 3-16897 13.16388 3.33101 13.84054 3.50139 14.55188 3.68053 15.29960 1.2 3.99082 12.24560 4.19483 12.87530 4-40933 13-53717 4.63487 14.23291 4.87198 14.96423 1-25 4.94218 11.89566 5.19485 12.50736 5-46075 13-15033 5-73977 13.82620 6.03341 14.53662 1-3 5.86309 II.4724O 6.16381 12.06234 6.47795 12.68244 6.80928 I3-33423 7.15765 14.01940 35 6.74784 10.9/841 7.09279 11.54294 7-45549 12.13633 7.83682 12.76006 8.23775 13.41572 4 7-59099 10.41673 7.97905 10.95235 8.38705 11-51541 8.81604 12.10723 9.26706 12.72933 45 8.38733 9.79082 8. 81610 10.29428 9.26691 10.82349 9.74090 "37975 10.23924 11.96447 5 9-13197 9.10456 9.59881 9.57273 10.08964 10.06483 10.60570 10.58212 11.14830 11.12585 55 9.82031 8.36217 10.32233 8.79215 10.85016 9.24413 11.40512 9.71923 11.98861 10.21863 .6 10.44810 7.56820 10.98222 7-95737 11-54379 8.36643 12.13423 8.79642 12.75501 9.24840 1.65 11.01147 6.72758 11.57440 7-07352 12.16623 7.43715 12.78852 7-81938 13.44278 8.22116 1-7 11.50695 5-84548 12.09520 6.14670 12.71369 6.46202 13-3639? 6.79414 14.04764 7.14324 1-75 11.93150 4.92735 12.54145 5.18072 13-18275 5-44705 13.85702 5.72700 14.56595 6.02127 1.8 12.28247 3.97883 12.91035 4-18343 I3-57054 4-39850 14.26465 4-62455 14.99442 4.86217 1.85 12.55773 3-00579 13.19970 3.16036 13-87465 3.32282 14.58432 3-49359 15.3304S 3.67311 1.9 12.75555 2.01422 13.40764 2.11780 14.09323 2.22666 14.81410 2.34110 15-57196 2.46139 1.95 12.87474 I.OIO22 I3-53290 1.06217 14.22498 1.11677 14.95250 1.17417 15.71746 1.23450 2.0 12.91456 o.oo 13.57476 o.oo 14.26891 o.oo 14.99874 o.oo 15.76607 o.oo Note. Negative quantities are in heavy type. Examples, cosh (3.45 + i 0.05) = 15.71746 + i 1.23450. cosh (3.25 + i 1.95) = 12.87474 + * 1.01022. TABLE VIII. HYPERBOLIC COSINES, cosh (x + iq) = u + iv. CONTINUED = 3-50 = 3-55 3.60 x = 3-65 x = 3.70 o 16.57282 o.oo 17.42102 o.oo 18.31278 o.oo 19.25033 o.oo 20.23601 o.oo 0.05 16.52173 1.29792 17.36731 1.36458 18.25632 1.43466 19.19100 1.50832 20.17362 1.58576 O.I 16.36878 2.58783 17.20653 2.72075 18.08732 2.86047 19.01331 3-00735 19.98688 3.16174 0.15 16.11491 3.86180 16.93970 4.06015 17.80680 4.26865 18.71843 4.48783 19.67690 4.71823 O.2 15.76170 5.11195 16.56840 5.37451 17.41650 5.65052 18.30814 5-94065 19.24560 6.24563 0.25 i5-3"3o 6.33059 16.09492 6.65574 16.91880 6.99754 17.78497 7.35683 18.69565 7-73453 0-3 14.76650 7.51020 15-52225 7.89594 16.31680 8.30143 17.1527.6 8.72766 18.03040 9-17573 o-3S 14.13065 8.64360 14.85386 9-08745 15.61421 9-55412 16.41360 10.04469 17.25404 10.56037 0.4 13.40770 9.72361 14.09390 10.22293 14-81535 10.74789 15-57383 11.29972 16.37127 11.87990 0-45 12.60210 10.74357 13.24705 11.29540 13-92515 11.87545 14.63806 12.48520 15.38760 13.12620 o.S 11.71820 11.69740 12.31852 12.29820 12.94910 12.92978 13.61203 13-59365 14.30902 14-29155 o-55 10.76320 12.57910 11.31405 13.22520 11.89320 13.90436 12.50208 14.61830 13.14223 15.36878 0.6 9.74126 13.38330 10.23982 14.07066 10.76400 14.79326 11.31505 15-55281 11.89444 16.35127 0.65 8.65928 14.10490 9.10246 14.82933 9.56840 15.59090 10.05827 16.39143 10.57330 17.23295 0.7 7-52391 14.73960 7.90898 15.49665 8.31383 16.29246 8.73946 17.12900 9.18696 18.00840 0-75 6.34215 15.28340 6.66674 16.06840 7.00800 16.89356 7.36678 17.76096 7-74399 18.67280 0.8 5.12129 15-73300 5-38339 16.54105 5.65896 17.39050 5.94868 18.28342 6.25327 19.22208 0.85 3.86885 16.08555 4.06686 16.91175 4-27503 17.78022 4.49390 18.69316 4.72401 19.65290 0.9 2.59256 16.33896 2.72525 17.17817 2.86475 18.06033 3.01142 18.98765 3.16561 19.96246 o-9S 1.30029 16.49163 1.36684 17-33870 1.43680 18.22900 1.51036 19.16506 1.58770 20.14900 I.O o.oo 16.54263 0.00 17.39230 0.00 18.28546 o.oo 19.22434 0.00 20.21129 1.05 1.30029 16.49163 1.36684 17.33870 1.43680 18.22900 1.51036 19.16506 1.58770 20.14900 i.i 2.59256 16.33896 2.72525 17.17817 2.86475 18.06033 3.01142 18.98765 3.16561 19.96246 i-iS 3.86885 16.08555 4.06686 16.91175 4.27503 17.78022 4.49390 18.69316 4.72401 19.65290 1.2 5.12129 15-73300 5-38339 16.54105 5.65896 17.39050 5.94868 18.28342 6.25327 19.22208 I-2S 6.34215 15.28340 6.66674 16.06840 7.00800 16.89356 7.36678 17.76096 7-74399 18.67280 3 7.52391 14.73960 7.90898 15.49665 8.31383 16.29246 8.73946 17.12900 9.18696 18.00840 35 8.65928 14.10490 9.10246 14.82933 9.56840 15.59090 10.05827 16.39143 io.5733o 17.23295 4 9.74126 13-38330 10.23982 14.07066 10.76400 14.79326 11.31505 15-55281 11.89444 16.35127 45 10.76320 12.57910 11.31405 13.22520 11.89320 13.90436 12.50208 14.61830 13.14223 15.36878 5 11.71820 11.69740 12.31852 12.29820 12.94910 12.92978 13.61203 13-59365 14.30902 14-29155 55 12.60210 10.74357 13.24705 11.29540 13.92515 11-87545 14.63806 12.48520 15.38760 13.12620 .6 13.40770 9.72361 14.09390 10.22293 14.81535 10.74789 15.57383 11.29972 16.37127 11.87990 65 14.13065 8.64360 14.85386 9.08745 15.61421 9-55412 16.41360 10.04469 17.25404 10.56037 7 14.76650 7.51020 15.52225 7-89594 16.31680 8.30143 17.15216 8.72766 18.03040 9-!7573 i-7S 15.31130 6.33059 16.09492 6.65574 16.91880 6.99754 17.78497 7.35683 18.69565 7-73453 1.8 15.76170 S-n^S 16.56840 5-37451 17.41650 5-65052 18.30814 5-94065 19.24560 6.24563 1.85 16.11491 3.86180 16.93970 4.06015 17.80680 4.26865 18.71843 4.48783 19.67690 4.71823 1.9 16.36878 2.58783 17.20653 2.72075 18.08732 2.86047 19.01331 3-00735 19.98688 3.16174 i-95 16.52173 1.29792 17.36731 1.36458 18.25632 1.43466 19.19100 1.50832 20.17362 I-58576 2.O 16.57282 0.00 17.42102 0.00 18.31278 o.oo 19.25033 o.oo 20.23601 o.oo Note. Negative quantities are in heavy type. Examples, cosh (3.50 + * 0.70) = 7.52391 + * 14.73960 . cosh (3.60 + * 1.55) = -13-92515 + * ".87545- [72] TABLE VIII. HYPERBOLIC COSINES, cosh (* + iq) = u + iv. CONTINUED * = 3-75 3-80 3-85 x = 3.90 3-95 o 21.27230 o.oo 22.36178 o.oo 23.50717 o.oo 24-71135 o.oo 25-97731 o.oo 0.05 21.20670 1.66716 22.29283 1-75273 23.43470 1.84268 24.63516 1.93724 25.89724 2.03665 O.I 21.01038 3-32404 22.08646 3-49465 23.21775 3.67400 24.40710 3.86254 25-65749 4.06074 0.15 20.68452 4.96043 21.74391 5-21503 22.85766 5-48267 24.02856 5.76402 25-25957 6.05979 O.2 20.23113 6.56624 21.26731 6.90325 22.35664 7.25754 23.50188 7.62997 24.70590 8.02149 0.25 19.65301 8.13156 20.65958 8.54892 21.71778 8.98766 22.83030 9.44887 23.99991 9-93373 0-3 i8.95373 9.64674 19.92448 10.14188 20.94503 10.66233 22.01797 11.20952 23-14597 11.78474 o-3S 18.13760 11.10246 19.06655 11.67230 20.04315 12.27134 21.06988 12.90106 22.14931 13-56305 0.4 17.20963 12.48971 18.09105 13.13076 19.01770 13.80465 19.99190 14-51307 21.01610 I5-25776 0.45 16.17556 13.79998 1 7.00402 14.50828 17.87500 15.25286 18.79065 16.03558 19.75331 16.85842 o-S 15.04177 15.02516 15.81216 15-79634 16.62208 16.60702 17-47355 17.45924 18.36873 18.35512 o-SS 13.81524 16.15770 14.52281 16.98701 15.26668 17.85880 16.04874 18.77526 16.87098 19.73867 0.6 12.50353 17.19062 13-14392 18.07296 13.81716 19.00048 14.52497 19-97556 15.26910 21.00052 0.65 11.11473 18.11756 11.68400 19.04746 12.28246 20.02501 12.91164 21.05272 13-57315 22.13290 0.7 9.65741 18.93280 10.15203 19.90455 10.67203 20.92608 11.21871 21.99993 11.79366 23.12881 o.7S 8.14055 19.63131 8.55748 20.63891 8.99580 21.69813 9.45662 22.81160 9.94109 23.98212 0.8 6.57350 20.20879 6.91017 21.24603 7.26411 22.33640 7.63623 23.48262 8.02744 24.68760 0.85 4.96592 20.66167 5.22025 21.72216 5.48764 22.83696 5.76875 24.00888 6.06429 25.24084 0.9 3.32772 20.98716 3-49815 22.06437 3-67733 23.19673 3-86571 24.38710 4.06375 25.63849 0-95 1.66900 21.18327 1.75448 22.27052 1-84435 23-41348 1.93883 24.61500 2.03816 25-87805 I.O o.oo 21.24878 0.00 22.33941 o.oo 23.48589 0.00 24.69110 o.oo 25.95806 05 1.66900 21.18327 1.75448 22.27052 1.84435 23.41348 1.93883 24.61500 2.03816 25.87805 .1 3.32772 20.98716 3.49815 22.06437 3.67733 23.19673 3.86571 24.38710 4.06375 25.63849 IS 4.96592 20.66167 5-22025 21.72216 5.48764 22.83696 5.76875 24.00888 6.06429 25.24084 .2 6.57350 20.20879 6.91017 21.24603 7.26411 22.33640 7.63623 23.48262 8.02744 24.68760 2 S 8.14055 19-63131 8.55748 20.63891 8.99580 21.69813 9.45662 22.81160 9.94109 23.98212 3 9.65741 18.93280 10.15203 19.90455 10.67203 20.92608 11.21871 21.99993 11.79366 23.12881 i-3S 11.11473 18.11756 11.68400 19.04746 12.28246 20.02501 12.91164 21.05272 13.57315 22.13290 1.4 12.50353 17.19062 13.14392 18.07296 13.81716 19.00048 14.52497 19-97556 15.26910 21.00052 MS 13.81524 16.15770 14.52281 16.98701 15.26668 17.85880 16.04874 18.77526 16.87098 19.73867 *$ 15.04177 15.02516 15.81216 15-79634 16.62208 16.60702 17.47355 17.45924 18.36873 18.35512 x -55 16.17556 13.79998 17.00402 14.50828 17.87500 15.25286 18.79065 16.03558 19.75331 16.85842 1.6 17.20963 12.48971 18.09105 13.13076 19.01770 13.80465 19.99190 14.51307 21.01610 15-25776 1.65 18.13760 11.10246 19.06655 11.67230 20.04315 12.27134 21.06988 12.90106 22.14931 13-56305 .? 18.95373 9.64674 19.92448 10.14188 20.94503 10.66233 22.01797 11.20952 23.14597 11.78474 1-75 19.65301 8.13156 20.65958 8.54892 21.71778 8.98766 22.83030 9.44887 23.99991 9-93373 1.8 20.23113 6.56624 21.26731 6.90325 22.35664 7.25754 23.50188 7.62997 24.70590 8.02149 1.85 20.68452 4.96043 21.74391 5-21503 22.85766 5.48267 24.02856 5.76402 25.25957 6.05979 1.9 21.01038 3-32404 22.08646 3-49465 23.21775 3.67400 24.40710 3.86254 25.65749 4.06074 i-9S 21.20670 1.66716 22.29283 1-75273 23.43470 1.84268 24.635*6 1.93724 25.89724 2.03665 2.0 21.27230 o.oo 22.36178 o.oo 23.50717 o.oo 24.71135 o.oo 35.97731 0.00 Note. Negative quantities are in heavy type. Examples, cosh (3.90 + i 0.25) = 22.83030 + i 9.44887. cosh (3.75 + i 1.25) = -8.14055 + i 19.63131. [73] TABLE IX. HYPERBOLIC TANGENTS, tanh (* + iq) = u + iv x = 0.05 x = 0.15 X = 0.2 o.oo o.oo 0.04996 o.oo 0.09967 o.oo 0.14889 o.oo 0.19738 o.oo 0.05 o.oo 0.07870 0.05027 0.07850 0.10028 0.07792 0.14979 0.07695 0.19855 0.07562 O.I o.oo 0.15838 0.05121 0.15798 0.10214 0.15677 0.15254 0.15479 0.20213 0.15207 0.15 o.oo 0.24008 0.05283 0.23944 0.10535 0.23755 0.15727 0.23446 0.20828 0.23021 0.2 o.oo 0.32492 0.05522 0.32402 0.11008 0.32136 0.16422 0.31698 0.21732 0.31098 0.25 o.oo 0.41421 0.05850 0.41300 0.11657 0.40940 0.17377 0.40350 0.22970 0-39543 o-3 0.00 0.50953 0.06289 0.50792 0.12522 0.50317 0.18647 0.49538 0.24613 0.48477 0-35 o.oo 0.61280 0.06865 0.61070 0.13659 0.60446 0.20310 0.59427 0.26758 0.58044 0.4 o.oo 0.72654 0.07623 0.72378 0.15149 0.71557 0.22484 0.70222 0.29549 0.68417 0-45 o.oo 0.85408 0.08624 0.85040 0.17113 0.83951 0.25339 0.82186 0.33192 0.79813 o.S o.oo I.OOOO 0.09967 0.99503 0.19738 0.98033 0.29131 0.95663 0.37995 0.92501 0-55 o.oo 1.17085 0.11804 1-16395 0.23313 1.14365 0.34258 1. 11113 0.44423 1.06819 0.6 o.oo 1.37638 0.14392 1.36649 0.28315 1-33754 0.41357 1.29164 0.53203 1-23185 0.65 o.oo 1.63185 0.18179 1.61702 0.35567 1.57401 0.51496 1.50674 0.65502 1.42088 0.7 o.oo 1.96261 0.24007 1.93900 0.46575 1.87150 0.66202 1.75880 0.83268 1.64005 0-75 o.oo 2.41421 0.33624 2.37365 0.64333 2.25941 0.90034 2.09061 1.09837 1.89083 0.8 o.oo 3.07768 0.51109 2.99911 0.95397 2.78504 1.28858 2.48723 1.50982 2.16055 0.85 o.oo 4.16530 0.87867 3.98246 1.56000 3-51765 1.97316 2.94167 2.16111 2.38860 0.9 o.oo 6.31375 1.85674 5.72808 2.91746 4.47780 3.22989 3-27758 3-15925 2.37676 0-95 o.oo 12.70620 5.79801 9.05499 6.21808 4-83133 5.28217 2.71349 4.39854 1.67517 I.O o.oo 00 20.01667 o.oo 10.03331 o.oo 6.71659 o.oo 5.06649 o.oo 1.05 o.oo 12.70620 5.79801 9.05499 6.21808 4.83133 5.28217 2.71349 4.39854 1.67517 i.i o.oo 6.31375 1.85674 5.72808 2.91746 4.47780 3.22989 3.27758 3-15925 2.37676 i.iS o.oo 4.16530 0.87867 3.98246 1.56000 3.51765 1.97316 2.94167 2.16111 2.38860 1.2 o.oo 3.07768 0.51109 2.99911 0.95397 2.78504 1.28858 2.48723 1.50982 2.16055 I-2S o.oo 2.41421 0.33624 2.37365 0.64333 2.25941 0.90034 2.09061 1.09837 1.89083 1-3 o.oo 1.96261 0.24007 1.93900 0.46575 1.87150 0.66202 1.75880 0.83268 1.64005 i-3S o.oo 1.63185 0.18179 1.61702 0.35567 1.57401 0.51496 1.50674 0.65502 1.42088 1.4 o.oo 1.37638 0.14392 1.36649 0.28315 1-33754 0.41357 1.29164 0.53203 1.23185 1-45 0.00 1.17085 0.11804 1.16395 0.23313 1.14365 0.34258 I. IXII3 0.44423 1.06819 i-S o.oo I.OOOO 0.09967 0.99503 0.19738 0.98033 0.29131 0.95663 0-37995 0.92501 i-S5 o.oo 0.85408 0.08624 0.85040 0.17113 0.83951 0.25339 0.82186 0.33192 0.79813 1.6 o.oo 0.72654 0.07623 0.72378 0.15149 0.71557 0.22484 0.70222 0.29549 0.68417 1.65 o.oo 0.61280 0.06865 0.61070 0.13659 0.60446 0.20310 0.59427 0.26758 0.58044 i-7 o.oo 0.50953 0.06289 0.50792 0.12522 0.50317 0.18647 0.49538 0.24613 0.48477 i-75 o.oo 0.41421 0.05850 0.41300 0.11657 0.40940 0.17377 0.40350 0.22970 0.39543 1.8 o.oo 0.32492 0.05522 0.32402 0.11008 0.32136 0.16422 0.31698 0.21732 0.31098 1.85 o.oo 0.24008 0.05283 0.23944 0.10535 0.23755 0.15727 0.23446 0.20828 0.23021 1.9 o.oo 0.15838 0.05121 0.15798 0.10214 0.15677 0.15254 0.15479 0.20213 0.15207 i-9S o.oo 0.07870 0.05027 0.07850 0.10028 0.07792 0.14979 0.07695 0.19855 0.07562 o.oo o.oo 0.04996 o.oo 0.09967 o.oo 0.14889 o.oo 0.19738 o.oo Note. Negative quantities are in heavy type. Examples, tanh (o + i 0.95) = o 4- i 12.70620. tanh (o 4- i 1.45) = o i 1.17085. [74] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 0.25 = 0.3 o-35 x = 0.4 x = 0.45 o 0.05 O.I 0.15 O.2 0.24492 o.oo 0.24635 0.07395 0.25069 0.14866 0.25814 0.22490 0.26907 0.30351 0.29131 o.oo 0.29296 0.07199 0.29799 0.14464 0.30660 0.21864 0.31921 0.29471 0.33638 o.oo 0.33322 0.06975 0.34384 0.14007 0.35346 0.21153 0.36750 0.28475 0-37995 o.oo 0.38196 0.06728 0.38808 0.13503 0-39853 0.20373 0.41376 0.27384 0.42190 o.oo 0.42405 0.06462 0.43056 0.12961 0.44169 0.19534 0.45784 0.26216 0.25 0-3 0-35 0.4 0-45 0.28402 0.38540 0.30377 0.47162 0.32947 0.56335 0.36272 0.66200 0.40582 0.76919 0.33640 0.37362 0.35903 0.45623 0.38833 0.54348 0.42600 0.63638 0.47444 0.73604 0.38658 0.36035 0.41161 0.43898 0.44383 0.52131 0.48497 0.60802 0-53739 0.69969 0.43438 0.34585 0.46130 0.42022 0-49575 0.49737 0.53941 0.57764 0.59450 0.66116 0.47964 0.33039 0.50796 0.40033 0-54397 0.47216 0.58924 0.54593 0.64580 0.62138 o-S o-SS 0.6 0.65 0.7 0.46212 0.88682 0.53655 1.01699 0.63656 1. 16180 0.77356 1.32269 0.96528 1.49862 0.53705 0.84355 0.61869 0.95983 0.72640 1.08513 0.87037 1. 21810 1.06521 1.35360 0.60437 0.79671 0.69041 0.89893 0.80176 1.00519 0.94684 1. 1 1 212 1.13666 1. 21222 0.66404 0.74770 0.75200 0.83632 0.86357 0.92478 1.00528 1.00856 1.18469 1.07919 0.71630 0.69779 0.80407 0.77365 0.91321 0.84608 1.04844 0.91003 1.21438 0.95708 o.7S 0.8 0.85 0.9 o-9S 1.23914 1.68151 1-63553 1-84484 2.20223 1.91863 2.95122 1.75011 3.72316 1.11789 1.33091 1.47820 1.69121 1.56140 2.16210 1.54177 2.71602 1.31829 3.21905 0.79096 1.38412 I.29O2O 1.70028 I.3I745 2.08309 1.24667 2.49443 I.OI6I3 2.83603 0.58484 1.40896 1. 12181 1.68068 1.11235 1.98936 1.01695 2.29855 0.79978 2.53928 0.44728 1.41397 0.97400 1.64487 0.94186 1.89366 0.83750 2.12957 0.64107 2.30471 0.35122 I.O 1.05 i.i i-iS 1.2 4.08299 o.oo 3.72316 1.11789 2.95122 1.75011 2.20223 1.91863 1-63553 1.84484 3.43274 o.oo 3.21905 0.79096 2.71602 1.31829 2.16210 1.54177 1.69121 1.56140 2.97287 o.oo 2.83603 0.58484 2.49443 1.01613 2.08309 1.24667 1.70028 1.31745 2.63193 o.oo 2.53928 0.44728 2.29855 0.79978 1.98936 .01695 1.68068 .11235 2.37024 o.oo 2.30471 0.35122 2.12957 0.64107 1.89366 0.83750 1.64487 0.94186 1-25 i-3 1-35 1.4 1-45 1.23914 1.68151 0.96528 1.49862 0.77356 1.32269 0.63656 1. 16180 0.53655 1-01699 1.33091 1.47820 1.06521 1.35360 0.87037 1. 21810 0.72640 1.08513 0.61869 0.95983 1.38412 1.29020 1.13666 1. 21222 0.94684 I.II2I2 0.80176 I.OO5I9 0.69041 0.89893 1.40896 .12181 1.18469 .07919 1.00528 .00856 0.86357 0.92478 0.75200 0.83632 1.41397 0.97400 1.21438 0.95708 1.04844 0.91003 0.91321 0.84608 0.80407 0.77365 i-5 i-55 1.6 1.65 i-7 0.46212 0.88682 0.40582 0.76919 0.36272 0.66200 0.32947 0.56333 0.30377 0.47162 0.53705 0.84355 0.47444 0.73604 0.42600 0.63638 0.38833 0.54348 0-35903 0.45623 O.6O437 0.79671 0-53739 0.69969 0.48497 0.60802 0.44383 0.52131 0.41161 0.43898 0.66404 0.74770 0.59450 0.66 1 16 0.53941 0.57764 0.49575 0.49737 0.46130 0.42022 0.71630 0.69779 0.64580 0.62138 0.58924 0.54593 0.54397 0.47216 0.50796 0.40033 i-7S 1.8 1.85 1.9 J-95 0.28402 0.38540 0.26907 0.30351 0.25814 0.22490 0.25069 0.14866 0.24635 0.07395 0.33640 0.37362 0.31921 0.29471 0.30660 0.21864 0.29799 0.14464 0.29296 0.07199 0.38658 0.36035 0.36750 0.28475 0.35346 0.21153 0.34384 0.14007 0.33822 0.06975 0.43438 0.34585 0.41376 0.27384 0-39853 0.20373 0.38808 0.13503 0.38196 0.06728 0.47964 0.33039 0.45784 0.26216 0.44169 0.19534 0.43056 0.12961 0.42405 0.06462 2.O 0.24492 o.oo 0.29131 o.oo 0-33638 o.oo 0-37995 0.00 0.42190 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.4 + i 0.4) = 0.53941 + i 0.57764. tanh (0.45 + i 1.75) = 0.47964 - i 0.33039. [75] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = + w. CONTINUED o-5 = 0.55 x = 0.6 x = 0.65 = 0.7 o 0.05 O.I 0.15 0.2 0.46211 o.oo 0.46436 0.06181 0.47119 0.12390 0.48281 0.18651 0.49964 0.24990 0.50052 o.oo 0.50284 0.05889 0.50987 0.11796 0.52183 0.17737 0.53910 0.23725 0.53704 o.oo 0.53941 0.05590 0.54657 0.11189 0.55872 0.16804 0.57620 0.22437 0.57167 o.oo 0.57405 0.05288 0.58125 0.10576 0.59344 0.15863 0.61094 0.21144 0.60437 o- 0.60674 0.04984 0.61390 0.09962 0.62602 0.14925 0.64336 0.19858 0.25 0-3 0-35 0.4 0-45 0.52227 0.31424 0.55151 0.37967 0.58846 0.44616 0.63452 0.51350 0.69149 0.58116 0.56223 0.29765 0.59196 0.35856 0.62928 0.41979 0.67541 0.48093 0.73188 0.54121 0-59953 0.28085 0,62935 0.33731 0.66653 0.39344 0.71212 0.44869 0.76736 0.50211 0.63419 0.26404 0.66377 0.31618 0.70039 0.36748 0.74493 0.41714 0.79836 0.46428 0.66631 0.24741 0.69534 0.29540 0.73105 0.34205 0.77413 0.38662 0.82533 0.42806 o-S 0-55 0.6 0.65 0.7 0.76159 0.64805 0.84752 0.71229 0.95230 0.77067 1.07907 0.81812 1.23020 0.84688 0.80050 0.59933 0.88332 0.65320 0.98247 0.69957 1.09973 0.73362 1.23587 0.74858 0.83365 0.55229 0.91249 0.59707 1.00521 0.63346 1.11262 0.65676 1.23436 0.66157 0.86173 0.50738 0.93607 0.54434 1.02195 0.57227 1.11962 0.58738 1.22793 0.58492 0.88535 0.46492 0.95480 0.49522 1.03389 0.51635 1. 12222 0.52508 I.2I827 0.51757 0-75 0.8 0.85 0.9 o-9S 1.40579 0.84585 1.60095 0.80073 1.80225 0.69623 1.98505 0.52197 2.11599 0.28167 1.38926 0.73549 1.55398 0.68387 1.71785 0.58390 1.86183 0.43071 1.96180 0.22977 1.36782 0.64076 1.50695 0.58681 1.64135 0.49366 1.75601 0.35949 1.83417 0.19009 1.34386 0.55951 1.46173 0.50588 1.57271 0.42040 1.66532 0.30300 1.72736 0.15910 1.31896 0.48976 1.41909 0.43803 I.5II48 0.36034 1.58713 0.25755 I.637II 0.13449 I.O 1.05 i.i i-iS 1.2 2.16395 -o 2.11599 0.28167 1.98505 0.52197 1.80225 0.69623 1.60095 0.80073 1.99792 o.oo 1.96180 0.22977 1.86163 0.43071 1.71785 0.58390 1-55398 0.68387 1.86202 o.oo 1.83417 0.19009 1.75601 0.35949 1.64135 0.49366 1.50695 0.58681 1.74926 o.oo 1.72736 0.15910 1.66532 0.30300 1.57271 0.42040 1.46173 0.50588 1.65462 o.oo 1.63711 0.13449 1.58713 0.25755 1.51148 0.36034 1.41909 0.43803 I.2 S i-3 i-3S 1.4 i. 45 1.40579 0.84585 1.23020 0.84688 1.07907 0.81812 0.95230 0.77067 0.84752 0.71229 1.38926 0.73549 1.23587 0.74858 1.09973 0.73362 0.98247 0.69957 0.88332 0.65320 1.36782 0.64076 1.23436 0.66157 1.11262 0.65676 1.00521 0.63346 0.91249 0.59707 1.34386 0.55951 1.22793 0.58492 1.11962 0.58738 1.02195 0.57227 0.93607 0.54434 1.31896 0.48976 1.21827 0.51757 1. 12222 0.52508 1.03389 0.51635 0.95480 O.49522 i-5 i-55 1.6 1.65 i-7 0.76159 0.64805 0.69149 0.58116 0.63452 0.51350 0.58846 0.44616 o-SS^i 0.37967 0.80050 0.59933 0.73188 0.54121 0.67541 0.48093 0.62928 0.41979 0.59196 0.35856 0.83365 0.55229 0.76736 0.50211 0.71212 0.44869 0.66653 0.39344 0.62935 0.33731 0.86173 0.50738 0.79836 0.46428 0.74493 0.41714 0.70039 0.36748 0.66377 0.31618 0.88535 0.46492 0.82533 0.42806 0.77413 0.38662 0.73105 0.34205 0.69534 0.29540 1-75 1.8 1-85 1.9 i-9S 0.52227 0.31424 0.49964 0.24990 0.48281 0.18651 0.47119 0.12390 0.46436 o. 06181 0.56223 0.29765 0.53910 0.23725 0.52183 0.17737 0.50987 0.11796 0.50284 0.05889 0-59953 0.28085 0.57620 0.22437 0.55872 0.16804 0.54657 0.11189 0.53941 0.05590 0.63419 0.26404 0.61094 0.21144 0.59344 0.15863 0.58125 0.10576 0.57405 0.05288 0.66631 O.2574I 0.64336 0.19858 O.626O2 O.I4925 0.61390 0.09962 0.60674 0.04984 2.O 0.46211 o.oo 0.50052 o.oo 0-53704 o.oo 0.57167 o.oo 0.60437 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.6 + i 0.6) = 1.00521 + i 0.63346. [76] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 0.75 = 0.85 * = 0.9 x = 0.95 o 0.05 O.I 0.15 0.2 0.63515 o.oo 0.63749 0.04684 0.64456 0.09354 0.65649 0.13997 0.67352 0.18592 0.66403 o.oo 0.66633 0.04388 0.67325 0.08758 0.68490 0.13089 0.70148 0.17357 0.69107 o.oo 0.69330 0.04099 0.70002 0.08176 0.71132 0.12206 0.72736 0.16160 0.71629 o.oo 0.71845 0.03820 0.72494 0.07614 0.73582 0.11354 0.75123 0.15008 0.73978 o.oo 0.74185 0.03551 0.74807 0.07073 0.75850 0.10537 0.77321 0.13906 0.25 0-3 o-3S 0.4 o-45 0.69595 0.23112 0.72420 0.27516 0.75872 0.31749 0.80005 0.35735 0.84871 0.39368 0.72325 0.21528 0.75051 0.25559 0.78364 0.29392 0.82300 0.32949 0.86893 0.36128 0.74832 O.2OOOI 0.77446 0.23683 0.80603 0.27146 0.84328 0.30314 0.88638 0.33091 0.77130 0.18537 0.79620 0.21893 0.82611 0.25018 0.86117 0.27837 0.90143 0.30261 0.79231 0.17143 0.81592 0.20198 0.84411 0.23013 0.87695 0.25520 0.91438 0.27634 0-5 0-55 0.6 0.65 0.7 0.90515 0.42510 0.96963 0.44978 1.04203 0.46543 1. 12161 0.46934 1.20665 0.45847 0.92167 0.38798 0.98122 0.40796 1.04722 0.41926 1.11872 0.42057 1.19395 0.40661 0.93541 0.35357 0.99018 0.36966 1.05015 0.37751 1.11427 0.37527 1. 18081 0.36108 0.94681 0.32181 0.99700 0.33469 1.05136 0.33985 1.10885 0.33580 1.16765 0.32108 0.95624 0.29259 I.O02II 0.30285 1.05129 0.30593 1.10271 0.30064 1.15485 0.28588 0-75 0.8 0.85 0.9 0-95 1.29416 0.42977 1.37961 0.38084 1.45701 0.31065 1.51945 0.22051 1.56023 0.11463 1.27012 0.37806 1.34331 0.33238 1.40862 0.26920 1.46062 0.19000 1.49428 0.09840 1.24723 0.33335 1.31017 0.29108 1.36562 0.23434 1.40931 0.16461 1-43735 0.08499 1.22572 0.29458 1.28006 0.25573 1.32742 0.20483 1.36438 0.14330 1.38796 0.07380 1.20569 0.26087 1.25279 0.22532 1.29344 0.17968 1.32493 0.12528 1.34490 0.06438 I.O 1.05 i.i i-iS 1.2 1.57443 o.oo 1.56023 0.11463 1.51945 0.22051 1.45701 0.31065 1.37961 0.38084 1.50594 o.oo 1.49428 0.09840 1.46062 0.19000 1.40862 0.26920 1.34331 0.33238 1.44703 o.oo 1-43735 0.08499 1.40931 0.16461 1.36562 0.23434 1.31017 0.29108 1.39606 o.oo 1.38796 0.07380 1.36438 0.14330 1.32742 0.20483 1.28006 0.25573 1.35175 o.oo 1.34490 0.06438 1.32493 0.12528 1.29344 0.17968 1.25279 0.22532 1-25 i-3 i-3S 1.4 I-4S 1.29416 0.42977 1.20665 0.45847 1. 12161 0.46934 1.04203 0.46543 0.96963 0.44978 1.27012 0.37806 1.19395 0.40661 1.11872 0.42057 1.04722 0.41926 0.98122 0.40796 1.24723 0.33335 1. 18081 0.36108 1.11427 0.37527 1.05015 0.37751 0.99018 0.36966 1.22572 0.29458 1.16765 0.32108 1.10885 0.33580 1.05136 0.33985 0.99700 0.33469 1.20569 0.26087 1.15485 0.28588 1.10271 0.30064 1.05129 0.30593 I.OO2II 0.30285 i-5 i-5S 1.6 1.65 i-7 0.90515 0.42510 0.84871 0.39368 0.80005 0.35735 0.75872 0.31749 0.72420 0.27516 0.92167 0.38798 0.86893 0.36128 0.82300 0.32949 0.78364 0.29392 0.75051 0.25559 0.93541 0.35357 0.88638 0.33091 0.84328 0.30314 0.80603 0.27146 0.77446 0.23683 0.94681 0.32181 0.90143 0.30261 0.86117 0.27837 0.82611 0.25018 0.79620 0.21893 0.95624 0.29259 0.91438 0.27634 0.87695 0.25520 0.84411 0.23013 0.81592 0.20198 i-75 1.8 1.85 1.9 I-9S 0.69595 0.23112 0.67352 0.18592 0.65649 0.13997 0.64456 0.09354 0.63749 0.04684 0.72325 0.21528 0.70148 0.17357 0.68490 0.13089 0.67325 0.08758 0.66633 0.04388 0.74832 O.200OI 0.72736 0.16160 0.71132 0.12206 0.70002 0.08176 0.69330 0.04099 0.77130 0.18537 0.75123 0.15008 0.73582 0.11354 0.72494 0.07614 0.71845 0.03820 0.79231 0.17143 0.77321 0.13906 0.75850 0.10537 0.74807 0.07073 0.74185 0.03551 2.0 0.63515 o.oo 0.66403 o.oo 0.69107 o.oo 0.71629 o.oo 0.73978 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.95 + * o) = 0.73978 + *o. tanh (0.9 + i 1.9) = 0.72494 10.07614. [77] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 1.05 x = 1.15 0.05 O.I 0.15 O.2 0.76159 o.oo 0-76357 0.03293 0.76950 0.06556 0.77943 0.09757 0.79341 0.12858 0.78181 o.oo 0.78368 0.03048 0.78932 0.06065 0.79873 0.09016 0.81195 0.11867 0.80050 o.oo 0.80227 0.02816 0.80760 0.05599 0.81648 0.08317 0.82893 0.10932 0.81775 o.oo 0.81943 0.02597 0.82444 0.05160 0.83279 0.07658 0.84447 0.10054 0.83365 o.oo 0.83522 0.02390 0.83992 0.04748 0.84775 0.07041 0.85867 0.09233 0.25 o-3 Q-3S 0.4 0-45 0.81151 0.15821 -833 7 7 0.18598 0.86022 0.21133 0.89086 0.23361 0.92554 0.25205 0.82901 0.14575 0.84991 0.17096 0.87464 0.19377 0.90311 0.21356 o-935 J 5 0.22966 0.84495 0.13405 0.86450 0.15692 0.88753 0.17742 0.91392 0.19501 0.94344 0.20906 0.85945 0.12310 0.87768 0.14383 0.89907 0.16226 0.92345 0.17789 0.95058 0.19017 0.87264 0.11288 0.88958 0.13166 0.90938 0.14823 0.93186 0.16213 0.95674 0.17287 0-5 o.SS 0.6 0.65 0.7 0.96403 0.26580 1.00585 0.27392 1.05030 0.27542 1.09632 0.26933 1.14516 0.25486 0.97045 0.24130 1.00852 0.24767 1.04864 0.24798 1.08984 0.24144 1.13084 0.22747 0.97574 0.21892 1.01034 0.22389 1.04654 0.22331 1.08342 0.21658 1.11984 0.20326 0.98010 0.19852 1.01151 0.20236 1.04415 0.20115 1.07718 0.19441 1.10957 0.18182 0.98368 0.17996 1.01217 0.18288 1.04160 0.18123 1.07119 0.17461 1.10003 0.16281 0.75 0.8 0.85 0.9 0-95 1.18715 0.23145 1.22812 0.19904 1.26319 0.15812 1.29017 0.10993 1.30721 0.05638 1.17009 0.20572 1.20585 0.17623 1.23624 0.13955 1.25949 0.09677 1.27410 0.04956 1.15445 0.18315 1.18575 0.15637 1.21219 0.12347 1.23231 0.08544 1.24492 0.04369 1.14015 0.16330 1.16763 0.13902 1.19346 0.10950 1.20821 0.07563 1.21914 0.03863 1.12709 0.14580 1.15129 0.12380 1.17152 0.09730 1. 1 8680 0.06709 1.19631 0.03424 I.O i. 05 i.i i-iS 1.2 1.31304 o.oo 1.30721 0.05638 1.29017 0.10093 1.26319 0.15812 1.22812 0.19904 1.27908 o.oo 1.27410 0.04956 1.25949 0.09677 1.23624 0.13955 1.20585 0.17623 1.24922 o.oo 1.24492 0.04369 1.23231 0.08544 1.21219 0.12347 1.18575 0.15637 1.22286 o.oo 1.21914 0.03863 1.20821 0.07563 1.19346 0.10950 1.16763 0.13902 1.19954 o.oo 1.19631 0.03424 1. 18680 0.06709 1.17152 0.09730 1.15129 0.12380 1-25 J -3 1-35 1.4 i-45 1.18715 0.23145 1.14516 0.25486 1.09632 0.26933 1.05030 0.27542 1.00585 0.27392 1.17009 0.20572 1.13084 0.22747 1.08984 0.24144 1.04864 0.24798 1.00852 0.24767 1.15445 0.18315 1.11984 0.20326 1.08342 0.21658 1.04654 0.22331 1.01034 0.22389 1.14015 0.16330 1.10957 0.18182 1.07718 0.19441 1.04415 0.20115 1.01151 0.20236 1.12709 0.14580 1.10003 0.16281 1.07119 0.17461 1.04160 0.18123 1.01217 0.18288 i-5 i-SS 1.6 1.65 i-7 0.96403 0.26580 0.92554 0.25205 0.89086 0.23361 0.86022 0.21133 0.83377 0.18598 0.97045 0.24130 0.93515 0.22966 0.90311 0.21356 0.87464 0.19377 0.84991 0.17096 0.97574 0.21892 0.94344 0.20906 0.91392 0.19501 0.88753 0.17742 0.86450 0.15692 0.98010 0.19852 0.95058 0.19017 0.92345 0.17789 0.89907 0.16226 0.87768 0.14383 0.98368 0.17996 0.95674 0.17287 0.93186 0.16213 0.90938 0.14823 0.88958 0.13166 1-75 1.8 1.85 1.9 i-95 0.81151 0.15821 0.79341 0.12858 0.77943 0.09757 0.76950 0.06556 0.76357 0.03293 0.82901 0.14575 0.81195 0.11867 0-79873 0.09016 0.78932 0.06065 0.78368 0.03048 0.84495 0.13405 0.82893 0.10932 0.81648 0.08317 0.80760 0.05599 0.80227 0.02816 0.85945 0.12310 0.84447 0.10054 0.83279 0.07658 0.82444 0.05160 0.81943 0.02597 0.87264 0.11288 0.85867 0.09233 0.84775 0.07041 0.83992 0.04748 0.83522 0.02390 2JO 0.76159 o.oo 0.78181 o.oo 0.80050 o.oo 0.81775 o.oo 0.83365 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.2 + to. 75) = 1.12709 + 1*0.14580. [78] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u+ iv. CONTINUED * = 1.25 * = 1.3 * = i-35 x = 1.4 * = 1.45 0.84828 o.oo 0.86172 o.oo 0.87405 o.oo 0.88535 o.oo 0.89569 o.oo 0.05 0.84975 0.02197 0.86309 0.02017 0.87533 0.01849 0.88653 0.01693 0.89678 0.01549 O.I 0.85414 0.04363 0.86719 0.04003 0.87913 0.03668 0.89006 0.03357 0.90005 0.03070 0.15 0.86145 0.06464 0.87398 0.05927 0.88544 0.05428 0.89591 0.04965 0.90545 0.04537 0.2 0.87162 0.08468 0.88344 0.07756 0.89421 0.07097 0.90401 0.06486 0.91293 0.05923 0.25 0.88461 0.10339 0.89548 0.09458 0.90535 0.08644 0.91429 0.07892 0.92240 0.07199 o-3 0.90032 0.12039 0.90791 0.10997 0.91875 0.10036 0.92663 0.09151 0-93375 0.08338 0-35 0.91861 0.13528 0.92686 0.12336 0.93425 0.11240 0.94087 0.10234 0.94680 0.09312 0.4 0.93928 0.14765 0.94585 0.13437 0.95166 O.I222I 0.95680 0.11108 0.96137 0.10092 0-45 0.96207 0.15706 0.96669 0.14262 0.97069 O.I2945 0.97417 0.11745 0.97719 0.10654 o-S 0.98661 0.16307 0.98903 0.14773 O.99IOI 0.13381 0.99263 0.12117 0.99396 0.10971 0-55 1.01244 0.16528 1.01243 0.14937 I.OI2I9 0.13499 1.01181 0.12199 1.01133 0.11026 0.6 1.03897 0.16332 1.03634 0.14722 1-03375 0.13275 1.03125 0.11972 1.02885 0.10801 0.65 1.06550 0.15691 1.06013 0.14109 I.055IO 0.12693 1.05042 0.11425 1.04607 0.10288 0.7 1.09121 0.14591 1.08308 0.13088 1.07560 O.II749 1.06875 0.10555 1.06248 0.09488 o-7S 1.11521 0.13034 1.10439 0.11665 1.09457 0.10450 1.08565 0.09371 1.07756 0.08411 0.8 1.13656 0.11042 1.12328 0.09862 I.III3I 0.08820 1.10052 0.07896 1.09078 0.07077 0.85 1.15434 0.08662 1.13895 0.07724 I.I25I2 0.06897 1.11277 0.06167 1. 10166 0.05521 0.9 1.16772 0.05964 1.15070 0.05311 I.I355I 0.04738 1.12192 0.04232 1.10976 0.03785 0-95 1.17603 0.03041 1.15799 0.02706 I.I4I92 O.O24I2 1.12758 0.02153 1.11476 0.01925 I.O 1.17885 o.oo 1.16047 o.oo 1.14410 o.oo 1.12950 o.oo 1.11646 o.oo 1.05 1.17603 0.03041 1.15799 0.02706 1.14192 0.02412 1.12758 0.02153 1.11476 0.01925 i.i 1.16772 0.05964 1.15070 0.05311 1.13551 0.04738 1.12192 0.04232 1.10976 0.03783 i-iS 1.15434 0.08662 1.13895 0.07724 1.12512 0.06897 1.11277 0.06167 1. 10166 0.05521 1.2 1.13656 0.11042 1.12328 0.09862 1.11131 0.08820 1.10052 0.07896 1.09078 0.07077 1-25 1.11521 0.13034 1.10439 0.11665 1.09457 0.10450 1.08565 0.09371 1.07756 0.08411 i-3 1.09121 0.14591 1.08308 0.13088 1.07560 0.11749 1.06875 0.10555 1.06248 0.09488 i-35 1.06550 0.15691 1.06013 0.14109 1.05510 0.12693 1.05042 0.11425 1.04607 0.10288 1.4 1.03897 0.16332 1.03634 0.14722 1-03375 0.13275 1.03125 0.11972 1.02885 0.10801 1-45 1.01244 0.16528 1.01243 0.14937 1.01219 0.13499 1.01181 0.12199 1.01133 0.11026 i-5 0.98661 0.16307 0.98903 0.14773 0.99101 0.13381 0.99263 0.12117 0.99396 0.10971 i-55 0.96207 0.15706 0.96669 0.14262 0.97069 0.12945 0.97417 0.17745 0.97719 0.10654 1.6 0.93928 0.14765 0.94585 0.13437 0.95166 O.I222I 0.95680 0.11108 0.96137 0.10092 1.65 0.91861 0.13528 0.92686 0.12336 0.93425 O.II240 0.94087 0.10234 0.94680 0.09312 i-7 0.90032 0.12039 0.90791 0.10997 0.91875 O.IO036 0.92663 0.09151 0-93375 0.08338 i-7S 0.88461 0.10339 0.89548 0.09458 0.90535 0.08644 0.91429 0.07892 0.92240 0.07199 1.8 0.87162 0.08468 0.88344 0.07756 0.89421 O.O7O97 0.90401 0.06486 0.91293 0.05923 1.85 0.86145 0.06464 0.87398 0.05927 0.88544 O.O5428 0.89591 0.04965 0.90545 0.04537 1.9 0.85414 0.04363 0.86719 0.04003 0.87913 0.03668 0.89006 0.03357 0.90005 0.03070 I- 95 0.84975 0.02197 0.86309 0.02017 0.87533 0.01849 0.88653 0.01693 0.89678 0.01549 2.0 0.84828 0.00 0.86172 o.oo 0.87405 o.oo 0.88535 0.89569 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.4 + i 0.8) = 1.10052 + i 0.07896. tanh (1.3 + * 1.3) = 1.08308 * 0.13088. [79] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED - 1.5 = i-55 x = 1.6 = 1.65 x = 1.7 0.05 O.I 0.15 O.2 0.90515 o.oo 0.90616 O.OI4I5 0.90917 O.O28O4 0.9I4I5 0.04143 0.92104 0.05404 0.91379 o.oo 0.91471 0.01292 0.91749 0.02560 0.92208 0.03779 0.92842 0.04927 0.92167 o.oo 0.92253 0.01178 0.92508 0.02334 0.92929 0.03445 0.93511 0.04488 0.92886 o.oo 0.92964 0.01074 0.93199 0.02127 0.93586 0.03138 0.94119 0.04086 0.93541 o.oo 0.93613 0.00979 0.93828 0.01937 0.94183 0.02857 0.94671 0.03718 O.25 o-3 o-35 0.4 o-45 0.92975 0.06563 0.94017 0.07593 O.952I2 0.08468 0.96542 0.09165 0.97983 0.09660 0.93641 0.05978 0.94595 0.06909 0.95689 0.07697 0.96903 0.08320 0.98214 0.08758 0.94245 0.05442 0.95118 0.06284 0.96117 0.06993 0.97223 0.07551 0.98415 0.07938 0.94791 0.04951 0.95589 0.05712 0.96501 0.06351 0.97509 0.06850 0.98592 0.07193 0.95285 0.04502 0.96015 0.05190 0.96846 0.05766 0.97763 0.06213 0.98748 0.06517 0-5 0-55 0.6 0.65 0.7 0.99506 0.09933 I.OIO76 0.09965 I.O2657 0.09746 I.O42O4 0.09268 1.05675 0.08534 0.99595 0.08092 1.01016 0.09008 1.02441 0.08796 1.03834 0.08353 1.05152 0.07680 0.99668 0.08139 1.00954 0.08142 1.02240 0.07940 1.03492 0.07530 1.04676 0.06915 0.99728 0.07367 1.00891 0.07361 1.02052 0.07169 1.03179 0.06791 1.04242 0.06230 0-99777 0.06667 1.00829 0.06655 1.01877 0.06474 1.02892 0.06126 1.03847 0.05614 0-75 0.8 0.85 0.9 0-95 1.07022 0.07554 I.O82OO 0.06349 1.09167 0.04947 1.09886 O.O339O I.IO329 O.OI723 1.06357 0.06790 1.07408 0.05700 1.08269 0.04438 1.08909 0.03038 1.09302 0.01544 1-05755 0.06107 1.06693 0.05121 1.07461 0.03984 1.08030 0.02726 1.08380 0.01385 1.05209 0.05495 1.06049 0.04604 1.06734 0.03579 1.07241 0.02448 1.07554 0.01243 1.04714 0.04948 1.05466 0.04142 1.06079 0.03218 1.06533 O.02200 1.06811 0.01117 I.O 1.05 i.i i.i5 1.2 I.I0479 ' I.I0329 0.01723 1.09886 O.O339O 1.09167 O.O4947 I.O82OO O.OO349 1.09436 o.oo 1.09302 0.01544 1.08909 0.03038 1.08269 0.04438 1.07408 0.05700 1.08500 o.oo 1.08380 0.01385 1.08030 0.02726 1.07461 0.03984 1.06693 0.05121 1.07659 o.oo 1.07554 0.01243 1.07241 0.02448 1.06734 0.03579 1.06049 0.04604 1.06906 o.oo 1.06811 0.01117 1.06533 0.02200 1.06079 0-03218 1.05466 0.04142 1.25 i-3 i-35 1.4 1-45 I.O7O22 O.O75S4 1.05675 0.08534 I.O4204 0.09268 I.O2657 0.09746 I.OIO76 O.O9905 1.06357 0.06790 1.05152 0.07680 1.03834 0.08353 1.02441 0.08796 1.01016 0.09008 1.05755 0.06107 1.04676 0.06915 1.03492 0.07530 1.02240 0.07940 1.00954 0.08142 1.05209 0.05495 1.04242 0.06230 1.03179 0.06791 1.02052 0.07169 1.00891 0.07361 1.04714 0.04948 1.03847 0.05614 1.02892 0.06126 1.01877 0.06474 1.00829 0.06655 i-5 i-55 1.6 1.65 i-7 0.99506 0.09933 0.97983 0.09660 0.96542 O.O9I65 O.952I2 0.08468 0.94017 0.07593 0.99595 0.08992 0.98214 0.08758 0.96903 0.08320 0.95689 0.07697 0.94595 0.06909 0.99668 0.08139 0.98415 0.07938 0.97223 0.07551 0.96117 0.06993 0.95118 0.06284 0.99728 0.07367 0.98592 0.07193 0.97509 0.06850 0.96501 0.06351 0.95589 0.05712 0-99777 0.06667 0.98748 0.06517 0.97763 0.06213 0.96846 0.05766 0.96015 0.05190 i-75 1.8 1.85 1.9 i-95 0.92975 0.06563 O.92IO4 O.O5404 0.91415 0.04143 0.90917 O.O28O4 0.90616 O.OI4I5 0.93641 0.05978 0.92842 0.04927 0.92208 0.03779 0.91749 0.02560 0.91471 0.01292 0.94245 0.05442 0.93511 0.04488 0.92929 0.03445 0.92508 0.02334 0.92253 0.01178 0.94791 0.04951 0.94119 0.04086 0.93586 0.03138 0.93199 0.02127 0.92964 0.01074 0.95285 0.04502 0.94671 0.03718 0.94183 0.02857 0.93828 0.01937 0.93613 0.00979 2.0 0.90515 o.oo 0.91379 o.oo 0.92167 o.oo 0.92886 o.oo 0.93541 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.7 + *Q-7) = 1.03847 + i 0.05614. tanh (1.7 + i Q-7) tanh (1.6 + i 1.6) 0.97223 0.07551. [80] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 1.75 x = 1.85 x = 1.9 x = 1.95 o 0.05 O.I 0.15 0.2 0.94138 o.oo 0.94204 0.00891 0.94400 0.01763 0.94725 0.02600 0.95172 0.03382 0.94681 o.oo 0.94741 0.00811 0.94921 0.01604 0.95218 0.02364 0.95626 0.03074 0.95175 o.oo 0.95230 0.00737 0.95394 0.01459 0.95666 0.02149 0.96038 0.02793 0.95624 o.oo 0.95674 0.00670 0.95825 0.01326 0.96072 0.01952 0.96412 0.02537 0.96032 o.oo 0.96078 0.00609 0.96215 0.01204 0.96441 0.01773 0.96750 0.02303 0.2S 0-3 o-3S 0.4 0-45 Q-95733 0.04092 0.96399 0.04714 0.97156 0.05233 0.97991 0.05634 0.98884 0.05904 0.96139 O.037j8 0.96746 0.04280 0.97435 0.04748 0.98194 0.05107 0.99005 0.05348 0.96506 0.03376 0.97059 0.03885 0.97686 0.04307 0.98376 0.04629 0.99111 0.04843 0.96838 0.03065 0.97342 0.03525 0.97912 0.03905 0.98538 0.04195 0.99206 0.04386 0.97138 0.02782 0.97597 0.03198 0.98116 0.03541 0.98684 0.03801 0.99289 0.03972 o-5 o-5S 0.6 0.65 0.7 0.99818 0.06034 1.00769 0.06017 1.01714 0.05848 1.02629 0.05528 1.03488 0.05061 0.99851 0.05461 1.00711 0.05440 1.01565 0.05283 1.02389 0.04989 1.03162 0.04564 0.99878 0.04942 1.00656 0.04919 1.01427 0.04773 1.02170 0.04504 1.02866 0.04118 0.99900 0.04472 1.00604 0.04448 1.01300 0.04313 1.01970 0.04067 1.02596 0.03716 0.99918 0.04047 1.00555 0.04022 1.01184 0.03897 1.01788 0.03673 1.02352 0.03354 0-75 0.8 0.85 0.9 0-95 1.04266 0.04457 1.04940 0.03729 1.05489 0.02895 1.05895 0.01978 1.06143 0.01004 1.03861 0.04016 1.04466 0.03358 1.04958 0.02606 1.05320 0.01780 1.05543 0.00903 1.03494 0.03621 1.04037 0.03026 1.04478 0.02347 1.04804 0.01602 1.05003 0.00813 1.03162 0.03265 1.03650 0.02727 1.04046 0.02115 1.04337 0.01443 1.04516 0.00732 1.02862 0.02946 1.03300 0.02459 1.03656 0.01906 1.03918 0.01301 1.04078 0.00659 I.O 1.05 i.i 1.13 1.2 1.06228 o.oo 1.06143 0.01004 1.05895 0.01978 1.05489 0.02895 1.04940 0.03729 1.05619 o.oo 1.05543 0.00903 1.05320 0.01780 1.04958 0.02606 1.04466 0.03358 1.05070 o.oo 1.05003 0.00813 1.04804 0.01602 1.04478 0.02347 1.04037 0.03026 1.04576 o.oo 1.04516 0.00732 1.04337 0.01443 1.04046 0.02115 1.03650 0.02727 1.04131 o.oo 1.04078 0.00659 1.03918 0.01301 1.03656 0.01906 1.03300 0.02459 I.2S i-3 i-35 1.4 1-45 1.04266 0.04457 1.03488 0.05061 1.02629 0.05528 1.01714 0.05848 1.00769 0.06017 1.03861 0.04016 1.03162 0.04564 1.02389 0.04989 1.01565 0.05283 1.00711 0.05440 1.03494 0.03621 1.02866 0.04118 1.02170 0.04504 1.01427 0.04773 1.00656 0.04919 1.03162 0.03265 1.02596 0.03716 1.01970 0.04067 1.01300 0.04313 1 .00604 0.04448 1.02862 0.02946 1.02352 0.03354 1.01788 0.03673 1.01184 o.o'3897 1.00555 0.04022 i-5 i-5S 1.6 1.65 i-7 0.99818 0.06034 0.98884 0.05904 0.97991 0.05634 0.97156 0.05233 0.96399 0.04714 0.99851 0.05461 0.99005 0.05348 0.98194 0.05107 0.97435 0.04748 0.96746 0.04280 0.99878 0.04942 0.99111 0.04843 0.98376 0.04629 0.97686 0.04307 0.97059 0.03885 0.99900 0.04472 0.99206 0.04386 0.98538 0.04195 0.97912 0.03905 0.97342 0.03525 0.99918 0.04047 0.99289 0.03972 0.98684 0.03801 0.98116 0.03541 0-97597 0.03198 i-7S 1.8 1.85 1.9 i-95 0-95733 0.04092 0.95172 0.03382 0.94725 0.02600 0.94400 0.01763 0.94204 0.00891 0.96139 0.03718 0.95626 0.03074 0.95218 0.02364 0.94921 0.01604 0.94741 0.00811 0.96506 0.03376 0.96038 0.02793 0.95666 0.02149 0.95394 0.01459 0.95230 0.00737 0.96838 0.03065 0.96412 0.02537 0.96072 0.01952 0.95825 0.01326 0.95674 0.00670 0.97138 0.02782 0.96750 0.02303 0.96441 0.01773 0.96215 0.01204 0.96078 0.00609 2.0 0.94138 o.oo 0.94681 o.oo 0.95175 o.oo 0.95624 o.oo 0.96032 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.85 + i 0.85) = 1.04478 + i 0.02347. tanh (1.95 + * 1.25) = 1.02862 i 0.02946. [81] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 2.05 x = 2.15 0.05 O.I 0.15 0.2 0.96403 o.oo 0.96445 0.00553 0.96570 O.OIO94 0-96775 0.01610 0.97058 0.02090 0.96740 o.oo 0.96778 0.00502 0.96892 0.00993 0.97079 0.01461 0.97336 0.01897 0.97045 o.oo 0.97080 0.00456 0.97184 0.00901 0.97354 0.01326 0.97588 0.01721 0.97323 o.oo 0-97354 0.00413 0.97449 0.00817 0.97604 0.01203 0.97816 0.01561 0.97574 o.oo 0.97603 0.00375 0.97689 0.00741 0.97830 0.01091 0.98023 0.01415 0.2S o-3 -35 0.4 o-4S 0.97411 0.02524 0.97827 0.02900 0.98299 0.03209 0.98815 0.03444 0.99364 0.03596 0.97657 0.02289 0.98036 0.02630 0.98464 0.02909 0.98932 0.03120 0.99430 0.03256 0.97880 0.02076 0.98224 0.02384 0.98613 0.02636 0.99037 0.02826 0.99489 0.02948 0.98081 0.01882 0.98394 0.02161 0.98747 0.02388 0.99132 0.02559 0.99541 0.02669 0.98264 0.01706 0.98548 0.01958 0.98868 0.02163 0.99217 0.02317 0.99587 0.02416 o-S o-SS 0.6 0.65 0.7 0.99933 0.03662 1.00509 0.03638 1.01077 0.03523 1.01623 0.03318 1.02131 0.03028 0.99945 0.03314 1.00466 0.03290 1.00979 0.03184 1.01471 0.02998 1.01930 0.02734 0.99955 0.02998 1.00426 0.02976 .00890 0.02878 .01334 0.02709 .01748 0.02469 0.99963 0.02713 1.00389 0.02691 .00808 0.02602 .OI2IO 0.02448 .01583 0.02231 0.99970 0.02455 1.00355 0.02434 1.00734 0.02353 I.OI097 O.O22I2 I.OI434 O.O2OI5 0-75 0.8 0.85 0.9 Q-95 1.02589 0.02658 1.02984 0.02218 1.03304 0.01719 1-03539 0.01172 1.03683 0.00594 1.02343 0.02399 1.02698 0.020OI 1.02986 O.OI55O 1.03197 0.01057 1.03327 0.00536 .02120 0.02166 .02440 0.01806 1.02699 0.01399 1.02889 0.00954 1.03005 0.00483 .01919 O.OI956 .022O7 O.OI63I .0244O O.OI262 .02611 O.OO86I .02715 0.00436 1.01736 0.01767 1.01996 O.OI472 1.02206 O.OII40 1.02360 O.O0777 1.02454 0.00394 I.O 1.05 i.i i-iS 1.2 1.03731 o.oo 1.03683 0.00594 1.03539 0.01172 1.03304 0.01719 1.02984 0.02218 1.03370 o.oo 1.03327 0.00536 1.03197 0.01057 1.02986 0.01550 1.02698 O.O2OOI 1.03045 o.oo 1.03005 0.00483 1.02889 0.00954 1.02699 0.01399 1.02440 0.01806 1.02751 o.oo 1.02715 0.00436 1.02611 0.00861 1.02440 0.01262 1.02207 0.01631 1.02486 o.oo 1.02454 0.00394 1.02360 0.00777 1.02206 0.01140 1.01996 0.01472 1-25 !-3 i-3S 1.4 i. 45 1.02589 0.02658 1.02131 0.03028 1.01623 0.03318 1.01077 0.03523 1.00509 0.03638 1.02343 0.02399 I.OI930 0.02734 I.OI47I 0.02998 I.O0979 0.03184 I.OO466 0.0329O I.O2I2O O.O2I66 1.01748 0.02469 I.OI334 0.02709 1.00890 0.02878 I.OO426 O.02976 1.01919 0.01956 1.01583 0.02231 I.OI2IO 0.02448 I.Oo8o8 0.02602 1.00389 O.O269I 1.01736 0.01767 1.01434 0.02015 I.OIO97 O.022I2 1.00734 0.02353 1.00355 0.02434 i-5 i-SS 1.6 1.65 i-7 0-99933 0.03662 0.99364 0.03596 0.98815 0.03444 0.98299 0.03209 0.97827 0.02900 0.99945 0.03314 0.99430 0.03256 0.98932 O.O3I2O 0.98464 O.02909 0.98036 O.O263O 0.99955 0.02998 0.99489 0.02948 0.99037 O.O2826 0.98613 0.02636 0.98224 0.02384 0.99963 0.02713 0.99541 0.02669 0.99132 0.02559 0.98747 O.02388 0.98394 0.02161 0.99970 0.02455 0.99587 O.O24I6 0.99217 O.O23I7 0.98868 O.02I63 0.98548 O.OI958 1-75 1.8 1.85 1.9 i -95 0.97411 0.02524 0.97058 0.02090 0.96775 0.01610 0.96570 0.01094 0.96445 0.00553 0-97657 0.02289 0.97336 0.01897 0.97079 O.OI46I 0.96892 O.OO993 0.96778 O.OO5O2 0.97880 O.O2O76 0.97588 O.OI72I 0.97354 0.01326 0.97184 jo. 0090 1 0.97080 0.00456 0.98081 0.01882 0.97816 0.01561 0.97604 0.01203 0.97449 0.00817 0.97354 0.00413 0.98264 O.OI7O6 0.98023 O.OI4I5 0.97830 O.OIO9I 0.97689 O.0074I 0.97603 0.00375 2.0 0.96403 o.oo 0.96740 o.oo 0.97045 o.oo 0.97323 o.oo 0.97574 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.2 + * o) = o 97574 + i o. tanh (2.15 -f i 1.15) = 1.02440 0.01262. [82] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 2.25 x = 2.3 = 2.35 2.4 * = 2.45 o 0.05 O.I 0.15 O.2 0.97803 o.oo 0.97829 0.00340 0.97907 0.00672 0.98035 0.00989 0.98210 0.01283 0.98010 o.oo 0.98034 0.00308 0.98105 0.00610 0.98221 0.00897 0.98380 0.01163 0.98197 o.oo 0.98367 o.oo 0.98219 0.00280 0.98387 0.00253 0.98283 0.00553 0.98446 0.00501 0.98389 0.00813 0.98541 0.00736 0.98534 0.01054 0.98673 0.00955 0.98522 o.oo 0.98540 0.00230 0.98592 0.00454 0.98680 0.00667 0.98799 0.00865 0.25 -3 -35 0.4 0-45 0.98429 0.01547 0.98687 0.01774 0.98977 0.01960 0.99294 0.02098 0.99629 0.02187 0.98579 0.01401 0.98813 0.01607 0.99076 0.01775 0.99363 0.01900 0.99667 0.01979 0.98714 0.01270 0.98836 0.01150 0.98926 0.01456 0.99029 0.01319 0.99165 0.01607 0.99245 0.01456 0.99425 0.01720 0.99481 0.01557 0.99700 0.01791 0.99730 0.01621 0.98947 0.01042 0.99121 0.01194 0.99317 0.01318 0.99531 0.01410 O-99757 0.01468 o-5 0-55 0.6 0.65 0.7 O.99975 O.O2222 1.00324 O.O22O2 I.OO666 O.O2I27 I.OO994 O.O2OOO I.OI298 O.OI82I 0.99980 O.O2OIO 1.00295 0.01992 I.O06O4 O.OI924 1.00900 O.Ol8o8 I.OII75 0.01646 0.99984 0.01819 0.99986 0.01646 1.00269 0.01802 1.00245 0.01630 .00549 0.01740 1.00498 0.01573 .00816 0.01634 1.00739 0.01478 .01064 0.01487 1.00963 0.01345 0.99989 0.01489 1.00222 0.01474 1.00451 0.01423 1.00693 0.01336 1.00872 0.01216 0-75 0.8 0.85 0.9 0-95 I.OI57I 0.01596 I.OI8O5 O.OI33O I.OI994 O.OIO29 I.02I33 O.OO7OI I.022I8 0.00355 I.OI42I O.OI442 I.OI632 O.OI2OI I.OI8O3 0.00929 I.OI928 0.00633 I.O2006 O.OO322 .01286 0.01303 1.01164 0.01178 .01477 0.01085 1.01336 0.00981 1.01631 0.00839 1 - OI 475 0.00758 1.01744 0.00572 1.01576 0.00517 1.01812 0.00290 1.01639 0.00262 1.01053 0.01064 1.01208 0.00886 1.01333 0.00685 1.01425 0.00467 1.01482 0.00237 .0 5 .1 IS .2 1.02247 o.oo 1.02218 0.00355 1.02133 0.00701 1.01994 0.01029 1.01805 0.01330 1.02031 o.oo I.O20O6 O.OO322 I.OI928 O.OO633 I.OI8O3 O.OO929 I.OI632 O.OI2OI 1.01836 o.oo 1.01659 o.oo 1.01812 0.00290 1.01639 0.00262 1.01744 0.00572 1.01576 0.00517 1.01631 0.00839 1.01475 0.00758 1.01477 0.01085 1.01336 0.00981 1.01500 o.oo 1.01482 0.00237 1.01425 0.00467 1.01333 0.00685 1. 01208 0.00886 25 3 35 4 J-45 1.01571 0.01596 1.01298 0.01821 I.OO994 O.O2OOO I.00666 0.02127 1.00324 O.O22O2 I.OI42I O.OI442 I.OII75 0.01646 I.OO9OO O.OlSoS 1.00604 O.OI924 I.OO295 O.OI992 1.01286 0.01303 1.01164 0.01178 1.01064 0.01487 1.00963 0.01345 1.00816 0.01634 1.00739 0.01478 1.00549 0.01740 1.00498 0.01573 1.00269 0.01802 1.00245 0.01630 1.01053 0.01064 1.00872 0.01216 1.00693 0.01336 1.00451 0.01423 1.00222 O.OI474 i-5 i-55 1.6 1.65 i-7 0-99975 0.02222 0.99629 O.02I87 0.99294 O.O2098 0.98977 O.OI96O 0.98687 O.OI774 0.99980 O.O20IO 0.99667 O.OI979 0.99363 O.OI9OO 0.99076 O.OI775 0.98813 O.OI6O7 0.99984 0.01819 0.99986 0.01646 0.99700 0.01791 0.99730 0.01621 0.99425 0.01720 0.99481 0.01557 0.99165 0.01607 0.99245 0.01456 0.98926 0.01456 0.99029 0.01319 0.99989 0.01489 0.99757 0.01468 0.99531 O.OI4IO 0.99317 O.OI3I8 O.99I2I O.OII94 1-75 1.8 1.85 1.9 1-95 0.98429 O.OI547 0.98210 O.OI283 0.98035 0.00989 0.97907 O.OO672 0.97829 O.OO340 0.98579 O.OI4OI 0.98380 O.OII63 0.98221 O.O0897 0.98105 O.OO6IO 0.98034 O.OO3O8 0.98714 0.01270 0.98836 0.01150 0.98534 0.01054 0.98673 0.00955 0.98389 0.00813 0.98541 0.00736 0.98283 0.00553 0.98446 0.00501 0.98219 0.00280 0.98387 0.00253 0.98947 O.OI042 0.98799 0.00865 0.98680 0.00667 0.98592 O.OO454 0.98540 O.OO23O 2.0 0.97803 o.oo 0.98010 o.oo 0.98197 o.oo 0.98367 o.oo 0.98522 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.25 + 0.25) = 0.98429 + i 0.01547. tanh (2.45 + * 1.45) = 1.00222 * 0.01474. [83] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED = 2.5 * = 2.55 * = 2.6 x = 2.65 x = 2.7 0.05 O.I 0.15 O.2 o. 8661 o.oo 0.98678 O.O0208 0.98726 O.004II 0.98805 O.006O5 0.98913 0.00784 0.98788 o.oo 0.98803 0.00189 0.98846 0.00373 0.98918 0.00548 0.99016 0.00710 0.98903 o.oo 0.98916 O.OOI7I 0.98956 0.00337 O.9902I 0.00496 0.99109 0.00643 0.99007 o.oo 0.99019 0.00155 0.99055 0.00306 0.99113 0.00449 0.99194 0.00582 0.99101 o.oo 0.99112 0.00140 0.99144 0.00277 0.99198 0.00407 0.99270 0.00527 0.25 o-3 o-3S 0.4 Q-4S 0.99047 O.00944 0.99205 0.01082 0.99383 O.OII93 0.99576 O.OI276 0.99781 0.01328 0.99138 0.00855 0.99281 0.00979 0.99442 0.01080 0.99617 0.01155 0.99802 O.OI2O2 O.9922O O.OO774 0.99350 0.00887 0.99495 0.00978 0.99654 O.OI046 0.99822 O.OI088 0.99294 0.00701 0.99412 0.00803 0.99544 0.00886 0.99688 0.00947 0.99839 0.00985 0.99361 0.00635 0.99468 0.00727 0.99588 0.00802 0.99718 0.00857 0.99855 0.00891 o-5 o-SS 0.6 0.65 0.7 0.99991 0.01348 1.00202 O.OI334 I.004O9 O.OI287 I.Oo6o6 O.OI2O8 1.00789 O.OIO99 0.99993 O.OI2I9 I.OOI84 O.OI207 I.OO37I O.OII64 1.00549 0.01093 I.OO7I4 0.00994 0.99994 O.OIIO3 I.OOI67 O.OIO92 1.00336 0.01053 1.00498 0.00988 1.00647 0.00898 0.99995 0.00998 1.00151 0.00988 1.00304 0.00952 1.00450 0.00894 1.00585 0.00812 0.99996 0.00903 1.00137 0.00893 1.00276 0.00862 1.00408 0.00808 1.00530 0.00735 0-75 0.8 0.85 0.9 o-9S 1.00953 0.00962 I.OIO93 O.0001 I. OI2O6 0.00619 I.OI289 0.00422 I.OI34O O.O02I4 I.O0862 O.OO87O 1.00989 O.OO726 I.OIOgi 0.00560 1.01166 0.00381 i.oi2ii 0.00193 I.O078o O.00786 1.00895 0.00654 I.OOgS? 0.00506 I.OI054 0.00345 1.01096 O.OOI75 1.00706 0.00711 1.00809 0.00592 1.00893 -457 1.00954 0.00312 1.00991 0.00158 1.00639 0.00643 1.00732 0.00535 1.00807 0.00413 1.00862 0.00282 1.60896 0.00143 I.O 1.05 i.i i-iS 1.2 1.01357 o.oo I.OI34O O.OO2I4 1.01289 O.00422 1.01206 0.00619 1.01093 0.00801 1.01227 - 1.01211 O.OOI93 I.OII66 O.0038I I.OIOgi O.OO56O 1.00989 O.OO726 I. OHIO O.OO 1.01096 0.00175 1.01054 0.00345 1.00987 0.00506 1.00895 0.00654 1.01003 o.oo 1.00991 0.00158 1.00954 0.00312 1.00893 0.00457 1.00809 0.00592 1.00907 o.oo 1.00896 0.00143 1.00862 0.00282 1.00807 0.00413 1.00732 0.00535 1-25 i-3 i-3S 1.4 145 1.00953 0.00962 1.00789 0.01099 1.00606 0.01208 1.00409 0.01287 1.00202 O.OI334 I.OO862 O.0087O I.OO7I4 O.OO994 I.OO549 O.OIO93 I.O037I O.OII64 I.OOl84 O.O12O7 1.00780 0.00786 1.00647 0.00898 1.00498 0.00988 1.00336 0.01053 1.00167 0.01092 1.00706 0.00711 1.00585 0.00812 1.00450 0.00894 1.00304 0.00952 1.00151 0.00988 1.00639 0.00643 1.00530 0.00735 1.00408 0.00808 1.00276 0.00862 1.00137 0.00893 i-5 i-55 1.6 1.65 i-7 0.99991 O.OI348 0.99781 O.OI328 0.99576 O.OI276 -99383 O.OII93 0.99205 O.OI082 O-99993 O.OI2I9 0.99802 O.OI2O2 0.99617 O.OII55 0.99442 O.OI08O 0.99281 O.OO979 0.99994 0.01103 0.99822 0.01088 0.99654 0.01046 0.99495 0.00978 0.99350 0.00887 0.99995 0.00998 0.99839 0.00985 0.99688 0.00947 0.99544 0.00886 0.99412 0.00803 0.99996 0.00903 0.99855 0.00891 0.99718 0.00857 0.99588 0.00802 0.99468 0.00727 i-7S 1.8 1.85 1.9 i-9S 0.99047 0.00944 0.98913 0.00784 0.98805 O.00605 0.98726 O.OO4II 0.98678 O.00208 0.99138 0.00855 0.99016 O.OO7IO 0.98918 O.OO548 0.98846 O.00373 0.98803 O.OOI89 0.99220 0.00774 0.99109 0.00643 0.99021 0.00496 0.98956 0.00337 0.98916 0.00171 0.99294 0.00701 0.99194 0.00582 0.99113 0.00449 0.99055 0.00306 0.99019 0.00155 0.99361 0.00635 0.99270 0.00527 0.99198 0.00407 0.99144 0.00277 0.99112 0.00140 2.O 0.98661 o.oo 0.98788 o.oo 0.98903 o.oo 0.99007 o.oo 0.99101 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.60 + i 0.35) = 0.99495 + * 0.00978. tanh (2.70 + * 1.35) = 1.00408 i 0.00808. [84] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED x = 2.75 x = 2.80 x = 2.85 = 2.90 x = 2.95 0.99186 o.oo 0.99263 o.oo 0-99333 o.oo 0.99396 o.oo 0.99454 o.oo 0.05 0.99196 0.00127 0.99272 0.00115 0.99341 0.00104 0.99404 0.00094 0.99460 0.00085 O.I 0.99225 0.00251 0.99299 0.00227 0.99365 0.00206 0.99426 0.00186 0.99480 0.00168 0.15 0.99274 0.00368 0.99343 0.00334 0.99405 0.00302 0.99462 0.00273 0.99513 0.00248 0.2 0.99340 0.00477 0.99403 0.00432 0.99459 0.00391 0.99511 0.00354 0-99557 0.00321 0.25 0.99422 0.00575 0.99477 0.00519 0.99527 0.00471 0.99572 0.00426 0.99613 0.00386 0-3 0.99519 0.00658 0.99564 0.00596 0.99606 0.00539 0.99644 0.00488 0.99678 0.00442 o-3S 0.99627 0.00726 0.99663 0.00657 0.99695 0.00594 0.99724 0.00538 0.99750 0.00487 0.4 0.99745 0.00775 0.99769 0.00702 0.99791 0.00635 0.99811 0.00575 0.99830 0.00520 0-45 0.99869 0.00806 0.99882 0.00730 0.99893 0.00660 0.99904 0.00598 0.99913 0.00541 o-5 0.99997 0.00817 0.99997 0.00740 0.99998 0.00669 0.99998 0.00606 0.99999 0.00548 o-55 1.00125 0.00808 1.00113 0.00731 1.00103 0.00662 1.00093 0.00599 1.00084 0.00542 0.6 1.00250 0.00779 1.00226 0.00705 1.00205 0.00638 1.00186 0.00577 1.00168 0.00522 0.65 1.00369 0.00731 1.00334 0.00661 1.00303 0.00598 1.00274 0.00541 1.00248 0.00489 0.7 1.00479 0.00664 1.00434 0.00601 1.00393 0.00544 I - 00 35S 0.00480 1.00322 0.00445 o.7S 1.00578 0.00581 1.00523 0.00525 1.00473 0.00475 1.00428 0.00430 1.00387 0.00389 0.8 1.00662 0.00484 1.00599 0-00437 1.00542 0.00396 1.00490 0.00358 1.00444 0.00324 0.85 .00730 0.00374 1. 0066 1 0.00338 1.00598 0.00306 1.00540 0.00276 1.00489 0.00250 0.9 .00780 0.00255 1.00706 0.00230 1.00638 0.00208 1.00577 0.00188 1.00522 0.00170 o-95 .00810 0.00129 1.00733 0.00117 1.00663 0.00105 1.00600 0.00095 1.00542 0.00086 I.O .00821 o.oo 1.00742 o.oo 1.00671 o.oo 1.00607 o.oo 1.00549 o.oo '1.05 .00810 0.00129 1.00733 0.00117 1.00663 0.00105 1.00600 0.00095 1.00542 0.00086 i.i .00780 0.00255 1.00706 0.00230 1.00638 0.00208 1.00577 0.00188 1.00522 0.00170 i-iS .00730 0.00374 1. 00661 0.00338 1.00598 0.00306 1.00540 0.00276 1.00489 0.00250 1.2 1.00662 0.00484 1.00599 0.00437 1.00542 0.00396 1.00490 0.00358 1.00444 0.00324 I-2S 1.00578 0.00581 1.00523 0.00525 1.00473 0.00475 1.00428 0.00430 1.00387 0.00389 i-3 1.00479 0.00664 1.00434 0.00601 1.00393 0.00544 1 -003 5 5 0.00480 1.00322 0.00445 i-35 1.00369 0.00731 1.00334 0.00661 1.00303 0.00598 1.00274 0.00541 1.00248 0.00489 1.4 1.00250 0.00779 1.00226 0.00705 1.00205 0.00638 1.00186 0.00577 i .00168 0.00522 i -45 1.00125 0.00808 1.00113 0.00731 1.00103 0.00662 1.00093 0.00599 1.00084 0.00542 i-5 0.99997 0.00817 0.99997 0.00740 0.99998 0.00669 0.99998 0.00606 0.99999 0.00548 i-55 0.99869 0.00806 0.99882 0.00730 0.99893 0.00660 0.99904 0.00598 0.99913 0.00541 1.6 0.99745 0.00775 0.99769 0.00702 0.99791 0.00635 0.99811 0.00575 0.99830 0.00520 1.65 0.99627 0.00726 0.99663 0.00657 0.99695 0.00594 0.99724 0.00538 0.99750 0.00487 i-7 0.99519 0.00658 0.99564 0.00596 0.99606 0.00539 0.99644 0.00488 0.99678 0.00442 1-75 0.99422 0.00575 0.99477 0.00519 0.99527 0.00471 0.99572 0.00426 0.99613 0.00386 1.8 0.99340 0.00477 0.99403 0.00432 0.99459 0.00391 0.99511 0.00354 0-99557 0.00321 1.85 0.99274 0.00368 0-99343 0.00334 0.99405 0.00302 0.99462 0.00273 0.99513 0.00248 1.9 0.99225 0.00251 0.99299 0.00227 0.99365 0.00206 0.99426 0.00186 0.99480 0.00168 1-95 0.99196 0.00127 0.99272 0.00115 0.99341 0.00104 0.99404 0.00094 0.99460 0.00085 2.O 0.99186 O.OO 0.99263 o.oo 0-99333 o.oo 0.99396 o.oo 0.09454 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.9 + i 0.9) = 1.00577 + * 0.00188. tanh (2.95 + i 1.95) = 0.99460 * 0.00085. [85] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED = 3-0 * = 3-5 x = 3.10 x = 3.20 0.05 O.I 0.15 O.2 0.99505 o.oo 0.99512 O.OOO77 0.99530 0.00153 0.99559 0.00224 0.99599 O.O029O 0.99552 o.oo 0.99558 0.00070 0.99574 0.00138 0.99601 0.00203 0.99637 0.00263 0.99595 o.oo 0.99600 0.00063 0.99615 0.00125 0.99639 0.00184 0.99672 0.00238 0.99633 o.oo 0.99638 0.00057 0.99651 0.00113 0.99673 0.00166 0.99703 0.00215 0.99668 o.oo 0.99672 0.00052 0.99684 O.OOIO2 0.99704 0.00150 0.99731 0.00195 0.25 0-3 0-35 0.4 0-45 0.99649 0.00349 0.99708 O.O040O 0.99774 O.OO44I 0.99846 O.0047I 0.99921 0.00489 0.99683 0.00316 0.99736 0.00362 0.99796 0.00399 0.99861 0.00426 0.99929 0.00443 0.99713 0.00286 0.99761 0.00328 0.99815 0.00361 0.99874 0.00386 0.99936 0.00401 0.99740 0.00259 0.99784 0.00297 0.99833 0.00327 0.99886 0.00349 0.99942 0.00363 0.99765 0.00234 0.99805 0.00268 0.99849 0.00296 0.99897 0.00316 0.99947 0.00328 0-5 0-55 0.6 0.65 0.7 0.99999 0.00496 I.OOO76 0.00490 I.OOI52 0.00472 1.00224 0.00443 I.OO29I O.OO4O2 0.99999 0.00449 1.00069 0.00443 1.00138 0.00427 1.00203 0.00401 1.00263 0.00364 0.99999 0.00406 1.00063 0.00401 1.00125 0.00387 1.00184 0.00362 1.00238 0.00329 0.99999 0.00367 1.00057 0.00363 1.00113 0.00350 1.00166 0.00328 1.00216 0.00298 0.99999 0.00332 1.00051 0.00328 I.OOI02 O.OO3I6 1.00151 0.00297 1.00195 0.00269 0-75 0.8 0.85 0.9 0-95 I.0035I 0.00352 I.OO4OI O.OO293 1.00443 O.00226 1.00473 0.00154 1.00491 0.00078 1.00317 0.00318 1.00363 0.00265 1.00400 0.00205 1.00426 0.00139 1.00444 0.00071 1.00287 0.00288 1.00329 0.00239 1.00362 0.00185 1.00387 0.00126 1.00402 0.00064 1.00260 0.00260 1.00297 0.00217 1.00328 0.00167 1.00350 0.00114 1.00363 0.00058 1.00235 0.00236 1.00269 0.00196 1.00297 0.00151 1.00317 0.00103 1.00329 0.00052 .0 OS .1 IS .2 1.00497 o.oo 1.00491 0.00078 1.00473 0.00154 1.00443 0.00226 1.00401 0.00293 1.00450 o.oo 1.00444 0.00071 1.00426 0.00139 1.00400 0.00205 1.00363 0.00265 1.00407 o.oo 1.00402 0.00064 1.00387 0.00126 1.00362 0.00185 1.00329 0.00239 1.00368 o.oo 1.00363 0.00058 1.00350 0.00114 1.00328 0.00167 1.00297 0.00217 1.00333 o.oo 1.00329 0.00052 1.00317 0.00103 1.00297 0.00151 1.00269 0.00196 25 3 35 1.4 i-45 1.00351 0.00352 1.00291 0.00402 1.00224 0.00443 1.00152 0.00472 1.00076 0.00490 1.00317 0.00318 1.00263 0.00364 1.00203 0.00401 1.00138 0.00427 1.00069 0.00443 1.00287 0.00288 1.00238 0.00329 1.00184 0.00362 1.00125 0.00387 1.00063 0.00401 1.00260 0.00260 1. 00216 0.00298 1.00166 0.00328 1.00113 0.00350 1.00057 0.00363 1.00235 0.00236 1.00195 0.00269 1.00151 0.00297 I.OOI02 O.O03I6 1.00051 0.00328 i-5 i-SS 1.6 1.65 i-7 0.99999 0.00496 0.99921 0.00489 0.99846 0.00471 0.99774 0.00441 0.99708 0.00400 0.99999 0.00449 0.99929 0.00443 0.99861 0.00426 0.99796 0.00399 0.99736 0.00362 0.99999 0.00406 0.99936 0.00401 0.99874 0.00386 0.99815 0.00361 0.99761 0.00328 0.99999 0.00367 0.99942 0.00363 0.99886 0.00349 0.99833 0.00327 0.99784 0.00297 0.99999 0.00332 0.99947 0.00328 0.99897 0.00316 0.99849 0.00296 0.99805 0.00268 i-75 1.8 1.85 1.9 i-95 0.99649 0.00349 0.99599 0.00290 0-99559 0.00224 0.99530 0.00153 0.99512 0.00077 0.99683 0.00316 0.99637 0.00263 0.99601 0.00203 0.99574 0.00138 0.99558 0.00070 0.99713 0.00286 0.99672 0.00238 0.99639 0.00184 0.99615 0.00125 0.99600 0.00063 0.99740 0.00259 0.99703 0.00215 0.99673 0.00166 0.99651 0.00113 0.99638 0.00057 0.99765 0.00234 0.99731 0.00195 0.99704 0.00150 0.99684 O.OOI02 0.99672 0.00052 2.O 0-99505 o.oo 0.99552 o.oo 0.99595 o-o o-99 6 33 o- 00 0.99668 o.oo Note. Negative quantities are in heavy type. Examples. tanh (3.0 + * i.oo) tanh (3.0 + * 1.50) 1.00497 + io. 0.99999 ~~ 0.00496. [86] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) =u+ iv. CONTINUED = 3-25 * = 3-30 * = 3-35 x = 3.40 = 3-45 o 0.99700 o.oo 0.99728 o.oo 0.99754 o.oo 0.99777 o.oo 0.99799 o.oo 0.05 0.99704 0.00047 0.99732 0.00042 0.99757 0.00038 0.99780 O.OOO35 0.99801 0.00031 O.I 0.99715 0.00093 0.99742 0.00084 0.99766 0.00076 0.99788 0.00069 0.99809 0.00062 0.15 0.99732 0.00136 0.99758 0.00123 0.99781 O.OOII2 0.99802 o.ooioi 0.99821 0.00091 0.2 0.99757 0.00176 0.99780 0.00160 0.99801 0.00144 0.99820 0.00131 0.99837 0.00118 0.25 0.99787 O.O02I2 0.99808 0.00192 0.99826 O.OOI74 0.99843 0.00157 0.99857 0.00142 0-3 0.98823 0.00243 0.99840 O.OO22O 0.99855 0.00199 0.99869 0.00180 0.99881 0.00163 0-35 0.99863 0.00268 0.99876 O.O0242 0.99888 O.OO2I9 0.99899 0.00198 0.99908 0.00179 0.4 0.99907 O.O0286 0.99916 O.OO259 0.99924 O.OO234 O.9993I 0.00212 0.99938 0.00192 0-45 0.99953 0.00297 O.99957 O.O0269 0.99961 O.OO243 0.99965 O.OO22O 0.99968 O.OO2OO 0-5 .OOOOO O.OO3OI i.ooooo 0.00272 i.ooooo 0.00246 i.ooooo 0.00223 I.OOOOO O.002O2 0-55 .00047 O.O0297 1.00042 0.00269 1.00038 0.00243 I.OOO35 O.OO220 I.OO03I O.OO2OO 0.6 .OOO9I O.OO286 1.00084 O.O0259 1.00076 0.00234 1.00069 O.OO2I2 I.OOO62 O.OOI92 0.65 .00136 O.OO268 I.OOI23 O.OO243 I.OOII2 O.OO22O i.ooioi 0.00199 I.OOOgi O.OOlSo 0.7 .OOI77 O.00244 I.OOI6O O.OO22O I.OOI45 O.OOI99 I.OOI3I O.OOlSo 1.00118 0.00163 0.75 I.OO2I3 O.OO2I3 I.OOI92 O.OOI93 I.OOI74 O.OOI74 .OOI58 O.OOI58 I.OOI43 O.OOI43 0.8 I.OO244 O.OOI77 I.OO222 O.OOl6o I.OOI99 O.OOI45 .OOlSo O.OOI3I 1.00163 O.OOIig 0.85 1.00268 0.00137 I.OO242 O.OOI24 I.OO22O O.OOII2 .OOI99 O.OOIO3 I.OOlSo 0.00092 0.9 I.OO286 O.OOO93 1.00260 0.00084 I.OO234 O.OOO76 .OO2I2 O.OOO69 I.OOI92 0.00062 o-9S I.OO298 0.00047 1.00269 O.OOO43 I.OO243 O.OOO39 .OO22O O.OOO35 I.OO20O O.00032 I.O 1.00301 o.oo 1.00273 o.oo 1.00246 o.oo .OO223 - .00202 o.oo 1.05 1.00298 0.00047 1.00269 O.OOO43 I.OO243 O.OOO39 .OO22O 0.00035 .00200 0.00032 I.I 1.00286 0.00093 I.0026O 0.00084 I.OO234 O.OO076 .OO2I2 O.OOO69 .00192 0.00062 1-15 1.00268 0.00137 I.OO242 O.OOI24 I.OO22O O.OOII2 I.OOI99 O.OOIO3 .00180 0.00092 1.2 1.00244 0.00177 I.OO222 O.OOI6O 1.00199 O.OOI45 I.OOlSo O.OOI3I .00163 0.00119 1-25 1.00213 0.00213 I.OOI92 O.OOI93 I.OOI74 O.OOI74 I.OOI58 O.OOI58 .00143 0.00143 '3 1.00177 0.00244 I.OOl6o O.0022O I.OOI45 O.OOI99 I.OOI3I O.OOlSo .00118 0.00163 i-35 1.00136 0.00268 I.OOI23 O.O0243 I.OOII2 O.OO22O i.ooioi 0.00199 .00091 0.00180 1.4 1.00091 0.00286 1.00084 O.O0259 1.00076 O.OO234 1.00069 O.OO2I2 .00062 0.00192 i -45 1.00047 0.00297 I.OOO42 O.OO269 1.00038 O.OO243 I.OOO35 O.OO22O .OOO3I O.OO2OO i-5 i. ooooo 0.00301 i.ooooo 0.00272 i.ooooo 0.00246 i.ooooo 0.00223 I.OOOOO O.OO2O2 i-SS 0.99953 0.00297 O.99957 O.OO269 0.99961 O.OO243 0.99965 O.OO22O 0.99968 O.OO2OO 1.6 0.99907 0.00286 0.99916 O.OO259 0.99924 O.OO234 0.99931 O.OO2I2 0.99938 O.OOI92 1.65 0.99863 0.00268 0.99876 O.OO242 0.99888 O.OO2I9 0.99899 O.OOI98 0.99908 O.OOI79 i-7 0.99823 0.00243 0.99840 O.OO22O 0.99855 O.OOI99 0.99869 O.OOlSo 0.99881 0.00163 1-75 0.99787 O.O02I2 0.99808 O.OOI92 0.99826 O.OOI74 0.99843 O.OOI57 0.99857 O.OOI42 1.8 O-99757 O.OOI76 0.99780 O.OOI6O 0.99801 O.OOI44 0.99820 O.OOI3I 0.99837 O.OOIlS 1.85 O.99732 O.OOI36 0.99758 O.OOI23 O.9978l O.OOII2 0.99802 o.ooioi 0.99821 O.OOO9I i. 9 0.99715 0.00093 O.99742 O.O0084 0.99766 O.OO076 0.99788 O.O0069 0.99809 O.OOO62 i-9S 0.99704 O.O0047 O.99732 O.O0042 O.99757 O.O0038 0.99780 0.00035 0.99801 O.O003I 2.O O.997OO O.OO 0.99728 o.oo 0.99754 o.oo 0-99777 ' 0.99799 o.oo Note. Negative quantities are in heavy type. Examples, tanh (3.25 + to. 75) = 1.00213 + i 0.00213. tanh (3.30 + 1.50) = i.ooooo i 0.00272. [87] TABLE IX HYPERBOLIC TANGENTS, tanh (* + iq) =u + iv. CONTINUED - 3-50 = 3-55 x = 3.60 x = 3.70 0.05 O.I 0.15 O.2 0.99818 0.99820 0.99827 0.99837 0.99853 o.oo 0.00028 0.00056 0.00083 0.00107 0.99835 0.99837 0.99843 0-99853 0.99867 o.oo 0.00026 0.00051 0.00075 0.00097 0.99851 0.99853 0.99858 0.99867 0.99879 o.oo 0.00023 0.00046 0.00068 0.00088 0.99865 0.99867 0.99872 0.99880 0.99891 O.OO O.OOO2I 0.00042 O.OOo6l O.OOO79 0.99878 0.99879 0.99884 0.99891 0.99901 o.oo 0.00019 0.00038 0.00055 0.00072 0.25 o-3 0.35 0.4 0.45 0.99871 0.99893 0.99917 0.99944 0.99971 0.00129 0.00147 0.00162 0.00173 0.00180 0.99883 0.99903 0.99925 0.99949 0.99974 0.00117 0.00133 0.00147 0.00157 0.00163 0.99894 0.99912 0.99932 0.99954 0.99977 0.00105 O.OOI2I 0.00133 0.00142 0.00147 0.99904 0.99921 0-99939 0.99958 0-99979 0.00095 O.OOI09 O.OOI2O 0.00128 0.00133 0.99914 0.99928 0.99944 0.99962 0.99981 0.00086 0.00099 0.00109 0.00116 O.OOI2I o-S o-5S 0.6 0.65 0.7 I.OOOOO I 00028 1.00056 1.00083 1.00107 0.00182 0.00180 0.00174 0.00163 0.00148 I.OOOOO 1.00026 1.00051 1.00075 1.00097 0.00165 0.00163 0.00157 0.00147 0.00134 I.OOOOO 1.00025 1.00048 1.00068 1.00088 0.00149 0.00148 0.00142 0.00133 O.OOI2I I.OOOOO I.OO02I I.OOO42 I. OOO6 1 I.OOO79 0.00135 0.00133 O.OOI29 O.OO I 2O 0.00109 I.OOOOO 1.00019 1.00038 1.00055 1.00072 O.OOI22 O.OOI2I 0.00116 0.00109 0.00099 o.75 0.8 0.85 0.9 0-95 1.00129 1.00148 1.00163 1.00174 1.00180 0.00129 0.00107 0.00083 0.00056 0.00029 1.00117 1.00133 1.00147 1.00157 1.00163 0.00117 0.00097 0.00075 0.00051 0.00026 1.00106 I.OOI2I 1.00133 1.00142 1.00148 0.00106 0.00088 0.00068 0.00046 0.00023 I.OOO96 I.OOI08 I.OOI2O I.OOI29 I.OOI34 0.00096 0.00080 0.00061 0.00042 O.OOO2I 1.00086 1.00099 1.00109 1.00116 I.OOI2I 0.00086 0.00072 0.00056 0.00038 0.00019 I.O 1.05 i.i i-iS 1.2 1.00183 1.00180 1.00174 1.00163 1.00148 0.00 0.00029 0.00056 0.00083 0.00107 1.00165 1.00163 1.00157 1.00147 1.00133 o.oo 0.00026 0.00051 0.00075 0.00097 1.00149 1.00148 1.00142 1.00133 I.OOI2I 0.00 0.00023 0.00046 0.00068 0.00088 I.OOI35 I.OOI34 1.00129 I.OOI2O I.OOI08 O.OO O.OOO2I O.OOO42 O.OOO6l O.OOO8O I.OOI22 I.OOI2I 1.00116 1.00109 1 .00099 o.oo 0.00019 0.00038 0.00056 0.00072 1-25 i-3 i-35 1.4 1-45 1.00129 1.00107 1.00083 1.00056 1.00028 0.00129 0.00148 0.00163 0.00174 0.00180 1.00117 1.00097 1.00075 1.00051 1.00026 0.00117 0.00134 0.00147 0.00157 0.00163 1.00106 1.00088 1.00068 1.00048 1.00025 0.00106 O.OOI2I 0.00133 0.00142 0.00148 I.OOO96 I.OOO79 1. 000(5 1 I.OOO42 I.OO02I O.OOO96 O.OOIO9 O.OOI2O O.OOI29 O.OOI33 1 .00086 1.00072 1.00055 1.00038 1.00019 0.00086 0.00099 0.00109 o. 00116 O.OOI2I i-5 I-5S 1.6 1.65 i-7 I.OOOOO 0.99971 0.99944 0.99917 0.99893 0.00182 0.00180 0.00173 0.00162 0.00147 I.OOOOO 0.99974 0.99949 0.99925 0.99903 0.00165 0.00163 0.00157 0.00147 0.00133 I.OOOOO 0.99977 0.99954 0.99932 0.99912 0.00149 0.00147 0.00142 0.00133 O.OOI2I I.OOOOO 0.99979 0.99958 0.99939 0.99921 0.00135 0.00133 O.OOI28 O.OOI20 O.OOIO9 I.OOOOO 0.99981 0.99962 0-99944 0.99928 O.OOI22 O.OOI2I o. 00116 O.OOIO9 O.OOO99 1-75 1.8 1.85 1.9 i-95 0.99871 0.99853 0.99837 0.99827 0.99820 0.00129 0.00107 0.00083 0.00056 0.00028 0.99883 0.99867 0-99853 0.99843 0.99837 0.00117 0.00097 0.00075 0.00051 0.00026 0.99894 0.99879 0.99867 0.99858 0.99853 0.00105 0.00088 0.00068 0.00046 0.00023 0.99904 0.99891 0.99880 0.99872 0.99867 O.OOO95 O.OOO79 0.00061 O.OOO42 O.OO02I 0.99914 0.99901 0.99891 0.99884 0.99879 O.OO086 O.OOO72 0.00055 O.OOO38 O.OOOI9 2.O 0.99818 O.OO 0.99835 o.oo 0.99851 o.oo 0.99865 O.OO 0.99878 0.00 Note. Negative quantities are in heavy type. Examples, tanh (3.60 + i 0.80) = I.OOI2I + i O.OOO88. tanh (3.70 -+ * 1.70) = 0.99928 i 0.00099. [88] TABLE IX. HYPERBOLIC TANGENTS, tanh (x + iq) = u + iv. CONTINUED = 3-75 x = 3.80 = 3-85 3-90 = 3-95 o 0.99889 o.oo 0.99900 o.oo 0.99909 o.oo 0.99918 o.oo 0.99926 o.oo 0.05 0.99891 0.00017 0.99901 0.00016 0.99911 0.00014 0.99919 0.00013 0.99927 O.OOOI2 O.I 0.99895 0.00034 0.99905 0.00031 0.99914 0.00028 0.99922 0.00025 0.99930 0.00023 0.15 0.99901 0.00050 0.99911 0.00045 0.99919 0.00041 0.99927 0.00037 0.99934 0.00034 0.2 0.99911 0.00065 0.99919 0.00059 0.99927 0.00053 0.99934 0.00048 0.99940 0.00044 0.25 0.99922 0.00078 0.99929 0.00071 0.99936 0.00064 0.99942 0.00058 0.99948 0.00052 -3 0-99935 0.00089 0.99941 0.00081 0.99947 0.00073 0.99952 0.00066 0.99956 0.00060 0-35 0.99950 0.00099 0-99955 0.00089 0.99959 0.00081 0.99963 0.00073 0.99966 0.00066 0.4 0.99966 0.00105 0.99969 0.00095 0.99972 0.00086 0.99975 0.00078 0.99977 0.00071 0-45 0.99983 0.00109 0.99984 0.00099 0.99986 0.00089 0.99987 0.00081 0.99988 0.00073 o.S 1. 00000 O.OOIII I.OOOOO O.OOIOO I.OOOOO 0.00091 I.OOOOO 0.00082 I.OOOOO 0.00074 0-55 1.00017 0.00109 1.00016 0.00099 1.00014 0.00089 1.00013 0.00081 I.OOOI2 0.00073 0.6 1.00034 0.00105 1.00031 0.00095 1.00028 0.00086 1.00025 0.00078 1.00023 0.00071 0.65 1.00050 0.00099 1.00045 0.00089 1.00041 0.00081 1.00037 0.00073 1.00034 0.00066 0.7 1.00065 0.00090 1.00059 0.00081 1.00053 0.00073 1.00048 0.00066 1.00044 0.00060 0-75 1.00078 0.00078 1.00071 0.00071 1.00064 0.00064 1.00058 0.00058 1.00052 0.00052 0.8 1.00089 0.00065 1.00081 0.00059 1.00073 0.00053 1.00066 0.00048 1.00060 0.00044 0.85 1.00099 0.00050 1.00089 0.00045 1.00081 0.00041 1.00073 0.00037 1.00066 0.00034 0.9 1.00105 0.00034 1.00096 0.00031 1.00086 0.00028 1.00078 0.00025 1.00071 0.00023 o-9S 1.00109 0.00017 1.00099 0.00016 1.00089 0.00014 1.00081 0.00013 1.00073 O.OOOI2 I.O I.OOIII o.oo I.OOIOO o.oo 1.00090 o.oo 1.00082 o.oo 1.00074 O.OO 1.05 1.00109 0.00017 1.00099 0.00016 1.00089 0.00014 1.00081 0.00013 1.00073 0.00012 i.i 1.00105 0.00034 1.00096 0.00031 1.00086 0.00028 1.00078 0.00025 1.00071 O.OO023 i-iS 1.00099 0.00050 1.00089 0.00045 1.00081 0.00041 1.00073 0.00037 1.00066 O.OOO34 1.2 1.00089 0.00065 1.00081 0.00059 1.00073 0.00053 1.00066 0.00048 1.00060 O.OOO44 I.2S 1.00078 0.00078 1.00071 0.00071 1.00064 0.00064 1.00058 0.00058 1.00052 O.OO052 i-3 1.00065 0.00090 1.00059 0.00081 1.00053 0.00073 1.00048 0.00066 1.00044 O.OO06O i-35 1.00050 0.00099 1.00045 0.00089 1.00041 0.00081 1.00037 0.00073 1.00034 O.OO066 1.4 1.00034 0.00105 1.00031 0.00095 1.00028 0.00086 1.00025 0.00078 1.00023 O.OOO7I 1-45 1.00017 0.00109 1.00016 0.00099 1.00014 0.00089 1.00013 0.00081 I.OOOI2 O.OOO73 i-5 I.OOOOO O.OOIII I.OOOOO O.OOIOO I.OOOOO 0.00091 I.OOOOO 0.00082 I.OOOOO O.OO074 i-SS 0.99983 o.ooiog 0.99984 0.00099 0.99986 0.00089 0.99987 0.00081 0.99988 O.00073 1.6 0.99966 0.00105 0.99969 0.00095 0.99972 0.00086 0-99975 0.00078 0.99977 O.OOO7I 1.65 0.99950 0.00099 0.99955 00089 0.99959 0.00081 0.99963 0.00073 0.99966 O.OOO66 i-7 0-99935 0.00089 0.99941 0.00081 0.99947 0.00073 0-99952 0.00066 0.99956 O.OOO6O i-75 0.99922 0.00078 0.99929 0.00071 0.99936 0.00064 0.99942 0.00058 0.99948 O.OOO52 1.8 0.99911 0.00065 0.99919 0.00059 0.99927 0.00053 0.99934 0.00048 0.99940 O.OOO44 1.85 0.99901 0.00050 0.99911 0.00045 0.99919 0.00041 0.99927 0.00037 0.99934 O.OOO34 1.9 0.99895 0.00034 0.99905 0.00031 0.99914 0.00028 0.99922 0.00025 0.99930 O.OOO23 J-95 0.99891 0.00017 0.99901 0.00016 0.99911 0.00014 0.99919 0.00013 0.99927 O.OOOI2 2.O 0.99889 O.OO 0.99900 o.oo 0.99909 o.oo 0.99918 o.oo 0.99926 o.oo Note. Negative quantities are in heavy type. Examples, tanh (3.95 + i 0.95) = 1.00073 + 0.00012. tanh (3.95 -f i 1.05) = 1.00073 i 0.00012. [89] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y_ x = 0.05 x = 0.15 o o.ooo 9 0.05002 o.ooo 0.10017 o.ooo 0.15056 o.ooo 0.20134 o.ooo 0.05 0.07846 90 0.09305 57-593 0.12724 38.300 0.16978 27.861 0.21608 21.739 O.I 0.15643 90 0.16424 72.493 0.18576 57-819 0.21712 46.771 0.25497 38.746 0.15 0-23345 9 0.23874 78-245 0.25403 67-455 0.27779 58.195 0.30827 50.576 O.2 0.30902 90 0.31304 81.259 0.32485 72.947 0-34375 65.382 0.36882 58.723 O.25 0.38268 9 0.38594 83-123 0.39558 76.471 0.41124 70.229 0.43422 64.522 -3 0.45399 90 0.45672 84.400 0.46491 78.932 0.47831 73-7" 0.49663 68.825 0.35 0.52250 90 0.52487 85-339 0.53201 80.762 0.54376 76-344 0-55995 72.147 0.4 0.58778 9 0.58989 86.066 0.59626 82.189 0.60676 78.419 0.62131 74.802 0.45 0.64944 90 0-65135 86.652 0.65713 83.344 0.66667 80.111 0.67994 76.988 -5 0.70711 90 0.70803 87-I39 0.71417 84.308 0.72296 81.532 0.73521 78.835 o-55 0.76041 90 0.76202 87-557 0.76698 85-134 0.77527 82.753 0.78661 80.431 0.6 0.80902 00 0.81053 87.921 0.81520 85.858 0.82291 83.826 0.83370 81.839 0.65 0.85264 90 0.85407 88.246 0.85851 86.505 0.86583 84.787 0.87609 83.103 0.7 0.89101 90 0.89237 88.542 0.89662 87.093 0.90364 85.662 0.91347 84.257 o-7S 0.92388 90 0.92523 88.814 0.92930 87.636 0.93607 86.471 0.94556 85.326 0.8 0.95106 90 0.95237 89.070 0.95632 88.145 0.96290 87-231 0.97213 86.331 0.85 0.97237 90 0.97366 89-313 0.97752 88.629 0.98396 87-953 0.99300 87.287 0.9 0.98769 90 0.98895 89-547 0.99275 89.095 0.99909 88.649 1.00800 88.209 0-95 0.99692 90 0.99817 89-775 1.00194 89.550 1.00822 89.329 1.01704 89.110 I.O I.OOOOO 90 1.00125 90.000 1.00500 90.000 1.01127 90.000 1.02007 90.000 1.05 0.99692 90 0.99817 90.225 1.00194 90.450 1.00822 90.671 1.01704 90.890 i.i 0.98769 90 0.98895 90-453 0.99275 90.905 0.99909 9i-35i 1.00800 91.791 "5 0.97237 90 0.97366 90.687 0.97752 9*-37i 0.98396 92.047 0.99300 92.713 1.2 0.95106 90 0.95237 90.930 0.95632 9I-855 0.96290 92.769 0.97213 93.669 I-2S 0.92388 90 0.92523 91.186 0.92930 92-364 0.93607 93-529 0.94556 94.674 i-3 0.89101 90 0.89237 91.458 0.89662 92.907 0.90364 94.338 0.91347 95-743 i-35 0.85264 90 0.85407 91-754 0.85851 93-495 0.86583 95-213 0.87609 96.897 1.4 0.80902 90 0.81053 92.079 0.81520 94.142 0.82291 96.174 0.83370 98.161 i -45 0.76041 90 0.76202 92.443 0.76698 94.866 0.77527 97.247 0.78661 99.569 1-5 0.70711 90 0.70803 92.860 0.71417 95.692 0.72296 98.468 0.73521 101.165 1-55 0.64944 90 0.65135 93-348 0.65713 96.656 0.66667 99.889 0.67994 103.012 1.6 0.58778 90 0.58989 93-934 0.59626 97.811 0.60676 101.581 0.62131 105.198 1.65 0.52250 90 0.52487 94.661 0.53201 99-238 0.54376 103.656 0-55995 107-853 i-7 0.45399 90 0.45672 95.600 0.46491 101.068 0.47831 106.289 0.49663 111.175 i-7S 0.38268 90 0.38594 96.877 0.39558 103.529 0.41124 109.771 0.43242 115.478 1.8 0.30902 90 0.31304 98.741 0.32485 107.053 0-34375 114.618 0.36882 121.277 1.85 0.23345 90 0.23874 101.755 0.25403 112.545 0.27779 121.805 0.30827 129.424 1.9 0.15643 90 0.16424 107.507 0.18576 122.181 0.21712 133.229 0.25497 141.254 I- 95 0.07846 90 0.09305 122.407 0.12724 141.700 0.16978 152-139 0.21608 158.261 2.O o.oo 90 0.05002 180.000 0.10017 180.000 0.15056 180.000 0.20134 180.000 Example. sinh (0.15 + to. 15) = 0.27779 = 0.27779 /58.ii / .42*. TABLE X. HYPERBOLIC SINES, sinh (at + iq) = r /T. CONTINUED = 0.25 x = 0.3 = 0.35 * = 0.4 x = 0.45 0.25261 o.ooo 0.30452 0.000 0.35719 o.ooo 0.41075 o.ooo 0.46534 o.ooo 0.26452 17.814 0.31447 15.118 0.36571 13.169 0.41818 11.703 0.47191 10.567 0.29782 32.890 0.34235 28.533 0.38994 25.214 0-43953 22.629 0.49093 20.576 0.34396 44-943 0.38370 39-493 0.42671 35.516 0.47246 32.288 0.52062 29.642 o.399 J 3 52.992 0.43385 48.122 0.47231 44.008 0.51401 40.536 0.55860 37.601 0.45854 59-405 0.48906 54.882 0.52348 50.921 0.56140 47.471 0.60249 44-473 0.51954 64.327 0.54666 60.242 0.57766 56.843 0.61223 53-288 0.65012 50-374 0.58036 68.215 0.60476 64-575 0.63292 61.237 0.66462 58.200 0.69968 55-454 0.63977 7I-37I 0.66199 68.151 0.68781 65.157 0.71708 62.393 0.74969 59-856 0.69685 73-999 0.71730 71.166 0.74II9 68.503 0.76844 66.018 0.79895 63.712 0.75088 76.238 0.76989 73-759 0.79220 71.408 0.81775 69.196 0.84649 67.125 0.80127 78.185 0.81912 76.028 O.840I2 73.971 0.86426 72.021 0.89149 70.184 0.84754 79.911 0.86443 78.049 0.88436 76.267 0.90732 74.568 0-93330 72.958 0.88927 81.464 0-90539 79-878 0.92444 78.353 0.94642 76.893 0.97136 75.504 0.92612 82.887 0.94161 8i.557 0.95994 80.274 0.98113 79.043 1.00521 77.868 0-95779 84.207 0.97277 83.119 0.99052 82.068 1.01107 81.056 1.03446 80.087 0.98403 85-450 0.99862 84-593 I.OI592 83.762 1.03597 82.962 1.05880 82.194 1.00464 86.635 1.01894 85.999 1.03590 85.383 1-05557 84.788 1.07798 84.216 1.01949 87-779 1-03357 87-358 I.05O29 86.951 1.06970 86.556 1.09182 86.177 1.02843 88.896 1.04239 88.687 1.05898 88.484 1.07822 88.287 1.10017 88.098 1.03141 90.000 1-04534 90.000 i. 06 1 88 90.000 1.08107 90.000 1.10297 90.000 1.02843 91.104 1.04239 91.313 1.05898 91.516 1.07822 91-713 1.10017 91.902 1.01949 92.221 1-03357 92.642 1.05029 93.050 1.06970 93-444 1.09182 93-823 i .00464 93-365 1.01894 94.001 1.03590 94.617 1-05557 95-212 1.07798 0.98403 94-550 0.99862 95.407 1.01592 96.238 1-03597 97.038 1.05880 97.806 0-95779 95-793 0.97277 96.881 0.99052 97.932 1.01107 98.944 1.03446 99-9 J 3 0.92612 0.94161 98.443 0.95994 99.726 0.98113 100.957 1.00521 102.132 0.88927 98*536 0-90539 IOO.I22 0.92444 101.647 0.94642 103.107 0.97136 104.496 0-84754 100.090 0.86443 IOI.95I 0.88436 103.733 0.90732 105.432 0-93330 107.042 0.80127 101.815 0.81912 103.972 0.84012 106.029 0.86426 107.979 0.89149 109.816 0.75088 103.762 0.76989 IO6.24I 0.79220 108.592 0.81775 110.804 0.84649 112.875 0.69685 106.001 0.71730 108.834 0.74119 111.497 0.76844 113.982 0.79895 116.288 0.63977 108.629 0.66199 III.849 0.68781 114.843 0.71708 117.607 0.74969 120.144 0.58036 111.785 0.60476 115425 0.63292 118.763 0.66462 121.800 0.69968 124.546 0.51954 0.54666 119.758 0.57766 123.157 0.61223 126.712 0.65012 129.626 0-45854 120.595 0.48906 I25.II8 0.52348 129.079 0.56140 132.529 0.60249 I35.527 0.39913 127.008 0-43385 131.878 0.47231 135.992 0.51401 139.464 0.55860 142.399 0.34396 I35-057 > 0.38370 140.507 0.42671 144.484 0.47246 147.712 0.52062 150.358 0.29782 147.110 0.34235 151.467 0.38994 154.786 0-43953 I57-37I 0.49093 159.424 0.26452 162.186 0.31447 164.882 0.36571 166.831 0.41818 168.297 0.47191 169.433 0.25261 180.000 0.30452 iSo.OOO 0.35719 180.000 0.41075 180.000 0.46534 180.000 Example. sinh (0.40 + 0.25) = 0.56140 ( = 0.56140 /47.28'.i6". [90 TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r CONTINUED x = 0.5 = -55 x = 0.6 * = 0.65 0.05 O.I 0.15 0.2 0.52110 o.ooo 0.52697 9.665 0.54407 18.918 0.57100 27.453 0.60583 35.111 0.57815 o.ooo 0.58345 8.936 0.59894 17.559 0.62350 25.625 0.65555 32.990 0.63665 o.ooo 0.64147 8.337 0.65559 16.432 0.67810 24.086 0.70769 31.174 0.69675 o.ooo 0.70115 7.839 0.71409 15.486 0.73482 22.781 0.76220 29.613 0.75858 o.ooo 0.76263 7.419 0.77455 14.685 0.79369 21.665 0.81911 28.263 0.25 o-3 0-35 0.4 0-45 0.64652 41.871 0.69112 47-793 0-73793 52-980 0.78552 57-542 0.83266 61.584 0.69333 39-610 0.73510 45.511 0.77927 50.759 0.82447 55-437 0.86951 59.628 0.74282 37.642 0.78194 43-494 0.82361 48.769 0.86650 53-529 0.90946 57.838 0.79492 35.926 0.83160 41.710 0.87090 46.989 0.91156 51.803 0.95249 56.204 0.84965 34.426 0.88406 40.133 0.92112 45-397 0.95966 50.245 0.99861 54.716 0-55 0.6 0.65 0.7 0.87837 65.198 0.92182 68.462 0.96232 71-441 0.99927 74-189 1.03220 76-75 1 0.91338 63.411 0.95524 66.854 0-99437 70.016 1.03004 72.948 1.06214 75.693 0.95149 61.762 0.99171 65.360 1.02948 68.685 1.06411 71.784 1.09509 74.696 0.99270 60.245 .03134 63.976 .06769 67.445 .101 1 1 70.694 .13108 73.760 1.03704 58.853 1.07409 62.698 1.10904 66.294 1.14125 69.677 1.17019 72.884 0-75 0.8 0.85 0.9 0-95 1.06070 79.164 1.08446 81.461 1.10320 83.669 1.11672 85.814 1.12489 87.917 1.08987 78.287 1.11300 80.763 1.13127 83.148 1.14446 85.467 1.15244 87.744 1. 12200 77-459 1.14448 8O.IO2 1.16226 82.653 I.I75IO 85.121 I.I8287 87.580 .15715 76.678 .17892 79-477 .19623 82.185 .20871 84.826 1.21626 87.424 1.19541 75-946 1.21653 78.890 1.23327 81.744 1.24538 84.532 1.25271 87.277 I.O .1 IS .2 1.12763 90.000 1.12489 92.083 1.11672 94.186 1.10320 96.331 1.08446 98.539 1.15510 90.000 1.15244 92.256 ( 1.14446 94-533 1.13127 96.852 1.11300 99.237 1.18547 9O.OOO 1.18287 92.420 I.I75IO 94.879 I.I6226 97-347 1.14448 99.898 1.21879 90.000 1.21626 92.576 1.20871 95.174 1.19623 97.815 1.17892 100.523 1.25517 90.000 1.25271 92.723 1.24538 95-468 1.23327 98.256 1.21653 ioi.no 25 3 35 1.4 1-45 1.06070 100.836 1.03220 103.249 0.99927 105.811 0.96232 108.559 0.92182 111.538 1.08987 101.713 1.06214 104.307 1.03004 107.052 0.99437 109.984 0.95524 113.146 I.I22OO 102.541 1.09509 105.304 I.O64H IO8.2I6 1.02948 III.3I5 0.99174 114.640 1.15715 103.322 1.13108 106.240 i.ioni 109.306 1.06769 112.555 1.03134 116.024 1.19541 104.054 1.17019 107.116 1.14125 110.323 1.10904 113.706 1.07409 117.302 i-SS 1.6 1.65 0.87837 114.803 0.83266 118.416 0.78552 122.458 0.73793 127.020 0.69112 132.207 0.91338 116.589 0.86951 120.372 0.82447 124.563 0.77927 129.241 0.73510 134-489 0.95149 II8.238 0.90946 I22.I62 0.86650 126.471 0.82361 131.231 0.78194 136.506 0.99270 119.755 0.95249 123.796 0.91156 128.197 0.87090 133.011 0.83160 138.290 1.03704 121.148 0.99861 125.284 0.95966 129.755 0.92112 134.603 0.88406 139.867 1.8 1.85 1.9 0.64652 138.129 0.60583 144.889 0.57100 152.547 0.54407 161.082 0.52697 170.335 0-69333 140-390 0.65555 147-010 0.62350 154-375 0.59894 162.441 0.58345 171.064 0.74282 142.358 0.70769 148.826 0.67810 155.914 0.65559 163.568 0.64147 171.663 0.79492 144-074 0.76220 150.387 0.73482 157.219 0.71409 164.514 0.70115 172.161 0.84965 145.574 0.81911 151.737 0.79369 158.335 0.77455 165.315 0.76263 172.581 2.0 0.52110 180.000 0.57815 180.000 0.63665 iSo.OOO 0.69675 180.000 0.75858 180.000 Example. sinh (0.70 + i 1.70) = 0.88406 = 0.88406 / 130. $2'. 01*. [92] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED x = 0-75 x = 0.8 x = 0.85 x = 0.9 x = 0.95 9 r 7 r y r y r y r y o o o o.oo 0.82232 o.ooo 0.88811 0.000 0.95612 O.OOO 1.02652 o.oob 1.09948 O.OOO 0.05 0.82605 7.064 0.89157 6-759 0-95933 6.497 1.02951 6.270 1.10227 6.073 O.IO 0.83706 14.002 0.90178 I3-4I5 0.96883 12.909 1-03837 12.468 1.11056 12.085 0.15 0.85481 20.706 0.91828 19.877 0.98420 19.158 1.05273 18.529 1.12400 17.980 O.2 0.87846 27.093 0.94033 26.073 1.00481 25.l8l 1.07202 24-399 1.14209 23.712 0.25 0.90700 33-1" 0.96705 3 I -955 1.02985 30.938 1-09553 30-039 1.16418 29.245 0-3 0.93932 38.737 0.99742 37-500 1.05843 36.401 1.12243 35-426 1.19227 34-557 0.35 0.97427 43-974 1.03041 42.702 1.08957 4I.565 1.15184 40-547 1.21732 39-637 0.4 1.01079 48.840 1.06500 47-574 1.12234 46.433 1.18289 45-407 1.24674 44.483 Q-45 1.04785 53-363 1.10024 52.135 I.I5583 5I.O22 1.21471 50.014 1.27697 49.102 0-5 1.08453 57-578 1.13522 56.414 1.18918 55-353 1.24649 54.386 1.30723 53-507 o.SS I.I200I 61.522 1.16917 60.441 1.22163 59-450 1.27748 58.543 1.33682 57.714 0.6 I-I5356 65.228 1.20135 64.245 1.25246 63-399 1.30700 62.507 1-36505 6i.743 0.65 1.18457 68.733 1.23115 67-857 1.28107 67.048 1-33444 66.301 1.39136 65.613 0.7 I.2I248 72.067 1.25802 71.307 1.30693 70.602 1.35927 69-949 1.41519 69-347 0-75 1.23689 75.260 1.28152 74.621 I-32955 74.026 1.38105 73-474 1.43611 72.963 0.8 1.25726 78.339 1.30124 77-825 1-34858 77-344 1-39937 76.898 I-4537I 76.484 0.85 1.27346 81.330 1.31691 80.942 1.36369 80.580 I-4I395 80.242 1.46778 79-929 0.9 1.28520 84-255 1.32825 83.996 1.37466 83-754 1.42452 83-527 1.47797 83-316 o-9S 1.29230 87-138 I-335I3 87.008 1.38130 86.887, 1.43094 86.773 1.48415 86.668 I.O 1.29468 90.000 1-33743 90.000 I.38353 90.000 I-43309 90.000 1.48623 90.000 1.05 1.29230 92.862 I-335I3 92.992 1.38130 93-II3 1.43094 93.227 1.48415 93-332 i.i 1.28520 95-745 1.32825 96.004 1.37466 96.246 1.42452 96-473 1.47797 96.684 i-iS 1.27346 98.670 1.31691 99.058 1.36369 99.420 I-4I395 99.758 1.46778 100.071 1.2 1.25726 101.661 1.30124 102.175 1.34858 102.656 1-39937 103.102 I-4537I 103.516 I.2S 1.23689 104.740 1.28152 105-379 I.32955 105.974 1.38105 106.526 1.43611 107.037 i-3 I.2I248 107.933 1.25802 108.693 1.30693 109.398 I-35927 110.051 1.41519 110.653 i-35 I.I8457 111.267 1-23115 112.143 1.28107 112.952 1.33444 113.699 1.39136 114.387 4 I-I5356 114.772 1.20135 "5-755 1.25246 116.601 1.30700 117-493 1-36505 118.257 45 I.I2OOI 118.478 1.16917 119-559 1.22163 120.550 1.27748 121-457 1.33682 122.286 5 1.08453 122.422 1.13522 123.586 1.18918 124.647 1.24649 125.614 1.30723 126.493 55 1.04785 126.637 1.10024 127.865 1-15583 128.978 1.21471 129.986 .27697 130.898 .6 I.OIO79 131.160 1.06500 132.426 1.12234 I33-567 1.18289 134-593 .24674 I35-5I7 6S 0.97427 136.026 1.03041 137.298 1.08957 138.435 1.15184 139-453 .21732 140.363 7 0.93932 141.263 0.99742 142.500 1.05843 143-599 1.12243 144-574 19227 145-443 75 O.907OO 146.889 0.96705 148.045 1.02985 149.062 1.09553 149.961 .16418 150.755 .8 0.87846 152.907 0.94033 153-927 1.00481 154.819 1.07202 155.601 .14209 156.288 1.85 0.85481 159.294 0.91828 160.123 0.98420 160.842 1-05273 161.471 .12400 162.020 1.9 0.83706 165.998 0.90178 166.585 0.96883 167.091 1.03837 167.532 .11056 167.915 i-95 0.82605 172.936 0.89157 173.241 0-95933 I73-503 1.02951 I73.730 .10227 173.927 2.0 0.82232 ISO.OOO 0.888II iSo.OOO 0.95612 iSo.OOO 1.02652 iSo.OOO 1.09948 iSo.OOO Example. sinh (0.90 + i i.o) = 1.43309 /go . [93] TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r /r CONTINUED X = I.O x = 1.05 X I.I x = 1.15 X = 1.2 9 f y f y r y r y r y 0.05 O.I 0.15 O.2 1.17520 o.ooo 1.17782 5.900 1.18552 11.748 1.19816 17.496 1.21515 23.105 1.25386 o.ooo I.2563I 5.748 1.26358 11-453 1.27540 17.071 I.29I37 22.568 o L33565 o.ooo 1-33795 5-615 1.34478 11.192 1-35590 16.694 1.37093 22.092 1.42078 o.ooo 1.42294 5.497 1.42936 10.961 1.43983 16.361 1.45399 21.670 o 1.50946 o.ooo I.5II50 5.393 I.5I755 10.757 I.5274I 16.065 1.54077 21.294 0.25 0-35 0.4 0-45 1.23594 28.541 1.25984 33.784 1.28612 38.821 1.31400 43-65 1 1.34272 48.276 I.3I095 27.915 I-3335I 33-093 I-35837 38.090 1.38479 42.902 I.4I207 47-530 1.38939 27.359 1.41070 32.477 1.43421 37-435 1.45926 42.227 1.48517 46.855 1.47141 26.864 i.49 x 55 3 I -9 2 6 1.51381 36.847 1.53756 41.620 1.56217 46.245 I.5572I 26.421 1.57626 31.433 1-59733 3 6 -3 J 9 1.61987 41-073 1.64325 45-693 0-5 0-55 0.6 0.65 0.7 I.37I53 52-707 1.39976 56.957 1.42675 61.043 1.45193 64.981 1.47479 68.791 1.43950 51.981 1.46641 56.268 1.49220 60.403 1.51630 64.401 1.53820 68.280 1.51128 51.323 I.53694 55-640 1.56156 59.818 1.58460 63.870 1.60558 67.811 1.58701 50.725 1.61147 55.069 1.63497 59.284 1.65699 63.384 1.67705 67.380 1.66688 50.184 1.69018 54-549 1.71260 58.797 I.73363 62.939 1.75282 66.986 0-75 0.8 0.85 0.9 0-95 1.49487 72-491 1.51182 76.101 1.52532 79.638 I-535I3 83.122 1.54108 86.724 1-55747 72.056 J-57374 75-747 1.58671 79-369 1.59614 82.941 1.60187 86.479 1.62404 71.656 1-63965 75-421 1.65211 79.121 1.66117 82.774 1.66667 86.395 1.69472 71.287 1.70971 75.120 1.72166 78.893 1.73036 82.620 1.73564 86.317 I.76975 70-950 1.78409 74.844 1-79555 78-682 1.80389 82.478 1.80895 86.246 1.05 i.i 1.2 1.54308 90.000 1.54108 93.276 I-535I3 96-878 1.52532 100.362 1.51182 103.899 1.60379 90.000 1.60187 93-521 1.59614 97-059 1.58671 100.631 1.57374 104.253 1.66852 90.000 1.66667 93-605 1.66117 97.226 1.65211 100.879 1.63965 104.579 1.73741 90.000 I.73564 93-683 1.73036 97.380 1.72166 101.107 1.70971 104.880 i. 81066 90.000 1.80895 93-754 1.80389 97-522 1-79555 101.318 1.78409 105.156 1.25 1.4 i-45 1.49487 107.509 1.47479 111.209 1.45193 115.019 1.42675 118.957 1.39976 123.043 1.55747 107.944 1.53820 111.720 1.51630 115.599 1.49220 119.597 1.46641 123.732 1.62404 108.344 1.60558 112.189 1.58460 116.130 1.56156 120.182 1.53694 124.360 1.69472 108.713 1.67705 112.620 1.65699 116.616 1.63497 120.716 1.61147 124.931 1.76975 109.050 1.75282 113.014 1-73363 117-061 1.71260 121.203 1.69018 125.451 i-SS 1.6 1.65 1.37153 127.293 1.34272 131.724 1.31400 136.349 1.28612 141.179 1.25984 146.216 1.43950 128.019 1.41207 132.470 1.38479 137.098 1.35837 141.910 1 -3335 1 146.907 1.51128 128.677 1.48517 133.145 1.45926 137.773 1.43421 142.565 1.41070 147.523 1.58701 129.275 1.56217 133-755 I.53756 138.380 1.51381 143-153 1.49155 148.074 1.66688 129.816 1.64325 134-307 1.61987 138.927 1-59733 143-681 1.57626 148.567 1-75 1.8 1.85 1.9 1.23594 151.459 1.21515 156.895 1.19816 162.504 1.18552 168.252 1.17782 174.100 1.31095 152.085 1.29137 157.432 1.27540 162.929 1.26358 168.547 1.25631 174-252 1.38939 152.641 I-37093 I57-908 1.35590 163.306 1.34478 168.808 1-33795 174-385 1.47141 153.136 1.45399 158.330 1.47983 163.639 1.41936 169.039 1.42294 174-503 1.55721 153.579 1.54077 158.706 1.52741 163.935 1.51755 169.243 1.51150 174.607 2.00 1.17520 180.000 1.25386 180.000 I -33565 180.000 1.42078 180.000 1.50946 180.000 Example. sinh (1.20 + i 1.25) = 1.76975^109^.050 = 1.76995 /i09 .03'.oo*. [94] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED = 1.25 x = 1.3 * = i-3S x = 1.4 x = 1.45 o 0.05 O.I 0.15 O.2 1.60192 o.ooo 1.60384 5.301 1.60954 10.576 1.61884 15.803 1.63145 20.958 1.69838 o.ooo 1.70019 5.218 1.70557 10.415 1.71435 15.568 1.72627 20.659 1.79909 o.ooo 1.80080 5.145 1.80588 10.271 1.81417 15-359 1.82544 20.392 1.90430 o.ooo 1.90592 5.080 1.91072 10.143 1.91856 15.172 1.92921 20.153 2.01427 o.ooo 2.OI58O 5-021 2.02034 IO.028 2.02775 15.005 2.03784 19-939 0.25 o-35 0.4 0-45 1.64699 26.026 1.66501 30.991 1.68497 35.845 1.70635 40.580 1.72856 45-195 1.74096 25.673 1.75802 30.595 1.77694 35.418 1.79722 40.135 1.81832 44-745 1.83934 25.356 I.85549 30.240 I.87343 35-035 1.89268 39.735 1.91273 44-338 1.94237 25.073 1.95767 29.921 1.97468 34.689 1.99295 39.373 2.OI2OO 43.97O 2.O5O3O 24.8l8 2.06479 29.634 2.08093 34-379 2.09828 39.O47 2.II637 43.638 o-S 0-55 0.6 0.65 0.7 1.75104 49-693 1.77323 54-077 1.79462 58.354 1.81470 62.533 1.83304 66.625 1.83970 49.248 1.86084 53.648 1.88123 57.950 1.90040 62.163 1.91792 66.295 1.93306 48.845 i-953 I 9 53-258 1.97262 57-583 1.99091 61.825 2.00764 65.994 2.03135 48.480 2.05051 52.905 2.06903 57-249 2.08647 61.518 2.10244 65.719 2.13478 48.150 2.15302 52.584 2.17067 56.945 2.I873I 61.238 2.20254 65.469 0-75 0.8 0.85 0.9 o-95 1.84924 70.640 1.86297 74.590 1.87394 78.489 1.88193 82.348 1.88679 86.180 1.93341 70.356 1.94654 74-358 1.95704 78.311 1.96470 82.228 1.96935 86.120 2.02245 70.097 2.03500 74-145 2.04505 78.149 2.05238 82.118 2.05684 86.065 2.11658 69.861 2.12858 73.951 2.13819 78.00O 2.I452O 82.0l8 2.14947 86.014 2.2I6O4 69.645 2-22751 73-774 2.23669 77.864 2.24334 81.925 2.24747 85.968 I.O 1.05 i.i i.i5 1.2 1.88842 90.000 1.88679 93.820 1.88193 97.652 1.87394 101.511 1.86297 105.410 1.97091 90.000 1.96935 93.880 1.96470 97-772 1.95704 101.689 1.94654 105.642 2.05833 90.000 2.05684 93-935 2.05238 97.882 2.04505 101.851 2.03500 105.855 2.15090 9O.OOO 2.14947 93.986 2.14520 97-982 2.13819 I02.OOO 2.12858 IO6.O49 2.24884 90.000 2.24747 94-032 2.24334 98-075 2.23669 102.136 2.22751 106.226 1.25 i-35 1.4 1-45 1.84924 109.360 1.83304 113.375 1.81470 117.467 1.79462 121.646 1.77323 125.923 1.93341 109.644 1.91792 113.705 1.90040 117.837 1.88123 122.050 1.86084 126.352 2.02245 109.903 2.00764 114.006 1.99091 118.175 1.97262 122.417 1.95319 126.742 2.11658 IIO.I39 2.IO244 II4.28I 2.08647 118.482 2.06903 122.751 2.05051 127.095 2.21604 110.355 2.20254 H4-53 1 2.18731 118.762 2.17067 123.055 2.15302 127.416 1-55 1.6 1.65 1.75104 130.308 1.72856 134.805 1.70635 139.420 1.68497 144.155 1.66501 149.009 1.83970 130.752 1.81832 135.255 1.79722 139.865 1.77694 144.582 1.75802 149-405 1.93306 131.155 1.91273 135.662 1.89268 140.265 1.87343 144-965 1.85549 149.760 2.03135 131.520 2.OI20O 136.030 1.99295 140.627 1.97468 145-3" L95767 150.079 2.13478 131.851 2.11637 136.362 2.09828 140.953 2.08093 I45-62I 2.06479 150.366 1-75 1.8 1.85 1.9 1-95 1.64699 153.974 1.63145 159.042 1.61884 164.197 1.60954 169.424 1.60384 174.699 1.74096 154-327 1.72627 159-341 1.71435 164.432 1.70557 169.585 1.70019 174.782 1.83934 154-644 1.82544 159.608 1.81417 164.641 1.80588 169.729 1.80080 174.855 1.94237 I54.927 I.9292I 159.847 1.91856 164.828 1.91072 169.857 1.90592 174.920 2.05030 155.182 2.03784 160.061 2.02775 164.995 2.02034 169.972 2.01580 174.979 2.O 1.60192 180.000 1.69838 180.000 1.79909 180.000 1.90430 iSo.OOO 2.01427 180.000 Example. sinh (1.45 + i 1.70) = 2.06479 /I50.366 = 2.06479 /I SO . 2,2 '.01'. [95] TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r /V CONTINUED x = 1.50 X = 1 .60 x = 1.65 a; = 1.70 o 2.12928 o.ooo 2.24961 o.ooo 2-37557 o.ooo 2.50746 o.ooo 2.64563 O.OOO 0.05 2.13073 4.969 2.25098 4-923 2.37686 4.881 2.50869 4.843 2.64679 4.809 O.I 2.13502 9-925 2.25504 9.833 2.38071 9-751 2.51234 9.677 2.65025 9.610 0.15 2.14204 14-855 2.26169 14.721 2.38701 14.600 2.51831 14.492 2.65591 14-395 0.2 2-15159 19.746 2.27074 19-574 2.39558 19.419 2.52644 19.280 2.66361 19.155 0.25 2.16340 24.590 2.28193 24-385 2.40619 24.200 2.53650 24.034 2.67316 23.884 -3 2.17714 29.376 2.29496 29.144 2.41856 28.935 2.54824 28.747 2.68430 28.578 0-35 2.19245 34.099 2.30949 33-846 2-43235 33- 6l 9 2.56133 33-415 2.69673 33.229 0.4 2.20892 38.753 2.32513 38.488 2-44721 38.248 2-57544 38.032 2.71014 37.837 0-45 2.22612 43-337 2.34148 43.066 2.46274 42.820 2.59021 42.599 2.72418 42.398 0-5 2.24362 47.850 2.35812 47.580 2-47857 47-334 2.60527 47.113 2.73850 46.911 o-SS 2.26098 52-294 2.37465 52.030 2.49430 Si-79 1 2.62024 51.574 2.75274 51-378 0.6 2.27779 56.670 2.39066 56-420 2-50955 56.192 2.63476 55.986 2.76657 55-799 0.65 2.29365 60.984 2.40577 60.752 2-52395 60.542 2.64847 60.351 2.77963 60.178 0.7 2.30819 65.241 2.41976 65-033 2.53717 64.845 2.66108 64.673 2.79164 64-517 0-75 2.32107 69.448 2.43*93 69.268 2.54890 69.105 2.67226 68.956 2.80230 68.821 0.8 2.33202 73.611 2.44238 73-464 2-55887 73-329 2.68178 73.206 2.81138 73-094 0.85 2.34080 77.740 2.45076 77.626 2.56687 77-523 2.68941 77.429 2.81867 77-343 0.9 2.34720 81.842 2.45688 81.765 2.57272 81.695 2.69499 81.631 2.82399 Si-573 o-95 2.35110 85-925 2.46061 85-887 2.57627 85-851 2.69839 85.819 2.82723 85.790 I.O 2.35241 90.000 2.46186 90.000 2.57746 90.000 2.69951 90.000 2.82832 90.000 1.05 2.35110 94-075 2.46061 94-113 2.57627 94.149 2.69839 94.181 2.82723 94.210 i.i 2.34720 98.158 2.45688 98.235 2.57272 98-305 2.69499 98.369 2.82399 98.427 i-iS 2.34080 102.260 2.45076 102.374 2.56687 102.477 2.68941 102.571 2.81867 102.657 1.2 2.33202 106.389 2.44238 106.536 2.55887 106.671 2.68178 106.794 2.81138 106.906 1-25 2.32107 110.552 2.43*93 110.732 2.54890 110.895 2.67226 111.044 2.80230 111.179 i-3 2.30819 114-759 2.41976 114.967 2.53717 115-155 2.66108 "5-327 2.79164 115-483 i-3S 2.29365 119.016 2.40577 119.248 2-52395 119.458 2.64847 .119.649 2.77963 119.822 1.4 2.27779 123.330 2.39066 123.580 2-50955 123.808 2.63476 124.014 2.76657 124.201 1-45 2.26098 127.706 2.37465 127.970 2.49430 128.209 2.62024 128.426 2.75274 128.622 i-5 2.24362 132.150 2.35812 132.421 2.47857 132.666 2.60527 132.887 2.73850 133-089 i-55 2.22612 136.663 2.34148 136.934 2.46274 137.180 2.59021 137.401 2.72418 137.602 1.6 2.20892 141.247 2.32513 141.512 2.44721 141.752 2-57544 141.968 2.71014 142.163 1.65 2.19245 145.901 2.30949 146.154 2-43235 146.381 2-56133 146.585 2.69673 146.771 i-7 2.17714 150.624 2.29496 150.856 2.41856 151.065 2.54824 I5L253 2.68430 151.422 i-7S 2.16340 I55.4IO 2.28193 I55-6I5 2.40619 155.800 2.53650 155.966 2.67316 156.116 1.8 2-15159 160.254 2.27074 160.426 2.39558 160.581 2.52644 160.720 2.66361 160.845 1.85 2.14204 165.145 2.26169 165.279 2.38701 165.400 2.51831 165.508 2.65591 165.605 1.9 2.13502 170.075 2.25504 170.167 2.38071 170.249 2.51234 170.323 2.65025 170.390 i-95 2.13073 I75-03I 2.25098 175.077 2.37686 175-119 2.50869 I75-I57 2.64679 I75-I9I 2.O 2.12928 180.000 2.24961 180.000 2-37557 180.000 2.50746 180.000 2.64563 180.000 Example. sinh (1.55 + i 0.60) = 2.39066 /56.42O = 2.39066 /56. 25'. 12*. [96] TABLE X. HYPERBOLIC SINES, sinh (* + iq) = r fy. CONTINUED x = i-75 * = 1.8 x = 1.85 x = 1.9 x = i-95 9 r 7 r V r T r y r 7 o o o 2.79041 o.oo 2.94217 o.oo 3 .10129 o.oo 3.26816 0.00 3-44321 o.oo 0.05 2.79151 4-779 2.94322 4.752 3 .10228 4.727 3.26911 4-705 3.44410 4.685 O.I 2.79479 9-55 1 2-94633 9-497 3 10523 9.448 3.27191 9-405 3-44675 9-365 0.15 2.80016 I4-307 2.95142 14.228 3 ,11006 14.158 3.27649 14.094 3-45H2 14.036 0.2 2.80747 19.042 2-95835 18.941 3 11665 18.850 3.28274 !8. 7 67 3-45702 18.693 0.25 2.81653 23-750 2.96695 23.629 3' 12481 23.520 3.29049 23.421 3.46441 23-332 o-3 2.82710 28.425 2-97699 28.287 3' !3434 28.163 3-29955 28.051 3-47301 27.950 o-3S 2.83891 33-063 2.98821 32.912 3- 14500 32.776 3.30967 32-654 3.48263 32.543 0.4 2.85165 37.661 3.00031 37-501 3- 15650 37-357 3.32060 37-227 3-49302 37.110 0-45 2.86499 42.216 3.01300 42-053 3- 16856 41.904 3-33207 41.770 3-50393 41.649 o-5 2.87861 46.730 3-02595 46-565 3- 18088 46.416 3-34378 46.281 3-5 J 507 46.160 o-SS 2.89217 51.200 3-03885 5I-039 3- I 93 I 5 50-893 3-35546 50.761 3.52617 50.642 0.6 2.90532 55-630 3-05138 55-4/6 3' 20507 55-337 3.36681 55-211 3-53698 55.096 0.65 2.91777 60.020 3.06323 59-877 3- 21636 59-748 3-37756 59-63I 3-54721 59.524 0.7 2.92921 64-375 3-07413 64.246 3- 22675 64.1 29 3-38745 64.023 3-55663 63.927 0-75 2-93938 68.698 3.08382 68.586 3' 23598 68.484 3-39624 68.392 3-56500 68.308 0.8 2.94804 72.992 3.09206 72.900 3- 24384 72.816 3-40373 72.740 3-572I3 72.670 0.85 2.95498 77.265 3-09869 77-194 3- 25015 77.129 3-40975 77-071 3-57788 77.017 0.9 2.96006 81.520 3- J 0353 81.471 3- 25477 81.428 3-4i4i5 81.388 3.58207 81.352 0-95 2.96315 85-763 3.10648 85-738 3- 25759 85-716 3.41683 85.696 3.58462 85.678 I.O 2.96419 90.000 3.10747 90.000 3- 25853 90.000 34I773 90.000 3-58548 90.000 1.05 2.96315 94-237 3.10648 94.262 3' 25759 94.284 3.41683 94-304 3.58462 94.322 i.i 2.96006 98.480 3-10353 98.529 3. 25477 98.572 3-41415 98.612 3.58207 98.648 i-i5 2.95498 102.735 3.09869 102.806 3 25015 102.871 340975 102.929 3-57788 102.983 1.2 2.94804 107.008 3.09206 107.100 3 24384 107.184 3.40373 107.260 3-572I3 107.330 I-2S 2-93938 111.302 3.08382 111.414 3 23598 111.516 3-39624 111.608 3-56500 111.692 i-3 2.92921 115-625 3-074I3 115-754 3 .22675 115.871 3-38745 "5-977 3-55663 116.073 i-3S 2.91777 119.980 3-06323 120.123 3 .21636 120.252 3-37756 120.369 3-54721 120.476 i-4 2.90532 124.370 3-05138 124.524 3 .20507 124.663 3.36681 124.789 3-53698 124.904 i -45 2.89217 128.800 3-03885 128.961 3 I93I5 129.107 3-35546 129.239 3.52617 129.358 i-5 2.87861 133.270 3-02595 133-435 3 .18088 133-584 3-34378 I33-7I9 3-5I507 133.840 i-SS 2.86499 I37-784 3.01300 137-947 3 .16856 138.096 3-33207 138.230 3-50393 138.351 1.6 2.85165 142.339 3.00031 142.499 3 15650 142.643 3.32060 142.773 3-49302 142.890 1.65 2.83891 146.937 2.98821 147.088 3 .14500 147.224 3-30967 147.346 3.48263 147-457 i-7 2.82710 I5I-575 2-97699 I5L7I3 3 13434 151-837 3-29955 151.949 3-473 01 152.050 i-7S 2.81653 156.250 2-96695 156.371 3 .12481 156.480 3.29049 156.579 3.46441 156.668 1.8 2.80747 160.958 2-95835 161.059 3 .11665 161.150 3.28274 161.233 3-45702 161.307 1.85 2.80016 165.693 2.95142 165.772 3 .11006 165.842 3-27649 165.906 3-45ii2 165.964 1.9 2.79479 170.449 2-94633 170.503 3 10523 170.552 3.27191 170.595 3-44675 170.635 1-95 2.79151 175.221 2.94322 175.248 3 .10228 I75.273 3.26911 I75.295 3.44410 J75-3I5 2.O 2.79041 iSo.OOO 2.94217 iSo.OOO 3.IOI29 iSo.OOO 3.26816 iSo.OOO 3.44321 iSo.OOO Example. sinh (1.90 + * 2.0) = 3.26816 /i8o.o = 3.26816 \i8o.o. [97] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED X = 2.O * = 2.05 * = 2.1 * = 2.15 * = 2.2 g r i f 7 r 7 r V f V o o o o o 3.62686 o.ooo 2.81950 o.ooo 4.02186 o.ooo 4.23419 o.ooo 4.45711 o.ooo 0.05 3.62771 4-667 3.82039 4.651 4.02263 4.636 4.23491 4.623 4.45779 4.611 O.I 3.63023 9.330 3.82279 9.298 4.02490 9.269 4.23707 9.243 4.45985 9.220 0.15 3.63437 13.984 3.82671 13.938 4.02863 13.895 4.24061 13.857 4.46322 13.823 0.2 3.64000 18.626 3.83206 18.566 4.03371 18.511 4.24544 18.462 4.46780 18.418 0.25 3.64699 23.252 3.83870 23.180 4.04002 23.114 4.25145 23.055 4.47350 23.002 o-3 3.65517 27.858 3.84648 27.776 4.04740 27.701 4.25846 27.634 4.48017 27.573 0-35 3.66430 32.443 3.85515 32.353 4.05566 32.271 4.26630 32.197 4.48763 32.130 0.4 3.67418 37.004 3.86454 36.908 4.06459 36.821 4-27479 36.743 4.49570 36.671 0-45 3.68455 41.539 3.87440 41.440 4.07396 41-351 4.28371 41.270 4.50417 41.196 0-5 3.69515 46.049 3.88448 45-950 4.08355 45.859 4.29282 45.778 4.51285 45.703 o-SS 3.70572 50.533 3-89453 50-435 4.09311 50.347 4-3 OI 93 50-266 4.52150 50.193 0.6 3.71600 54-992 3.90432 54.898 4.10243 54.813 4.31079 54.736 4-52993 54-666 0.65 3.72574 59-427 3-91359 59-340 4.11125 59.260 4.31918 59.188 4-53793 59-123 0.7 3-73470 63.840 3.92213 63.761 4.11938 63.689 4.32693 63.624 4.54529 63.565 0-75 3.74268 68.232 3.92972 68.164 4.12661 68.101 4.33381 68.044 4.55184 67.993 0.8 3.74948 72.608 3.93620 72.551 4.13278 72-499 4.33968 72-452 4-55744 72.409 0.85 3-75495 76-969 3.94142 76.925 4.13774 76.885 4.34440 76.849 4.56167 76.816 0.9 3.75894 81.319 3.94521 81.289 4.14136 81.262 4.34785 81.237 4.56523 81.215 o.QS 3.76137 85.661 3-94753 85.646 4.14357 85.632 4.34996 85.620 4-56723 85.609 1.0 3.76220 oo.ooo 3.94832 90.000 4.14431 90.000 4-35o67 go- 000 4.56791 90.000 1.05 3.76137 94.339 3-94753 94-354 4-14357 94-368 4.34996 94-38o 4.56723 94.391 .1 3.75894 98.681 3.94521 98.711 4.14136 98-738 4.34785 98.763 4-56523 98-785 IS 3.75495 103.031 3.94142 103.075 4.13774 103.115 4.34440 103.151 4.56167 103.184 .2 3.74948 107.392 3.93620 107.449 4.13278 107.501 4.33968 107.548 4-55744 107.59! 25 3.74268 111.768 3.92972 111.836 4.12661 111.899 4.33381 111.956 4.55184 112.007 3 3.73470 116.160 3.92213 116.239 4.11938 116.311 4.32693 116.376 4.54529 116.435 35 3.72574 120.573 3.91359 120.660 4.11125 120.740 4.31918 120.812 4-53793 120.877 4 3.71600 125.008 3.90432 125.102 4.10243 125.187 4.31079 125.264 4.52993 125.334 145 3.70572 129.467 3.89453 129.565 4.09311 129.653 4.30193 129.734 4.52150 129.807 i-5 3.69515 133.951 3.88448 134-051 4-08355 134-141 4.29282 134.223 4.51285 134-297 i-SS 3.68455 138.461 3.87440 138.560 4.07396 138.649 4.28371 138.730 4.50417 138.804 1.6 3.67418 142.996 3.86454 143.092 4.06459 143-179 4.27479 143-257 4.49570 143-329 1.65 3.66430 147-557 3.85515 147-647 4.05566 147-729 4.26630 147.803 4.48763 147.870 i-7 3.65517 152.142 3.84648 152.224 4.04740 152.299 4.25846 152.366 4.48017 152.427 75 3.64699 156.748 3.83870 156.820 4.04002 156.886 4.25145 156.945 4.47350 156.998 .8 3.64000 161.374 3.83206 161.434 4.03371 161.489 4.24544 161.538 4.46780 161.582 85 3.63437 166.016 3.82671 166.062 4.02863 166.105 4.24061 166.143 4.46322 166.177 9 3.63023 170.670 3.82279 170.702 4.02490 170.731 4.23707 170.757 445985 170.780 95 3.62771 175.333 3.82039 175-349 4.02263 175.364 4.23491 175-377 4-45779 175-389 2.0 3.62686 180.000 3.81958 180.000 4.02186 180.000 4.23419 180.000 4.45711 180.000 Example.^ sinh (2.0 + i i.o) = 3.76220 /go . [98] TABLE X. HYPERBOLIC SINES, sinh (x + ig) = r fy. CONTINUED * = 2.25 x = 2.3 * = 2.35 x = 2.4 x = 245 4.69117 o.ooo 4.93696 o.ooo 5 19510 o.ooo 5.46623 o.ooo 5-75*03 o.ooo 0.05 4.69182 4.601 4-93759 4-591 5 19569 4.582 5.46679 4-574 5.75I56 4-567 O.I 4-69377 9.199 4-93944 9.180 5 I974S 9.163 5.46847 9.147 5-753i6 9-133 0.15 4.69697 13-792 4.94249 13-764 5- 20035 13.739 5.47122 13.716 5-75576 13-695 0.2 4.70134 18.378 4.94662 18.341 5 20428 18.309 5-47495 18.279 5-75932 18.252 0.2 S 4.70675 22.954 4-95I77 22.910 5 20917 22.871 5.47961 22.835 5-76375 22.803 o-3 4.71308 27.518 4.95780 27.469 5 21490 27.424 5-48505 27-383 5-76892 27-347 0-35 4.72017 32.070 4.96454 32.016 5 22131 31.966 549 II 5 31.922 5.77472 31.881 0.4 4.72784 36.608 4.97183 36.549 5 22825 36-497 549775 36449 5.78099 36.407 4S 4-73592 41.130 4-97950 41.070 5 23553 41.016 5.50468 40.966 5.78758 40.922 S 4.74415 45-636 4-98734 45-576 5 23728 45.521 5.5"77 45471 5-79434 45427 o-SS 4.75240 50.127 4.99518 50.068 5 25046 50.014 5-51887 49.965 5.80108 49.921 0.6 4.76042 54-603 5.00281 54-546 5 25772 54-494 5.52578 54-447 5.80765 54405 0.65 4.76802 59.064 5.01005 59-on 5 26461 58.962 5.53233 58.919 5-81389 58.879 0.7 4-775 3 63.512 5.01672 63-463 5 27096 63.419 5.53837 63.380 5.81964 63-344 0-75 4.78127 67.947 5.02265 67.904 5 27662 67.866 5-54375 67-831 5.82476 67.800 0.8 4.78661 72.371 5-02773 72.336 5 28143 72.304 5-54834 72.275 5-82913 72.249 0.85 4.79088 76.786 5.03181 76.759 5 28531 76.735 5-55204 76.712 5-83265 76.692 0.9 4.79402 81.195 5-03479 81.176 5 28816 81.160 5-55474 81.144 5-83522 81.131 o-95 4-79593 85-599 5.03661 85-589 5 28989 85.581 5.55639 85-573 5-83680 85.566 1.0 4-79657 90.000 5.03722 90.000 5 29047 90.000 5.55695 90.000 5.83732 90.000 1.05 4-79593 94.401 5.03661 94.411 5 28989 94.419 5.55639 94.427 5.83680 94434 i.i 4.79402 98.805 5-03479 98.824 5 28816 98.840 5-55474 98.856 5-83522 98.869 i-iS 4.79088 103.214 5.03181 103.241 5 28531 103.265 5-55204 103.288 5-83265 103.308 1.2 4.78661 107.629 5-02773 107.664 5 28143 107.696 5.54834 107.725 5-82913 107.751 I-2S 4.78127 112.053 5.02265 112.096 5 27662 112.134 5-54375 112.169 5.82476 112. 2OO i-3 4-77503 116.488 5.01672 "6.537 5 27096 116.581 5-53837 116.620 5.81964 116.656 i-35 4.76802 120.936 5.01005 120.989 5 26461 121.038 5-53233 121.081 5-81389 121. 121 14 4.76042 125-397 5.00281 125-454 5 25772 125.506 5.52578 125.553 5-80765 125.595 145 4.75240 129.873 4.99518 129.932 5 25046 129.986 5-51887 130-035 5.80108 130.079 -5 4-744I5 134-364 4-98734 134.424 5 23728 134-479 5-5"77 134.529 5-79434 134-573 55 4-73592 138.870 4-9795 138.930 5 23553 138.984 5.50468 I39.034 5-78758 139.078 .6 4.72784 143-392 4.97183 I4345I 5 22825 143-503 549775 I43.55I 5.78099 143-593 .65 4.72017 147.930 4-96454 147.984 S 22131 148.034 549"5 148.078 5-77472 I48.II9 7 4.71308 152.482 4.95780 152-531 5 21490 152.576 548505 152.617 5-76982 152.653 75 4-70675 157.046 4.95I77 157.090 5 20917 157.129 5.47961 157-165 5.76375 I57-I97 .8 4-70I34 161.622 4.94662 161.659 5 20428 161.691 547495 161.721 5-75932 161.748 85 4.69697 166.208 4.94249 166.236 5 20035 166.261 5.47122 166.284 5.75576 166.305 9 4.69377 170.801 4-93944 170.820 5 I974S 170.837 5.46847 170.853 5-753i6 170.867 1-95 4.69182 175.399 4-93759 175.409 5 19569 175.418 5.46679 175.426 5-75*56 175433 2.O 4.69117 180.000 4.93696 180.000 5 19510 180.000 546623 180.000 5-75I03 iSo.OOO Example. sinh (2.40 + * 0.4) = 549775 /36-449 = 5-49775 /36. 26^.56'. [99] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED x = 2.5 = 2.55 * = 2.6 x = 2.65 x = 2.7 o 0.05 O.I 0.15 O.2 6.05020 6.05071 6.05223 6.05471 6.05808 o.ooo 4.561 9.120 13.676 18.228 6.36451 6.36499 6.36644 6.36879 6.37201 o.ooo 4-555 9.109 13.660 18.206 6.69473 6.69518 6.69656 6.69880 6.70185 o.ooo 4-550 9.098 13.644 18.187 7.04169 7.04213 7-04343 7-04556 7.04847 0.000 4-545 9.089 13-631 18.169 7.40626 7.40668 7.40791 7.40994 7.41271 o.ooo 4-540 9.080 13.618 18.153 O.25 o-3 0-35 0.4 0-45 6.06229 6.06722 6.07273 6.07869 6.08496 22.774 27-314 3!-845 36.368 40.882 6.37601 6.38068 6.38592 6.39160 6-39755 22.748 27.284 31.812 36.333 40.845 6.70565 6.71011 6.71509 6.72048 6.72616 22.725 27.257 31.782 36.301 40.813 7.05208 7.05631 7.06105 7.06618 7.07158 22.703 27.232 3I-756 36.273 40.783 7.41614 7.42017 7-42467 742955 7.43468 22.684 27.210 36.246 40.756 o-S 0-55 0.6 0.65 0.7 6.09139 6.09781 6.10406 6.10999 6.11547 45-386 49.881 54-366 58-843 63.3" 6.40367 6.40977 6.41572 6.42137 6.42658 45-350 49.845 54-332 58.810 63.282 6.73197 6.73778 6-74343 6.74881 6.75376 45-316 49.812 54-300 58.781 63.255 7.07711 7.08264 7.08802 7-09313 7.09784 45.286 49.782 54.272 58.754 63.230 7-43994 7-445I9 745032 7-455I8 7.45966 45-259 49-755 54.246 58.730 63.209 o-75 0.8 0.85 0.9 0-95 6.12034 6.12450 6.12785 6.13030 6.13179 67.772 72.226 76.674 81.119 85.560 6.43121 6.43518 6.43836 6.44069 6.44212 67.746 72.204 76.658 81.107 85-554 6.75818 6.76195 6.76499 6.76720 6.76855 67.722 72.185 76.643 81.097 85.549 7.10204 7.10563 7.10851 7.11062 7.11191 67.701 72.167 76.629 81.088 85-544 7.46366 7.46708 7.46982 7-47I83 7.47306 67.682 72.152 76.617 81.080 85-540 I.OO 1.05 I.I 1-15 1.2 6.13229 6.13179 6.13030 6.12785 6.12450 90.000 94.440 98.881 103.326 107.774 6.44259 6.44212 6.44069 6.43836 6.43518 90.000 94-446 98-893 103.342 107.796 6.76901 6.76855 6.76720 6.76499 6.76195 90.000 94.451 98.903 103.357 107.815 7.11234 7.11191 7.11062 7.10851 7.10563 90.000 94.456 98.912 103.371 107.833 7-47347 7.47306 7-47i83 7.46982 7.46708 90.000 94.460 98.920 103-383 107.848 1-25 i-35 1.4 1-45 6.12034 6.11547 6.10999 6.10406 6.09781 112.228 116.689 121.157 125.634 130.119 6.43121 6.42658 6.42137 6.41572 6.40977 112.254 116.718 121.190 125.668 130.155 6.75818 6.75376 6.74881 6-74343 6-73778 112.278 116.745 121.219 125.700 130.188 7.10204 7.09784 7-093I3 7.08802 7.08264 112.299 116.770 121.246 125.728 130.218 7.46366 7.45966 745032 7-445I9 112.318 116.791 121.270 125.754 130.245 i-55 1.6 1-65 6.09139 6.08496 6.07869 6.07273 6.06722 134.614 139.118 143.632 148.155 152.686 6.40367 6-39755 6.39160 6.38592 6.38068 134-651 i39- I 55 143.667 148.188 152.716 6.73197 6.72616 6.72048 6.71509 6.71011 134.684 139.187 143.699 148.218 7.07711 7.07158 7.06618 7.06105 7.05631 I34-7I3 139.217 I43-727 148.244 152.768 7-43994 7.43468 7-42955 7.42467 7.42017 I34.74I 139.244 143-754 148.269 152.790 i-75 1.8 1-85 1.9 1-95 6.06229 6.05808 6.05471 6.05223 6.05071 157.226 161.772 166.324 170.880 175-439 6.37601 6.37201 6.36879 6.36644 6.36499 157-252 161.794 166.340 170.891 175-445 6.70565 6.70185 6.69880 6.69656 6.69518 157-275 161.813 166.356 170.902 175-450 7.05208 7.04847 7-04566 7-04343 7.04213 157.297 161.831 166.369 170.911 175-455 7.41614 7.41271 7.40994 7.40791 7.40668 157.316 161.847 166.382 170.920 175.460 2.0 6.05020 180.000 6.36451 180.000 6.69473 180.000 7.04169 180.000 7.40626 180.000 Example. sinh (2.6 + io.6) = 6.74343 /54-300 = 6.74343 /54-i8'oo". [ 100 ] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED x = 2.75 = 2.85 * = 2.9 x = 2.95 7.78935 o.ooo 8.19192 o.ooo 8.61497 o.ooo 9-05956 o.ooo 9.52681 o.ooo o.os 7.78975 4-537 8.19230 4-533 8.61532 4.530 9.05990 4-527 9-52713 4.525 O.I 7.79092 9-073 8.19341 9.066 8.61639 9.060 9.06091 9-054 9-52809 9.049 0.15 7.79285 13.607 8.19524 I3-596 8.61813 13-587 9.06257 13.579 9.52967 I3.57I 0.2 7-79547 18.138 8.19774 18.125 8.62051 18.113 9.06483 18.102 9.53182 18.093 0.25 7-79875 22.666 8.20085 22.650 8.62347 22.636 9.06764 22.623 9-53450 22.611 7.80257 27.190 8.20449 27.172 8.62691 27-155 9.07093 27.141 27.127 0-35 7.80685 31.709 8.20857 31.689 8.63080 31.671 9.07461 31.655 9-54"3 31.640 0.4 7.81149 36.223 8.21298 36.202 8.63500 36.183 9.07861 36.165 9-54492 36-150 0-45 7-81637 40.731 8.21762 40.710 8.63941 40.690 9.08281 40.671 9.54892 40.655 o-5 7.82138 45-234 8.22238 45.212 8.64395 45.192 9.08711 45.173 9-55301 45-157 0.55 7.82638 49-731 8.22713 49.709 8.64846 49.689 9.09142 49.671 9-557" 49.655 0.6 7-83125 54-222 8.23177 54-201 8.65287 54.182 9.09561 54.165 9.56109 54-149 0.65 7-83588 58.708 8.23617 58.688 8.65706 58.670 9.09959 58.654 9.56488 58.640 0.7 7-84015 63.189 8.24024 63.171 8.66092 63-155 9.10327 63.140 9-56838 63.127 0-75 7.84395 67-665 8.24385 67.649 8.66436 67.636 9.10655 67.622 9.57I50 67.611 0.8 7.84720 72.137 8.24694 72.124 8.66731 72.112 9.10934 72.102 9.57416 72.092 0.85 7.84980 76.606 8.24942 76-596 8.66967 76.587 9.11160 76.579 9-57630 76.571 0.9 7.85172 8l.O72 8.25124 81.065 8.67140 81.059 9.11324 81.053 9-57787 81.048 0-95 7.85288 85-536 8-25235 85-533 8.67246 85-530 9.11424 85.527 9.57882 85-524 I.O 7-85328 90.000 8.25273 90.000 8.67281 90.000 9.11458 90.000 9-579I5 90.000 1.05 7.85288 94.464 8-25235 94467 8.67246 94-470 9.11424 94-473 9-57882 94.476 i.i 7.85172 98.928 8.25124 98.935 8.67140 98.941 9.11324 98.947 9-57787 98.952 1.15 7.84980 103.394 8.24942 103.404 8.66967 103.413 9.11160 103.421 9-57630 103.429 1.2 7.84720 107.863 8.24694 107.876 8.66731 107.888 9.10934 107.898 9.57416 107.908 1-25 7.84395 "2-335 8.24385 "2.351 8.66436 112.364 9.10655 112.378 9-57I50 112.389 7.84015 116.811 8.24024 116.829 8.66092 116.845 9.10327 116.860 9.56838 116.873 1-35 7-83588 121.292 8.23617 121.312 8.65706 121.330 9.09959 121.346 9.56488 121.360 7-83125 125.778 8.23177 125.799 8.65287 125.818 9.09561 125-835 9.56109 125-851 1-45 7.82638 130.269 8.22713 130.291 8.64846 130.311 9.09142 130.329 9-557" I30.345 i-5 7.82138 134.766 8.22238 134.788 8.64395 134.808 9.08711 134.827 9-55301 134-843 1-55 7.81637 139.269 8.21762 139.290 8.63941 139.310 9.08281 139-329 9.54892 139-345 1.6 7.81149 143-777 8.21298 143.798 8.63500 143.817 9.07861 143-835 9.54492 143.850 1.65 7.80685 148.291 8.20857 148.311 8.63080 148.329 9.07461 148-345 9-54"3 148.360 7-80257 152.810 8.20449 152.828 8.62691 152.845 9.07093 152.859 152-873 i-75 7-79875 157-334 8.20085 I57-350 8.62347 157-364 9.06764 157-377 9-53450 I57-389 1.8 7-79547 161.862 8.19774 161.875 8.62051 161.887 9.06483 161.898 9.53182 161.907 1-85 7.79285 166.393 8.19524 166.404 8.61813 166.413 9.06257 166.421 9-52967 166.429 1.9 7.79092 170.927 8.19341 170.934 8.61639 170.940 9.06091 1 70.946 9.52809 170.951 1-95 7-7897S 175-463 8.19230 175467 8.61532 175.470 9.05990 175-473 9-52713 175-475 2.0 7-7893S 180.000 8.19192 180.000 8.61497 180.000 9-05956 180.000 9.52681 180.000 Example. sinh (2.95 + i 1.95) = 9-52713 /i75.475 i = 9-527I3 /I7S-28'.30*. [I0t] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /y. CONTINUED * = 3-o * = 3-05 x = 3.1 * = 3-15 X = 3.2 9 r y r y r y f y r y o 0.05 O.I 0.15 O.2 10.01787 o.ooo 10.01820 4.522 10.01910 9.044 10.02060 13.565 10.02260 18.084 10.53399 o.ooo 10.53430 4.520 - 10.53520 9.040 10.53660 13.559 I0 -538so 18.076 o 11.07645 o.ooo 11.07670 4.518 11.07750 9.036 11.07890 13.553 11.08080 18.068 11.64661 o.ooo 11.64690 4.516 11.64770 9.033 11.64890 13.548 11.65070 18.062 12.24588 o.ooo 12.24610 4.515 12.24690 9.030 12.24810 13.543 12.24980 18.056 0.25 0-35 0.4 0-45 10.02520 22.601 10.02820 27.115 10.03150 31.627 10.03510 36.135 10.03890 40.640 10.54090 22.591 10.54380 27.104 10.54690 31.615 10.55040 36.122 10.55400 40.627 11.08310 22.583 11.08580 27.094 11.08880 31.604 11.09200 36.111 11.09540 40.615 11.65290 22.575 11.65540 27.085 11.65830 31.594 11.66140 36.100 11.66470 40.604 12.25180 22.568 12.25430 27.077 12.25700 31.585 12.26000 36.091 12.26310 40.594 o-S 0-55 0.6 0.65 o-7 10.04280 45.142 10.04670 49.640 10.05050 54.135 10.05410 58.626 10.05743 63.114 10.55770 45-129 10.56140 49.627 10.56500 54-122 10.56840 58.614 10.57160 63.104 11.09900 45.116 11.10250 49.615 11.10600 54.110 11.10920 58.604 II.II22O 63.094 11.66810 45.105 11.67140 49.604 11.67470 54-100 11.67780 58.594 11.68060 63.085 12.26630 45.095 12.26950 49.594 12.27260 54.090 12.27550 58.585 12.27820 63.077 0-75 0.8 0.85 0.9 0-95 10.06040 67.600 10.06292 72.083 10.06500 76.564 10.06645 81.044 10.06737 85.522 10.57440 67.591 10.57680 72.075 10.57880 76.558 10.58020 81.040 10.58110 85.520 II.II49O 67.582 II.II72O 72.068 II.II900 76.553 II.I2O4O 81.036 II.I2I2O 85.518 11.68320 67.574 11.68540 72.062 11.68710 76.548 11.68840 81.032 11.68920 85.516 12.28070 67.567 12.28280 72.056 12.28440 76.543 12.28560 81.029 12.28640 85.515 1.0 1-05 i.i 1.2 10.06766 90.000 10.06737 94.478 10.06645 98.956 10.06500 103.436 10.06292 107.917 10.58135 90.000 10.58110 94.480 10.58020 98.960 10.57880 103.442 10.57680 107.925 II.I2I50 9O.OOO II.I2I2O 94.482 II.I2O4O 98.964 II.II900 103.447 II.II72O .107.932 11.68946 90.000 11.68920 94.484 11.68840 98.968 11.68710 103.452 11.68540 107.938 12.28665 90.000 12.28640 94.485 12.28560 98.971 12.28440 103.457 12.28280 107.944 1-25 1.4 i-45 10.06040 112.400 10.05743 116.886 10.05410 121.374 10.05050 125.856 10.04670 130.360 10.57440 112.409 10.57160 116.896 10.56840 121.386 10.56500 125.878 10.56140 130.373 II.II49O II2.4I8 II.II22O 116.906 II.I092O 121.396 II.I0600 125.890 11.10250 130.385 11.68320 112.426 11.68060 116.915 11.67780 121.406 11.67470 125.900 11.67140 130.396 12.28070 112.433 12.27820 116.923 12.27550 121.415 12.27260 125.910 12.26950 130.406 i-S I.6 3 1-65 10.04280 134.858 10.03890 139.360 10.03510 143.865 10.03150 148.373 10.02820 152.885 10.55770 134.871 10.55400 139.373 10.55040 143.878 10.54690 148.385 10.54380 152.896 II.O99OO 134.884 11.09540 139.385 II.O92OO 143.889 II.08880 148.396 11.08580 152.906 11.66810 134.895 11.66470 139.396 11.66140 143.900 11.65830 148.406 11.65540 152.915 12.26630 134.905 12.26310 139.406 12.26000 143.909 12.25700 148.415 12.25430 152.923 i!s 1.85 1.9 1.95 10.02520 157.399 10.02260 161.916 10.02060 166.435 10.01910 170.956 10.01820 175.478 10.54090 157.409 10.53850 161.924 10.53660 166.441 10.53520 170.960 10.53430 175-480 II.083IO 157.417 II.o8o8o 161.931 11.07890 166.447 11.07750 170.964 11.07670 175.482 11.65290 157-425 11.65070 161.938 11.64890 166.452 11.64770 170.967 11.64690 175.484 12.25180 157.432 12.24980 161.944 12.24810 166.457 12.24690 170.970 12.24610 175.485 2.O 10.01787 180.000 J o.53399 180.000 11.07645 iSo.OOO 11.64661 180.000 12.24588 180.000 Example. sinh (3.2 + i i.i) = 12.28560 /98.97i = 12.28560 /98.58'.i6 t [102] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r /T. CONTINUED = 3.25 = 3-3 = 3-35 * = 3-4 * = 3-45 o 12.87578 o.ooo 13-53788 o.ooo 14.23382 o.ooo 14.96536 o.ooo I5-73432 o-ooo 0.05 12.87600 4.514 13.53810 4.512 14.23403 4.511 14.96556 4.510 15-73450 4-509 O.I 12.87670 9.027 13.53878 9.024 14.23470 9.022 14.96616 9.020 15.73510 9.018 0.15 12.87790 13.539 13.53991 13.536 14.23573 13.532 14.96716 13.529 15.73603 13-526 O.2 12.87949 18.051 13.54141 18.046 14.23718 18.041 14.96855 18.038 15.73736 18.034 0.25 12.88146 22.561 13.54322 22.555 14.23897 22.550 14.97023 22.545 15.73898 22.541 o-3 12.88380 27.070 13.54550 27.063 14.24106 27.057 14.97225 27.052 15.74088 27.047 o-3S 12.88637 31.577 13.54796 31.569 14.24339 31.563 14.97448 31.557 15.74300 31.551 0.4 12.88910 36.082 13.55062 36.074 14.24595 36.067 14.97690 36.061 15.74529 36.055 o-45 12.89215 40.585 13.55346 40.577 14.24863 40.570 14.97945 40.563 15.74772 40.557 o-S 12.89518 45.086 13-55633 45.078 14.25137 45.071 14.98205 45.064 15.75020 45.058 0-55 12.89820 49.585 13.55918 49-577 14.25412 49.570 14.98466 49-563 15.75270 49-557 0.6 12.90117 54.082 13.56203 54-074 14.25614 54.067 14.98721 54.061 I5-755IO 54-055 0.65 12.90398 58.577 13.56470 58.569 14.25933 58.563 14.98960 58.557 I5.75740 58.551 0.7 12.90658 63.070 13.56718 63.063 14.26167 63.057 14.99186 63.052 I5-75950 63.047 0-75 12.90888 67.561 13.56936 67.555 14.26377 67.550 14.99385 67.545 15.76144 67.541 0.8 12.91085 72.051 I 3-57 I 23 72-046 14.26556 72.041 14-99555 72.038 15-76303 72-034 0.85 12.91240 76.539 I3-57275 76.535 14.26700 76.532 14.99692 76.529 15-76433 76-526 0.9 12.91360 81.027 13-57387 81.024 14.26805 81.022 14.99790 81.020 15-76530 81.018 o-9S 12.91430 85.514 13-57455 85.512 14.26870 85.511 14.99853 85.510 15-76587 85.509 I.O 12.91456 90.000 I3-57476 90.000 14.26891 90.000 14.99874 90.000 15.76607 90.000 1.05 12.91430 94.486 13-57455 94488 14.26870 94.489 14.99853 94.490 15-76587 94-491 i.i 12.91360 98.973 I3-57387 98-976 14.26805 98.978 14.99790 98.980 15-7653 98-982 i-iS 12.91240 103.461 I3-5727S 103.465 14.26700 103.468 14.99692 103.471 I5-76433 103.474 1.2 12.91085 107.949 13.57123 107.954 14.26556 107.959 14.99555 107.963 15-76303 107.966 1-25 12.90888 112.439 13-56936 112.445 14.26377 112.450 14.99385 112.455 15.76144 112.459 J-3 12.90658 116.930 13.56718 116.937 14.26167 116.943 14.99186 116.948 I5-75950 "6.953 i-3S 12.90398 121.423 13.56470 121.431 14.25933 121.437 14.98960 121.443 I5-75740 121.449 1.4 12.90117 125.918 13.56203 125.926 14.25614 125.933 14.98721 125.939 15.75510 125.945 i-4S 12.89820 130.415 13.55918 130.423 14.25412 130.430 14.98466 130.437 I5-75270 130.443 J -5 12.89518 134.914 I3-55633 I34-922 14.25137 134.929 14.98205 134.936 15.75020 134-942 i-55 12.89215 139.415 13.55346 139-423 14.24863 139.430 14.97945 139.437 15.74772 139-443 1.6 12.88910 143.918 13.55062 143.926 14.24595 143.933 14.97690 143.939 I5.74529 143-945 i.6 S 12.88637 148.423 13.54796 148.431 14.24339 148.437 14.97448 148.443 15.74300 148.449 i-7 12.88380 152.930 I3.54550 152.937 14.24106 152.943 14.97225 152.948 15.74088 152.953 i-75 12.88146 157.439 13.54322 157-445 14.23897 157.450 14.97023 157.455 15.73898 157-459 1.8 12.87949 161.949 13.54141 161.954 14.23718 161.959 14.96855 161.962 I5-73736 161.966 i.8 S 12.87790 166.461 I3-5399I 166.464 14.23573 166.468 14.96716 166.471 15-73603 166.474 1.9 12.87670 170.973 13.53878 170.976 14.23470 170.978 14.96616 170.980 15.73510 170.982 i-95 12.87600 175.486 1^.53810 175.488 14.23403 175-489 14.96556 175.490 I5-73450 I75-49I s.o 12.87578 180.000 13.53788 180.000 14.23382 180.000 14.96536 180.000 I5-73432 180.000 Example. sinh (3.3 + i 1.3) = 13.56718, = 13.56718 /i i6 ..s6'.i3*. [103] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r {y. CONTINUED = 3-5 = 3-55 x = 3.6 * = 3-65 = 3-7 o 16.54263 0.000 17.39230 o.ooo 18.28546 o.ooo 19.22434 o.ooo 20.21129 o.ooo 0.05 16.54281 4.508 17.39248 4-507 18.28560 4-507 19.22448 4.506 20.21140 4.506 O.I 16.54337 9.016 17.39300 9.015 18.28611 9.013 19.22496 9.012 20.21189 9.011 0.15 16.54428 13.524 17.39386 13-522 18.28694 13-519 19.22577 13.518 20.21268 13-516 0.2 16.54550 18.031 I7-39504 18.028 18.28805 18.025 19.22681 18.023 20.21365 18.021 O.25 16.54702 22.537 17.39650 22.534 18.28942 22.530 19.22813 22.527 20.21490 22.525 o-3 16.54885 27.042 17.39822 27.038 18.29109 27-035 19.22968 27.031 20.21639 27.028 o-3S 16.55087 31-547 17.40015 31-542 18.29292 31.538 19.23142 3I-S3S 20.21804 3I-S31 04 16.55307 36.050 17.40222 36.045 18.29490 36.041 I9-2333 1 36.037 20.21981 36-033 0-45 16.55538 40-552 17.40441 40.547 18.29699 40.542 I9-23530 40.538 20.22170 40-535 o-S 16.55774 45-052 17.40665 45-047 18.29912 45-043 19.23732 45-039 20.22365 45-035 0-55 16.56010 49-552 17.40900 49-547 18.30126 49.542 19.23940 49.538 20.22560 49-535 0.6 16.56240 54-050 17.41110 54-045 18.30335 54.041 19.24134 54-037 20.22750 54-033 0.65 16.56460 58.547 17.41310 58.542 18.30530 58.538 19.24323 58.535 20.22930 58.531 0.7 16.56660 63.042 17.41510 63.038 18.30715 63-035 19.24496 63-031 20.23090 63.028 0-7S 16.56840 67-537 17.41680 67-533 18.30880 67-530 19.24650 67-527 20.23240 67-525 0.8 16.56996 72.031 17.41830 72.028 18.31020 72.025 19.24790 72.023 20.23365 72.021 0.85 16.57120 76.524 17.41945 76.521 18.31130 76.519 19.24890 76.518 20.23460 76.516 0.9 16.57210 81.016 17.42030 81.015 18.31210 81.013 19.24970 81.012 20.23540 81.011 o-9S 16.57260 85.508 17.42083 85-507 18.31260 85-507 19.25015 85-506 20.23585 85.506 I.O 16.57282 90.000 17.42102 90.000 18.31278 90.000 19-25033 90.000 20.23601 90.000 1.05 16.57260 94.492 17.42083 94-493 18.31260 94-493 19.25015 94-494 20.23585 94.494 i.i 16.57210 98.984 17.42030 98.985 18.31210 98.987 19.24970 98.988 20.23540 98.989 "S 16.57120 103.476 I7-4I945 103.479 18.31130 103.480 19.24890 103.482 20.23460 103.484 1.2 16.56996 107.969 17.41830 107.972 18.31020 107.975 19.24790 107.977 20.23365 107.979 1-25 16.56840 112.463 17.41680 112.467 18.30880 112.470 19.24650 112.473 20.23240 "2.475 i-3 16.56660 116.958 17.41510 116.962 18.30715 116.965 19.24496 116.969 20.23090 116.972 1-35 16.56460 121.453 17.41310 121.458 18.30530 121.462 19.24323 121.465 20.22930 121.469 1.4 16.56240 125.950 17.41110 125-955 18.30335 125-959 19.24134 125.963 20.22750 125.967 i-4S 16.56010 130.448 17.40900 130.453 18.30126 130.458 19.23940 130.462 20.22560 130.465 i-5 16.55774 134.948 17.40665 134-953 18.29912 134-957 19.23732 134.961 20.22365 I34-965 i-SS 16.55538 139.448 17.40441 139-453 18.29699 I39-458 19.23530 139.462 20.22170 139-465 1.6 16.55307 I43-950 17.40222 143-955 18.29490 143-959 i9- 2 333i 143-963 20.21981 143.967 1.65 16.55087 148.453 17.40015 148.458 18.29292 148.462 19.23142 148.465 20.21804 148.469 i-7 16.54885 152.958 17.39822 152.962 18.29109 152.965 19.22968 152.969 20.21639 152.972 i-7S 16.54702 IS7-463 17.39650 157.466 18.28942 I57-470 19.22813 157-473 20.21490 157-475 1.8 16.54550 161.969 17.38504 161.972 18.28805 161.975 19.22681 161.977 20.21365 161.979 1.85 16.54428 166.476 17.39386 166.478 18.28694 166.481 19.22577 166.482 20.21268 166.484 1.9 16.54337 170.984 17.39300 170.985 18.28611 170.987 19.22496 170.988 20.21189 170.989 1-95 16.54281 175.492 17.39248 175-493 18.28560 175-493 19.2244^ 175-494 20.21140 175-494 2.O 16.54263 180.000 17.39230 180.000 18.28546 180.000 19.22434 180.000 20.21129 180.000 Example. sinh (3.65 -+- to. 25) = 19.22813 /22.527 = 19.22813 /22.3i'.37* [104] TABLE X. HYPERBOLIC SINES, sinh (x + iq) = r fy. CONTINUED X = 3-75 x = 3-8 x = 3.85 x = 3-9 * = 3-95 9 r 7 r y r y r 7 r 7 o o o o o o 0.05 O.I 0.15 0.2 21 21 21 21 21 .24878 .24891 24935 .25006 .25102 O.OOO 4.505 9.010 13-514 18.019 22.33941 22.33952 22-33995 22.34061 22.34153 O.OOO 4-504 9.009 I3-5I3 I8.0I7 23.48589 23.48601 23.48640 23.48704 23.48791 O.OOO 4.504 9.008 I8.0I5 24.69110 24.69120 24.69159 24.69223 24.69302 O.OOO 4.504 9.007 13-511 18.014 25.95806 25.95820 25-95854 25-95911 25-9599 1 O.OOO 4.503 9.007 18.013 0.25 0-3 0-35 0.4 0-45 21 21 21 21 21 .25221 25362 .25520 .25685 .25869 22.522 27.026 31-528 36.030 40.531 22.34270 22.34401 22.34550 22.34712 22.34883 22.52O 27.023 3 I -5 2 6 36.027 40.528 23.48900 23.49028 23-49H5 23.49322 23.49486 22.518 27.O2I 3I-523 36-025 40.526 24.69406 24.69528 24.69662 24.69809 24.69964 22.517 27.019 31-521 36.022 40.523 25.96090 25.96205 2S-96333 25.96471 25.96618 22.515 27.017 31.519 36.020 40.521 o-S 0-55 0.6 0.65 0.7 21 21 21 21 21 .26052 .26236 .26415 .26586 .26745 45-032 49-531 54-030 58.528 63.026 22.35060 22.35230 22-35403 22.35565 22.35716 45.029 49.528 54-027 58-526 63.023 23.49652 23.49820 23.49980 23.50136 23-50277 45.026 49.526 54-025 58.523 63.021 24.70121 24.70280 24.70440 24.70580 24.70720 45.023 49-523 54-022 58.521 63.019 25.96770 25.96920 25.97066 25.97206 25-97337 45-021 49-521 54-020 58-519 63.017 o-75 0.8 0.85 0.9 o-95 21 21 21 21 21 .26885 .27000 .27100 .27170 .27212 67-522 72.019 76.514 81.010 85.505 22.35850 22.35962 22.36055 22.36122 22.36163 67.520 72.017 76-513 81.009 85-504 23.50404 23-50512 23.50600 23.50664 23.50702 67.518 72.015 76.512 8I.008 85.504 24.70840 24.70940 24.71024 24.71090 24.71120 72.014 76.511 81.007 85.504 25.97450 25-97550 25-97630 25.97680 25.97720 67-5I5 72-013 76.510 81.007 85-503 I.O 1.05 i.i 1.2 21 21 21 21 21 .27230 .27212 .27170 .27100 .27000 90.000 94-495 98.990 103.486 107.981 22.36178 22.36163 22.36122 22.36055 22.35962 90.000 94.496 98.991 103.487 107.983 23-50717 23.50702 23.50664 23.50600 23-50512 90.000 94.496 98.992 103.488 107.985 24-71135 24.71120 24.71090 24.71024 24.70940 90.000 94.496 98.993 103.489 107.986 25.9773I 25.97720 25.97680 25.97630 25-97550 90.000 94-497 98.993 103.490 107.987 1.25 1.4 1-45 21 21 21 21 21 .26885 26745 .26586 .26415 .26236 112.478 116.974 121.472 125.970 130.469 22.35850 22.35716 22.35565 22.35403 22.35230 112.480 116.977 121.474 125-973 130.472 23.50404 23.50277 23.50136 23.49980 23.49820 112.482 II6.979 121.477 125-975 130.474 24.70840 24.70720 24.70580 24.70440 24.70280 112.483 116.981 121.479 125.978 130.477 25.97450 25-97337 25.97206 25.97066 25.96920 112.485 116.983 121.481 125.980 130.479 i-SS 1.6 -65 -7 21 21 21 21 21 .26052 .25869 .25685 .25520 .25362 134.968 139.469 143.970 148.472 152.974 22.35060 22.34883 22.34712 22.34550 22.34401 I34.97I 139.472 143-973 148.474 152.977 23.49652 23.49486 23.49322 23-49" 5 23.49028 134-974 139-474 143-975 148.477 152.979 24.70121 24.69964 24.69809 24.69662 24.69528 134.976 139-477 143.978 148.479 152.981 25.96770 25.96618 25.96471 25.96205 134-979 139-479 143.980 148.481 152.983 75 .8 85 9 1-95 21 21 21 21 21 .25221 .25102 .25006 24935 .24891 I57-478 161.981 166.486 170.990 175-495 22.34270 22.34153 22.34061 22-33995 22.33952 157.480 161.983 166.487 170.991 175.496 23.48900 23.48791 23.48704 23.48640 23.48601 157.482 161.985 166.488 I7O.992 175.496 24.69406 24.69302 24.69223 24.69159 24.69120 157.483 161.986 166.489 170.993 175.496 25.96090 25-9599I 25-959" 25-95854 25.95820 I57.485 161.987 166.490 170.993 175-497 2.O 21.24878 180.000 22.33941 180.000 23.48589 iSo.OOO 24.69110 180.000 25.95806 180.000 Example. sinh (3.90 + * 1.90) = 24.69159 /I70.993 = 24-69159 /I70.59' 35'. [105] TABLE XI. HYPERBOLIC COSINES, cosh (* + ig) = r /y x = 0.05 x, = 0.15 o 0.05 O.I 0.15 O.2 I.OOOOO O.OO 0.99692 o.oo 0.98769 o.oo 0.97237 o.oo 0.95106 o.oo 1.00125 o- 00 0.99817 0.225 0.98895 0.453 0.97366 0.687 0.95237 0.930 1.00500 o.oo 1.00194 0.450 0.99275 0.905 0.97752 1.371 0.95632 1.855 1.01127 o.oo 1.00822 0.671 0.99909 1.351 0.98396 2.047 0.96290 2.769 1.02007 o.oo 1.01704 0.890 1.00800 1.791 0.99300 2.713 0-97213 3- 66 9 O.25 0-3 0-35 0.4 0-45 0.92388 o.oo 0.89101 o.oo 0.85264 o.oo 0.80902 o.oo 0.76041 o.oo 0.92523 1.186 0.89237 1.458 0.85407 1.754 0.81053 2.079 0.76202 2.443 0.92930 2.364 0.89662 2.907 0.85851 3.495 0.81520 4.142 0.76698 4.866 0.93607 3.529 0.90364 4.338 0.86583 5.213 0.82291 6.174 0.77527 7.247 0.94556 4-674 0.91347 5-743 0.87609 6.897 0.83370 8.161 0.78661 9-569 o-5 0.55 0.6 0.65 0.7 0.70711 o.oo 0.64945 o.oo 0.58779 o.oo 0.52250 o.oo 0-45399 0-0 0.70803 2.860 0-65135 3-348 0.58989 3.934 0.52487 4.661 0.45672 5.600 0.71417 5.692 0.65713 6.656 0.59626 7.811 0.53201 9.238 0.46491 1 1. 068 0.72296 8.468 0.66667 9-889 0.60676 11.581 0.54376 13.656 0.47831 16.289 0.73521 11.165 0.67994 13.012 0.62131 15.198 0-55995 17-853 0.49663 21.175 o-7S 0.8 0.85 0.9 0-95 0.38268 o.oo 0.30902 o.oo 0.23345 o.oo 0.15643 o.oo 0.07846 o.oo 0.38594 6.877 0.31304 8.741 0.23874 11.755 0.16424 17.507 0.09305 32.407 0.39558 13-529 0.32485 17-053 0.25403 22.545 0.18576 32.181 0.12724 51.700 0.41124 19.771 0.34375 24.618 0.27779 3^805 0.21712 43.229 0.16978 62.139 0.43242 25.478 0.36882 31.277 0.30827 39.424 0.25497 5I-254 0.21608 68.261 I.O 1.05 i.i 1.2 o.oo 180 0.07846 180 0.15643 180 0.23345 180 0.30902 1 80 0.05002 90.000 0.09305 147-593 0.16424 162.493 0.23874 168.245 0.31304 171.259 0.10017 90.000 0.12724 128.300 0.18576 147.819 0.25403 157-455 0.32485 162.947 0.15056 90.000 0.16978 117.861 0.21712 136.771 0.27779 148.195 0.34375 155.382 0.20134 90.000 0.21608 111.739 0.25497 128.746 0.30827 140.576 0.36882 148.723 1-25 i-35 1.4 MS 0.38268 180 0-45399 180 0.52250 180 0.58779 180 0.64945 1 80 0.38594 173-123 0.45672 174.400 0.52487 175-339 0.58989 176.066 0.65135 176.652 0.39558 166.471 0.46491 168.932 0.53201 170.762 0.59626 172.189 0.65713 173-344 0.41124 160.229 0.47831 163.711 0.54376 166.344 0.60676 168.419 0.66667 170.111 0.43242 154.522 0.49663 158.825 0-55995 162.147 0.62131 164.802 0.67994 166.988 i-SS 1.6 1.65 0.70711 180 0.76041 1 80 0.80902 1 80 0.85264 1 80 0.89101 180 0.70803 177-139 0.76202 177.557 0.81053 177.921 0.85407 178.246 0.89237 178.542 0.71417 174.308 0.76698 175.134 0.81520 175.858 0.85851 176.505 0.89662 177.093 0.72296 171.532 0.77527 172.753 0.82291 173.826 0.86583 174.787 0.90364 175.662 0.73521 168.835 0.78661 170.431 0.83370 171-839 0.87609 173.103 0.91347 174-257 i-75 1.8 1.85 1.9 i-95 0.92388 180 0.95106 180 0.97237 180 0.98769 180 0.99692 180 0.92523 178.814 0.95237 179.070 0.97366 179-313 0.98895 179.547 0.99817 179-775 0.92930 177.636 0.95632 178.145 0.97752 178.629 0.99275 179.095 1.00194 179.550 0.93607 176.471 0.96290 177.231 0.98396 177.953 0.99909 178.649 1.00822 179.329 0.94556 175-326 0.97213 176.331 0.99300 177.287 1.00800 178.209 1.01704 179.110 2.O i.ooooo 180 1.00125 180.000 1.00500 180.000 1.01127 180.000 1.02007 180.000 Example. cosh (o.io + 10.55) = 0.65713 /6.6^6 = 0.65713 /6.39 '.22*. [106] TABLE XI. HYPERBOLIC COSINES, cosh (x + ig) = r Ay. CONTINUED X = 0.25 x = o-3 x = 0-35 x = 0.4 x 0.45 9 r 7 r y r y r y r y o o o o 1.03141 o.ooo 1-04534 o.ooo i. 06 1 88 o.ooo 1.08107 O.OOO 1.10297 O.OOO 0.05 1.02843 1.104 1.04239 1-313 1.05898 1.516 1.07822 I.7I3 1.10017 I.9O2 O.I 1.01949 2.221 1-03357 2.642 1.05029 3-050 1.06970 3-444 1.09182 3-823 0.15 1.00464 3-365 1.01894 4.001 1.03590 4.617 1-05557 5-212 1.07798 5-784 O.2 0.98403 4-550 0.99862 5-407 1.01592 6.238 1-03597 7.038 1.05880 7.806 0.25 0-95779 5-793 0.97277 6.881 0.99052 7-932 1.01107 8.944 1.03446 9-9I3 -3 0.92612 7-ii3 0.94161 8.443 0.95994 9.726 0.98113 10.957 1.00521 12.132 o-35 0.88927 8-536 0-90539 IO.I22 0.92444 11.647 0.94642 13.107 0.97136 14.496 0.4 0.84754 10.090 0.86443 11.951 0.88436 13-733 0.90732 I5-432 0-93330 17.042 0-45 0.80127 11.815 0.81912 I3-972 0.84012 16.029 0.86426 17.979 0.89149 I9.8l6 0-5 0.75088 13.762 0.76989 16.241 0.79220 18.592 0.81775 20.804 0.84469 22.875 o-55 0.69685 16.001 0.71730 18.834 0.74119 21-497 0.76844 23.982 0.79895 26.288 0.6 0.63977 18.629 0.66199 2 1 .849 0.68781 24.843 0.71708 27.607 0.74969 30.144 0.65 0.58036 21.785 0.60476 25-425 0.63292 28.763 0.66462 31.800 0.69968 34-546 0.7 0.51954 25-673 0.54666 29-758 0.57766 33-!57 0.61223 36.712 0.65012 39.626 0-75 0-45854 30.595 0.48906 35-II8 0.52348 39-079 0.56140 42.529 0.60249 45-527 0.8 o-399 J 3 37.008 0.43385 41-878 0.47231 45-992 0.51401 49.464 0.55860 52-399 0.85 0.34396 45-057 0.38370 50.507 0.42671 54-484 0.47246 57-712 0.52062 60.358 0.9 0.29782 57.110 0-34235 61.467 0.38994 64.786 0-43953 67-37I 0.49093 69.424 0-95 0.26452 72.186 0.31447 74.882 0.36571 76.831 0.41818 78.297 0.47191 79-433 I.O 0.25261 90.000 0.30452 90.000 0.35719 90.000 0.41075 90.000 0.46534 90.000 1.05 0.26452 107.814 0.31447 I05.II8 0.36571 103.169 0.41818 101.703 0.47191 100.567 i.i 0.29782 122.890 0.34235 "8.533 0.38994 115.214 0-43953 112.629 0.49093 110.576 US 0.34396 134-943 0.38370 129.493 0.42671 125.516 0.47246 122.288 0.52062 119.642 1.2 0.39913 142.992 0-43385 138.122 0.47231 134.008 0.51401 130.536 0.55860 127.601 1-25 0.45854 149.405 0.48906 144.882 0.52348 140.921 0.56140 I37.47I 0.60249 134-473 i-3 0.51954 I54-327 0.54666 150.242 0.57766 146.843 0.61223 143-288 0.65012 140.374 i-3S 0.58036 158.215 0.60476 154-575 0.63292 151-237 0.66462 148.200 0.69968 145-454 1.4 0.63977 161.371 0.66199 I58.I5I 0.68781 I55.I57 0.71708 152.393 0.74969 149-856 i-45 0.69685 163.999 0.71730 l6l.l66 0.74119 158.503 0.76844 156.018 0.79895 I53.7I2 i-5 0.75088 166.238 0.76989 163.759 0.79220 161.408 0.81775 159.196 0.84649 157-125 i-5S 0.80127 168.185 0.81912 166.028 0.84012 163.971 0.86426 162.021 0.89149 160.184 1.6 0.84754 169.910 0.86443 168.049 0.88436 166.267 0.90732 164.568 0.93330 162.958 1.65 0.88927 171.464 0-90539 169.878 0.92444 168.353 0.94642 166.893 0.97136 165.504 i-7 0.92612 172.887 0.94161 I7L557 0.95994 170.274 0.98113 169.043 1.00521 167.868 1-75 0-95779 174.207 0.97277 I73-H9 0.99052 172.068 1.01107 171.056 1.03446 170.087 1.8 0.98403 I75-450 0.99862 174-593 1.01592 173.762 1-03597 172.962 1.05880 172.194 1.85 i .00464 176.635 1.01894 175-999 1.03590 175-383 1-05557 174-788 1.07798 174.216 1.9 1.01949 177.779 1-03357 177.358 1.05029 176.950 1.06970 176.556 1.09182 176.177 i-9S 1.02843 178.896 1.04239 178.687 1.05898 178.484 1.07822 178.287 1.10017 178.098 2.0 1.03141 180.000 1-04534 180.000 1.06188 180.000 1.08107 180.000 1.10297 180.000 Example. cosh (0.40 + i 0.5) = 0.81775 = 0.81775 [107] TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED * = o.S X = 0-55 x = 0.6 x = 0.65 x = 0.7 q r 7 r 7 r 7 r 7 r 7 o o o o 1.12763 o.ooo i.iSSio o.ooo 1.18547 o.ooo 1.21879 O.OOO 1-25517 o.ooo 0.05 1.12489 2.083 1.15244 2.256 1.18287 2.420 1.21626 2.576 1.25271 2.723 O.I 1.11672 4.186 1.14446 4-533 1.17510 4.879 1.20871 5- J 74 1-24538 5.468 0.15 1.10320 6.331 1.13127 6.852 1.16226 7-347 1.19623 7-815 1.23327 8.256 O.2 1.08446 8.539 1.11300 9-237 1.14448 9.898 1.17892 10.523 1.21653 II. IIO 0.25 1.06070 10.836 1.08987 11.713 I.I22OO 12.541 I-I57I5 13.322 1.19541 14.054 o-3 1.03220 13.249 1.06214 14-307 1.09509 15-304 1.13108 16.240 1.17019 17.116 0-35 0.99927 15.811 1.03004 17.052 I.064II 18.216 I.IOIII 19.306 1.14125 20.323 0.4 0.96232 18.559 0-99437 19.984 1.02948 21-315 1.06769 22.555 1.10904 23.706 0-45 0.92182 21.538 0.95524 23.146 0.99174 24.640 1-03134 26.024 1.07409 27.302 o-S 0.87837 24.803 0.91338 26.589 0.95149 28.238 0.99270 29-755 1.03704 31.148 o-SS 0.83266 28.416 0.86951 30.372 0.90946 32.162 0.95249 33-796 0.99861 35.284 0.6 0.78552 32-458 0.82447 34-563 0.86650 36.471 0.91156 38.197 0.95966 39-755 0.65 0-73793 37.020 0.77927 39-241 0.82361 41.231 0.87090 43-on 0.92112 44.603 0.7 0.69112 42.207 0.73510 44.489 0.78194 46.506 0.83160 48.290 0.88406 49.867 0-75 0.64652 48.129 0.69333 50.390 O.74282 52.358 0.79492 54-074 0.84965 55-574 0.8 0.60583 54.889 0.65555 57.010 0.70769 58.826 0.76220 60.387 0.81911 61.737 0.85 0.57100 62.547 0.62350 64-375 0.67810 65.914 0.73482 67.219 0.79369 68-335 0.9 0.54407 71.082 0.59894 72.441 0.65559 73-568 0.71409 74.5I4 0-77455 75.3I5 o.QS 0.52697 80.335 0.58345 81.064 0.64147 81.663 0.70115 82.161 0.76263 82.581 I.O 0.52110 90.000 0.57815 90.000 0.63665 90.000 0.69675 90.000 0.75858 90.000 1.05 0.52697 99.665 0.58345 98-936 0.64147 98.337 0.70115 97.839 0.76263 97.419 i.i 0.54407 108.918 0.59894 107-559 0.65559 106.432 0.71409 105.486 0-77455 104.685 i-iS 0.57100 117.453 0.62350 115.625 0.67810 114.086 0.73482 112.781 0.79369 111.665 1.2 0.60583 125.111 0.65555 122.990 0.70769 121.174 0.76220 119.613 0.81911 118.263 I.2S 0.64652 131.871 0.69333 129.610 0.74282 127.642 0.79492 125.926 0.84965 124.426 i-3 0.69112 137.793 o.735io I35-5 1 ! 0.78194 133-494 0.83160 131.710 0.88406 130.133 i-35 0-73793 142.980 0.77927 140.759 0.82361 138.769 0.87090 136.989 0.92112 135-397 1.4 0.78552 147.542 0.82447 145-437 0.86650 I43-529 0.91156 141.803 0.95966 140.245 i. 45 0.83266 151-584 0.86951 149.628 0.90946 147.838 0.95249 146.204 0.99861 144.716 1.50 0.87837 155.198 0.91338 153-4" 0.95149 151.762 0.99270 150.245 1.03704 148.853 i-SS 0.92182 158.462 0.95524 156.854 0.99174 155-360 1-03134 I53.976 1.07409 152.698 i. 60 0.96232 161.441 0-99437 160.016 I.O2948 158.685 1.06769 157-445 1.10904 156-294 1.65 0.99927 164.189 1.03004 162.948 1.06411 161.784 I.IOIII 160.694 1.14125 159.677 1.70 1.03220 166.751 1.06214 165.693 1.09509 164.696 1.13108 163.760 1.17019 162.884 i-75 1.06070 169.164 1.08987 168.287 .I22OO 167.459 1.15715 166.678 1.19541 165.946 1.8 i .08446 171.461 1.11300 170.763 .14448 170.102 1.17892 169.477 1.21653 168.890 1.85 1.10320 173.669 1.13127 173.148 .16226 172.653 1.19623 172.185 1.23327 171.744 i. 9 1.11672 175.814 1.14446 175467 .17510 175.121 1.20871 174.826 1-24538 I74-532 i-9S 1.12489 177.917 1.15244 177-744 .18287 177.580 1.21626 177.424 1.25271 177.277 2.0 1.12763 180.000 1.15510 180.000 I.I8547 180.000 1.21879 180.000 1-25517 180.000 Example. cosh (0.65 + i i.o) = 0.69675 /go . [108] TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r fy. CONTINUED x = 0.75 = 0.85 x = o.g x = 0.95 o 1.29468 0.000 1 -33 743 o.ooo I-38353 o.ooo 1.43309 o.ooo 1.48623 o.ooo 0.05 1.29230 2.862 I-335I3 2.992 1.38130 3-"3 1.43094 3.227 1.48415 3-332 O.I 1.28520 5-745 1.32825 6.004 1.37466 6.246 1.42452 6.473 1.47797 6.684 0.15 1.27346 8.670 1.31691 9-058 1.36369 9.420 I-4I395 9-758 1.46778 10.071 O.2 1.25726 11.661 1.30124 12.175 1-34858 12.656 1-39937 13.102 1.45371 13-516 0.2S 1.23689 14.740 1.28152 15-379 I-32955 15-974 1.38105 16.526 1.43611 17.037 o-3 1.21248 17-933 1.25802 18.693 1.30693 19.398 I-35927 20.051 1.41519 20.653' 0.35 1.18457 21.267 1-23115 22.143 1.28107 22.952 1-33444 23.699 1.39136 24-387 0.4 1-15356 24.772 1.20135 25-755 1.25246 26.601 1.30700 27.493 1-36505 28.257 0-45 I.I200I 28.478 1.16917 29-559 1.22163 30-550 1.27748 31.457 1.33682 32.286 o-S 1.08453 32-442 1.13522 33-586 1.18918 34.647 1.24649 35-614 1.30723 36-493 0-55 1.04785 36-637 1.10024 37.865 1-15583 38.978 1.21471 39.986 1.27697 40.898 0.6 I.OI079 41.160 1.06500 42.426 1.12234 43-567 1.18289 44-593 1.24674 45-517 0.65 0.97427 46.026 1.03041 47.298 1.08957 48.435 1.15184 49-453 1.21732 50.363 0.7 0.93932 51-263 0.99742 52-500 1.05843 53-599 1.12243 54-574 1.19227 55-443 0-7S O.9O7OO 56.889 0.96705 58.045 1.02985 59.062 1-09553 59.961 1.16418 6o.755 0.8 0.87846 62.907 0.94033 63.927 1.00481 64.819 1.07202 65.601 1.14209 66.288 0.85 0.85481 69.294 0.91828 70.123 0.98420 70.842 1.05273 71.471 1.12400 72.020 0.9 0.83 706 75-998 0.90178 76.585 0.96883 77.091 1-03837 77-532 1.11056 77-9I5 0-95 0.82605 82.936 0.89157 83.241 0-95933 83-503 1.02951 83-730 1.10227 83.927 I.O O.82232 90.000 0.88811 90.000 0.95612 90.000 1.02652 90 ooo 1.09948 90.000 1.05 0.82605 97.064 0.89157 96.759 0-95933 96.497 1.02951 96.270 1.10227 96.073 i.i 0.83706 104.002 0.90178 103.415 0.96883 102.909 1-03837 102.468 1.11056 102.085 I.IS 0.85481 110.706 0.91828 109.877 0.98420 109.158 1.05273 108.529 I 12400 107.980 1.2 0.87846 117.093 0.94033 116.073 1.00481 115.181 1.07202 114.399 1.14209 113.712 1. 25 0.90700 123.111 0.96705 121.955 1.02985 120.938 1-09553 1 20.039 1.16418 119.245 i-3 0.93932 128.737 0.99742 127.500 1.05843 126.401 1.12243 125.426 1.19227 124-557 i-35 0.97427 133-974 1.03041 132.702 1.08957 i3!-565 1.15184 130.547 1.21732 129.637 1.4 I.OI079 138.840 1.06500 137-574 1.12234 136.433 1.18289 135.407 1.24674 134-483 1-45 1.04785 I43-363 1.10024 142-135 1-15583 141.022 1.21471 140.014 1.27697 139.102 i-5 1.08453 I47-578 1.13522 146.414 1.18918 145-353 1.24649 144.386 1.30723 I43-507 i-SS I.I2OOI 151-522 1.16917 150.441 1.22163 149.450 1.27748 148.543 1.33682 147.714 1.6 I-I5356 155.228 1.20135 I54-24S 1.25246 153-399 1.30700 152.507 1-36505 I5I-743 1.65 1.18457 158.733 i.23"5 I57-857 1.28107 157.048 1.33444 156.301 1.39136 I55-6I3 i-7 I.2I248 162.067 1.25802 161.307 1.30693 160.602 I-35927 159-949 1.41519 159-347 i-75 1.23689 165.260 1.28152 164.621 I-32955. 164.026 1.38105 163.474 1.43611 162.963 1.8 1.25726 168.339 1.30124 167.825 1-34858 167.344 1-39937 166.898 1.45371 166.484 1.85 1.27346 171-330 1.31691 170.942 1.36369 170.580 I-4I395 170.242 1.46778 169.929 1.9 I.2852O 174-255 1.32825 173.996 1.37466 173-754 1.42452 173-527 1.47797 I73-3I6 i -95 1.29230 177.138 I-335I3 177.008 1.38130 176.887 1.43094 176.773 1.48415 176.668 2.O 1.29468 180.000 1-33743 180.000 I-38353 180.000 1-43309 180.000 148623 180.000 Example. cosh (0.90 + * 0.5) = 1.24649 /35-6i4 = 1.24649 /35-36'.5o'. [109] TABLE XI. HYPERBOLIC COSINES, cosh (at + iq) = r /y. CONTINUED x = 1.05 x = 1.15 o 0.05 O.I 0.15 O.2 1.54308 o.ooo 1.54108 3.276 1.53513 6.878 1.52532 10.362 1.51182 13.899 1.60379 o.ooo 1.60187 3.521 1.59614 7.059 1.58671 10.631 1-57374 14-253 1.66852 o.ooo 1.66667 3-605 1.66117 7.226 1.65211 10.879 1.63965 14.579 1.73741 o.ooo 1.73564 3-683 1-73036 7-380 1.72166 11.107 1.70971 14.880 i. 81066 o.ooo 1.80895 3.754 1.80389 7.522 1-79555 11-318 1.78409 15.156 0.25 0-3 0-35 0.4 0-45 1.49487 17.509 1.47479 21.209 1.45193 25.019 1.42675 28.957 1.39976 33-043 1-55747 17-944 1.53820 21.720 1.51630 25.599 1.49220 29.597 1.46641 33-732 1.62404 18.344 1.60558 22.189 1.58460 26.130 1.56156 30.182 1.53694 34-360 1.69472 18.713 1.67705 22.620 1.65699 26.616 1-63497 30.716 1.61147 34.931 1.76975 19.050 1.75282 23.014 1-73363 27.061 1.71260 31.203 1.69018 35-451 o-5 0-55 0.6 0.65 0.7 1.37153 37.293 1.34272 41.724 1.31400 46.349 1.28612 51.179 1.25984 56.216 1.43950 38.019 1.41207 42.470 1.38479 47-098 I-35837 5i-9 10 I-3335I 56-907 1.51128 38.677 1.48517 43-H5 1.45926 47-773 1.43421 52-565' 1.41070 57-523 1.58701 39.275 1.56217 43-755 1.53756 48.380 1.51381 53-J53 i-49 I 55 58-074 1.66688 39.816 1.64325 44-307 1.61987 48.927 1-59733 53-681 1.57626 58.567 0-75 0.8 0.85 0.9 o-95 1.23594 61.459 1.21515 66.895 1.19816 72.504 1.18552 78.252 1.17782 84.100 1.31095 62.085 1.29137 67.432 1.27540 72.929 1.26358 78.547 1.25631 84.252 1.38939 62.641 1.37093 67.908 1-35590 73-306 1.34478 78.808 1-33795 84.385 1.47141 63.'i36 1.45399 68.330 1-43983 73.639 1.42936 79-039 1.42294 84.503 1.55721 63.579 1.54077 68.706 1.52741 73.935 I.5I755 79-243 1.51150 84.607 I.O 1.05 i.i i-iS 1.2 1.17520 90.000 1.17782 95.900 1.18552 101.748 1.19816 107.496 1.21515 113.105 1.25386 90.000 1.25631 95.748 1.26358 101.453 1.27540 107.071 1.29137 112.568 1 -33565 9 - o 1-33795 95-6i5 1.34478 101.192 1.35590 106.694 1.37093 112.092 1.42078 90.000 1.42294 95.497 1.42936 100.961 1.43983 106.361 1.45399 111-670 1.50946 90.000 1.51150 95.393 1.51755 100.757 1.52741 106.065 1.54077 111.294 1-25 i-3 i-35 1.4 i-45 1.23594 118.541 1.25984 123.784 1.28612 128.821 1.31400 133.651 1.34272 138.276 1.31095 117.915 1.33351 123.093 1.35837 128.090 1.38479 132.902 1.41207 137.530 1.38939 H7-359 1.41070 122.477 1.43421 127.435 1.45926 132.227 1.48517 136.855 1.47141 116.864 1.49155 121.926 1.51381 126.847 J -53756 131.620 1.56217 136.245 1.55721 116.421 1.57626 121.433 1.59733 126.319 1.61987 131.073 1.64325 135.693 i-5 i-SS 1.6 1.65 i-7 1.37153 142.707 1.39976 146.957 1.42675 151.043 1.45193 154.981 1.47479 I58.79 1 ' 1.43950 141.981 1.46641 146.268 1.49220 150.403 1.51630 154-401 1.53820 158.280 1.51128 141-323 1.53694 145.640 1.56156 149.818 1.58460 1.53.870 1.60558 157.811 1.58701 140.725 1.61147 145.069 1.63497 149.284 1.65699 153-384 1.67705 157-380 1.66688 140.184 1.69018 ^44-549 1.71260 148.797 I.73363 152.939 1.75282 156.986 i-7S 1.8 1.85 1.9 I-9S 1.49487 162.491 1.51182 166.101 1.52532 169.638 i-SSS^ 173-122 1.54108 176.724 1.55747 162.056 1.57374 165.747 1.58671 169.369 1.59614 172.941 1.60187 176.479 1.62404 161.656 1.63965 165.421 1.65211 169.121 1.66117 172.774 1.66667 176.395 1.69472 161.287 1.70971 165.120 1.72166 168.893 1.73036 172.620 1.73564 176.317 1.76975 160.950 1.78409 164.844 1-79555 168.682 1.80389 172.478 1.80895 176.246 2.0 1.54308 180.000 1.60379 180.000 1.66852 180.000 1.73741 180.000 i. 81066 180.000 Example. cosh (1.20 + io) = i. 81066 /o. [no] TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED x = 1.25 x = 1.3 = i-3S = 1.4 = 1-45 o 1.88842 o.ooo 1.97091 o.ooo 2.05833 o.ooo 2.15090 O.OOO 2.24884 0.000 0.05 1.88679 3.820 1.96935 3.880 2.05684 3-935 2.14947 3.986 2.24747 4.032 O.I I.88I93 7.652 1.96470 7.772 2.05238 7.882 2.14520 7.982 2-24334 8.075 0.15 1.87394 11.511 1.95704 11.689 2.04505 11.851 2.13819 I2.OOO 2.23669 12.136 O.2 1.86297 15.410 1.94654 15.642 2.03500 15-855 2.12858 16.049 2.22751 16.226 0.25 1.84924 19.360 1.93341 19.644 2.02245 19.903 2.11658 20.139 2.21604 20.355 -3 1.83304 23-375 1.91792 23.705 2.00764 24.006 2.10244 24.281 2.20254 24.531 o-35 I.8I4/O 27.467 1 .90040 27.837 1.99091 28.175 2.08647 28.482 2.18731 28.762 0.4 1.79462 31.646 1.88123 32.050 1.972612 32-417 2.06903 32.751 2.17067 33.055 0-45 I-77323 35-923 1.86084 26.352 1-95319 36.742 2.05051 37-095 2.15302 37.416 o-S I.75I04 40.308 1.83970 40.752 1.93306 4i-i55 2.03135 41.520 2.13478 41.851 0-55 1.72856 44.805 1.81832 45.255 1.91273 45.662 2.OI2OO 46.030 2.11637 46.362 0.6 I-70635 49.420 1.79722 49.865 1.89268 50.265 1.99295 50.627 2.09828 50.953 0.65 1.68497 54-155 1.77694 54.582 1.87343 54.96S 1.97468 55-3" 2.08093 55.621 0.7 I.6650I 59.009 1.75802 59.405 1.85549 59.760 1.95767 60.079 2.06479 60.366 0-75 1.64699 63.974 1.74096 64.327 1.83934 64.644 I.94237 64.927 2.05030 65.182 0.8 I-63I45 69.042 1.72627 69.341 1.82544 69.608 I.9292I 69.847 2.03784 70.061 0.85 1.61884 74-197 1-71435 74-432 1.81417 74.641 1.91856 74.828 2.02775 74-995 0.9 1.60954 79.424 1-70557 79-585 1.80588 79.729 I.9I072 79.857 2.02034 79.972 0-95 1.60384 84.699 1.70019 84.782 1 .80080 84-855 1.90592 84.920 2.01580 84.979 I.O 1.60192 90.000 1.69838 90.000 1.79909 90.000 1.90430 90.000 2.01427 90.000 1.05 1.60384 95-301 1.70019 95.218 1 .80080 95-145 1.90592 95.080 2.01580 95.021 i.i 1.60954 100.576 1-70557 100.415 1.80588 100.271 I.9IO72 100.143 2.02034 100.028 i-iS 1.61884 105.803 1-71435 105.568 1.81417 105-359 1.91856 105.172 2.02775 105.005 1.2 I-63I45 110.958 1.72627 110.659 1.82544 110.392 I.9292I 110.153 2.03784 109.939 I-2S 1.64699 116.026 1.74096 115-673 1.83934 115-356 1.94237 115-073 2.05030 114.818 i-3 1.66501 120.991 1.75802 120.595 1.85549 120.240 1.95767 119.921 2.06479 119.634 i-3S 1.68497 125.845 1.77694 125.418 1.87343 125-035 1.97468 124.689 2.08093 124.379 1.4 1.70635 130.580 1.79722 130.135 1.89268 129-735 1.99295 129.373 2.098^8 129.047 i -45 1.72856 135-195 1.81832 134.745 1.91273 I34-338 2.OI2OO I33.970 2.11637 133-638 i-5 I.75I04 139-693 1.83970 139.248 1.93306 138.845 2.03135 138.480 2.13478 138.150 i-SS 1.77323 144.077 1.86084 143.648 1.95319 143-258 2.05051 142.905 2.15302 142.584 1.6 1.79462 148.354 1.88123 147.950 1.97262 147-583 2.06903 147.249 2.17067 146.945 1.65 I.8I470 152.533 1 .90040 152.163 1.99091 151-825 2.08647 151-518 2.18731 151-238 i-7 1.83304 156.625 1.91792 156.295 2.00764 155-994 2.10244 I55.7I9 2.20254 155469 i-75 1.84924 160.640 1.93341 160.356 2.02245 160.097 2.II658 159.861 2.21604 I59.645 1.8 1.86297 164.590 1.94654 164.358 2.03500 164.145 2.12858 163.951 2.22751 163.774 1.85 1.87394 168.489 1.95704 168.311 2.04505 168.149 2.13819 168.000 2.23669 167.864 1.9 1.88193 172.348 1.96470 172.228 2.05238 172.118 2.14520 172.018 2-24334 171.925 I-9S 1.88679 176.180 1.96935 176.120 2.05684 176.065 2.14947 176.014 2.24747 175-968 2.0 1.88842 180.000 1.97091 180.000 2.05833 180.000 2.15090 180.000 2.24884 180.000 Example. cosh (1.35 -f i 1.30) = 1.85549 / 120. 240 = 1.85549 / 120. 14'. 24*. [in] TABLE XI. HYPERBOLIC COSINES, cosh (x + ig) = r /y. CONTINUED x = 1.5 = 1-55 x = 1.6 * = 1.65 x = 1.7 o 2.35241 o.ooo 2.46186 o.ooo 2.57746 o.ooo 2.69952 o.ooo 2.82832 o.ooo 0.05 2.35110 4.075 2.46061 4.113 2.57627 4.149 2.69839 4.181 2.82723 4.210 o.r 2.34720 8.158 2.45688 8.235 2.57272 8.305 2.69499 8.369 2.82399 8.427 0.15 2.34080 12.260 2.45076 12.374 2.56687 12.477 2.68941 12.571 2.81867 12.657 O.2 2.33202 16.389 2.44238 16.536 2.55887 16.671 2.68178 16.794 2.81138 16.906 0.25 2.32107 20.552 2.43193 20.732 2.54890 20.895 2.67226 21.044 2.80230 21.179 o-3 2.30819 24.759 2.41976 24.967 2.53717 25.155 2.66108 25.327 2.79164 25.483 o-35 2.29365 29.016 2.40577 29.248 2.52395 29.458 2.64847 29.649 2.77963 29.822 0.4 2-27779 33-33 2.39066 33.580 2-50955 33-808 2.63476 34.014 2.76657 34.201 0-45 2.26098 37.706 2.37465 37-970 2.49430 38.209 2.62024 38.426 2.75274 38.622 o-5 2.24362 42.150 2.35812 42.421 2.47857 42.666 2.60527 42.887 2.73850 43.089 0-55 2.22612 46.663 2.34148 46.934 2.46274 47.180 2.59021 47.401 2.72418 47.602 0.6 2.20892 51-247 2.32513 51.512 2.44721 51-752 2.57544 51.968 2.71014 52.163 0.65 2.19245 55-901 2.30949 56.154 2.43235 56.381 2.56133 56.585 2.69673 56.771 0.7 2.17714 60.624 2.29496 60.856 2.41856 61.065 2.54824 61.253 2.68430 61.422 o-7S 2.16340 65.410 2.28193 65.615 2.40619 65.800 2.53650 65.966 2.67316 66.116 0.8 2.15159 70-254 2.27074 70.426 2-39558 70.581 2.52644 70.720 2.66361 70.845 0.85 2.14204 75- x 45 2.26169 75.279 2.38701 75.400 2.51831 75.508 2-65591 75-605 0.9 2.13502 80.075 2.25504 80.167 2.38071 80.249 2.51234 80.323 2.65025 80.390 o-9S 2.13073 85.031 2.25098 85.077 2.37686 85.119 . 2.50869 85.157 2.64679 85.191 I.O 2.12928 90.000 2.24961 90.000 2-37557 90-000 2.50747 90.000 2.64563 90.000 1.05 2.13073 94-969 2.25098 94.923 2.37686 94.881 2.50869 94-843 2.64679 94.809 i.i 2.13502 99.925 2.25504 99.833 2.38071 99.751 2.51234 99.677 2.65025 99.610 i-iS 2.14204 104.855 2.26169 104.721 2.38701 104.600 2.51831 104.492 2.65591 104.395 1.2 2.15159 109.746 2.27074 109.574 2.39558 109.419 2.52644 109.280 2.66361 109.155 I.2S 2.16340 114-59 2.28193 114.385 2.40619 114.200 2.53650 114.034 2.67316 113.884 i~3 2.17714 119.376 2.29496 119.144 2.41856 118.935 2.54824 118.747 2.68430 118.578 35 2.19245 124.099 2.30949 123.846 2.43235 123.619 2.56133 123.415 2.69673 123.229 4 2.20892 128.753 2.32513 128.488 2.44721 128.248 2.57544 128.032 2.71014 127.837 45 2.22612 133.337 2.34148 133.066 2.46274 132.820 2.59021 132.599 2.72418 132.398 5 2.24362 137-850 2.35812 137.580 2-47857 137-334 2.60527 137.113 2.73850 136.911 55 2.26098 142.294 2.37465 142.030 2.49430 141-791 2.62024 141.574 2.75274 141.378 .6 2.27779 146.670 2.39066 146.420 2.50955 146.192 2.63476 145.986 2.76657 145.799 65 2.29365 150.984 2.40577 150.752 2.5 2 395 150.542 2.64847 150.351 2.77963 150.178 7 2.30819 155-241 2.41976 155.033 2.537I7 I54.845 2.66108 154.673 2.79164 154.517 75 2.32107 159.448 2.43193 159.268 2.54890 159.105 2.67226 158.956 2.80230 158.821 .8 2.33202 163.611 2.44238 163.464 2.55887 163.329 2.68178 163.206 2.81138 163.094 85 2.34080 167.740 2.45076 167.626 2.56687 167.523 2.68941 167.429 2.81867 167.343 9 2.34720 171.842 2.45688 171.765 2.57272 171.695 2.69499 171.631 2.82399 171.573 95 2.35110 175-925 2.46061 175.887 2.57627 175.851 2.69839 175.819 2.82723 175.790 2.0 2.35241 180.000 2.46186 180.000 2.57746 180.000 2.69952 180.000 2.82832 180.000 Example. cosh (1.6 + i 1.6) = 2.50955 /I46.i92 = 2.50955 /i46.n'.3i y . [112] TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r /y. CONTINUED x = 1-75 x = 1.8 x = 1.85 * = 1.9 X = i-95 9 r y r Y r 7 r 7 r Y o o 2.96419 o.ooo 3.10747 O.OOO 3. 25853 o.ooo 3.4I773 o.ooo 3-58548 o.ooo 0.05 2.96315 4.237 3.10648 4.262 3' 25759 4.284 3.41683 4-304 3.58462 4.322 O.I 2.96006 8.480 3-10353 8.529 3' 25477 8.572 3-41415 8.612 3.58207 8.648 0.15 2.95498 12-735 3.09869 12.806 3. 25015 12.871 3-40975 12.929 3-57788 12.983 0.2 2.94804 17.008 3.09206 17.100 3' 24384 17.184 3-40373 17.260 3-572I3 17.330 0.25 2.93938 21.302 3.08382 21.414 3' 23598 21.516 3-39624 21.608 3-56500 21.692 O.3 2.92921 25-625 3-074I3 25-754 3' 22675 25.871 3-38745 25-977 3-55663 26.073 o-3S 2.91777 29.980 3-06323 30.123 3' 21636 30.252 3-37756 30.369 3-54721 30.476 0.4 2.90532 34-370 3-05138 34-524 3' 20507 34.663 3.36681 34.789 3-53698 34.904 0-45 2.89217 38.800 3-03885 38.961 3' i93 J 5 39.107 3-35546 39-239 3.52617 39.358 o-5 2.87861 43.270 3-02595 43-435 3 ,18088 43.584 3-34378 43-719 3-51507 43.840 0-55 2.86499 47.784 3.01300 47-947 3' ,16856 48.096 3-33207 48.230 3-50393 48.351 0.6 2.85165 52-339 3.00031 52.499 3' 15650 52.643 3.32060 52.773 3-49302 52.890 0.65 2.83891 56.937 2.98821 57.088 3- 14500 57.224 3-30967 57.346 3-48263 57-457 0.7 2.82710 6i.575 2.97699 61.713 3. 13434 61.837 3-29955 61.949 3-47301 62.050 0-75 2.81653 66.250 2-96695 66.371 3' ,12481 66.480 3.29049 66.579 3.46441 66.668 0.8 2.80747 70.958 2-95835 7L059 3' ,11665 71.150 3.28274 71.233 3-45702 71-307 0-85 2.80016 75.693 2.95142 75-772 3' ,11006 75.842 3.27649 75.906 3-45II2 75-964 0.9 2.79479 80.449 2-94633 80.503 3 10523 80.552 3.27191 80.595 3-44675 80.635 o-95 2.79151 85.221 2.94322 85.248 3. ,10228 85-273 3.26911 85-295 3.44410 85-3I5 I.O 2.79041 90.000 2.94217 90.000 3- 10129 90.000 3.26816 90.000 3-44321 90.000 1.05 2.79151 94-779 2.94322 94-752 3 ,10228 94.727 3.26911 94-705 3.44410 94.685 i.i 2-79479 99-551 2.94633 99-497 3, 10523 99.448 3.27191 99.405 3-44675 99-365 1.15 2.80016 104.307 2.95142 104.228 3' ,11006 104.158 3-27649 104.094 3.45112 104.036 1.2 2.80747 109.042 2-95835 108.941 3' ,11665 108.850 3-28274 108.767 3-45702 108.693 1.25 2.81653 II3-750 2.96695 113.629 3 ,12481 113.520 3.29049 113.421 3.46441 II3-332 2.82710 118.425 2.97699 118.287 3' 13434 118.163 3-29955 118.051 3-47301 117.950 i-35 2.83891 123.063 2.98821 122.912 3- ,14500 122.776 3.30967 122.654 3.48263 122.543 2.85165 127.661 3.00031 127.501 3' 15650 127.357 3.32060 127.227 3-49302 127.110 1-45 2.86499 132.216 3.01300 132-053 3' 16856 131.904 3-33207 131.770 3-50393 131.649 1-5 2.87861 136.730 3-02595 136-565 3' ,18088 136.416 3-34378 136.281 3-5I507 136.160 i-SS 2.89217 141.200 3-03885 141.039 3' I93I5 140.893 3-35546 140.761 3-52617 140.642 1.6 2.90532 145.630 3-05138 I45-476 3. 20507 145.337 3.36681 145.211 3-53698 145.096 1-65 2.91777 150.020 3-06323 149.877 3- 21636 149.748 3-37756 149.631 3-54721 149.524 2.92921 154-375 3-07413 154.246 3' 22675 154.129 3-38745 154.023 3-55663 I53-927 1-75 2.93938 158.698 3.08382 158-586 3' 23598 158.484 3-39624 158.392 3-56500 158.308 1.8 2.94804 162.992 3.09206 162.900 3, 24384 162.816 3.40373 162.740 3-572I3 162.670 1.85 2.95498 167.265 3.09869 167.194 3. 25015 167.129 3-40975 167.071 3-57788 167.017 1.9 2.96006 171.520 3-10353 171.471 3' 25477 171.428 3-41415 171.388 3-58207 171-352 i-95 2.96315 I75-763 3.10648 I75-738 3' 25759 175.716 3.41683 175-696 3.58462 175-678 2.O 2.96419 180.000 3.10747 180.000 3 25853 180.000 3-4I773 180.000 3.58548 180.000 Example. cosh (1.95 + ' 1.25) = 3.4644I /ii3.33i ! = 3-46441 /H3.I9'. s [113] TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r /T. CONTINUED x = 2.05 a; = 2.15 o 0.05 O.I 0.15 0.2 3.76220 o.ooo . 3.76137 4-339 3.75894 8.68! 3-75495 13-031 3.74948 17-392 3.94832 o.ooo 3.94753 4.354 3.94521 8.711 3.94142 13.075 3.93620 17.449 4.14431 o.ooo 4-14357 4-368 4.14136 8.738 4-13774 13-115 4.13278 17.501 4.35067 o.ooo 4.34996 4.380 4.34785 8.763 4.34440 13.151 4.33968 17.548 4.56791 o.ooo 4.56723 4-391 4.56523 8.785 4.56167 13.184 4.55744 I7-59 1 0.25 0-35 0.4 0-45 3.74268 21.768 3.73470 26.160 3-72574 30-573 3.71600 35.008 3-70572 39467 3.92972 21.836 3.92213 26.239 3-9 r 359 30-660 3.90432 35.102 3-89453 39-565 4.12661 21.899 4.11938 26.311 4.11125 30.740 4.10243 35.187 4.09311 39.653 4.33381 21.956 4.32693 26.376 4.31918 30.812 4.31079 35-264 4.30193 39.734 4.55184 22.007 4.54529 26.435 4-53793 30-877 4.52993 35-334 4.52150 39.807 0-5 0-55 0.6 0.65 o-7 3.69515 43-951 3.68455 48.461 3.67418 52.996 3.66430 57-557 3.65517 62.142 3.88448 44-051 3.87440 48.560 3-86454 53-092 s-Sss^ 57-647 3.84648 62.224 4.08355 44.141 4.07396 48.649 4.06459 53.179 4.05566 57.729 4.04740 62.299 4.29282 44.223 4.28371 48.730 4.27479 53.257 4.26630 57.803 4.25846 62.366 4.51285 44.297 4.50417 48.804 4-49570 53-329 4.48763 57-870 4.48017 62.427 0-75 0.8 0.85 0.9 0-95 3.64699 66.748 3.64000 71.374 3-63437 76-016 3.63023 80.670 3.62771 85.333 3.83870 66.820 3.83206 71-434 3.82671 76.062 3.82279 80.702 3.82039 85.349 4.04002 66.886 4.03371 71.489 4.02863 76.105 4.02490 80.731 4.02263 85.364 4.25145 66.945 4.24544 71.538 4.24061 76-143 4.23707 80.757 4.23491 85.377 4.47350 66.998 4.46780 71.582 4.46322 76.177 4.45985 80.780 4-45779 85.389 I.O 1.05 i.i 1.2 3.62686 90.000 3.62771 94.667 3.63023 99.330 3.63437 103.984 3.64000 108.626 3.81958 90.000 3.82039 94-651 3.82279 99.298 3.82671 103.938 3.83206 108.566 4.02186 90.000 4.02263 94-636 4.02490 99.269 4.02863 103.895 4.03371 108.511 4.23419 90.000 4.23491 94.623 4.23707 99.243 4.24061 103.857 4.24544 108.462 4.45711 90.000 4-45779 94-6ii 4.45985 99.220 4.46322 103.823 4.46780 108.418 1.25 1.4 i-45 3.64699 113.252 3.65517 117-858 3.66430 122.443 3.67418 127.004 3.68455 I3I-539 3.83870 113.180 3.84648 117.776 3.85515 122.353 3.86454 126.908 3.87440 131.440 4.04002 113.114 4.04740 117.701 4.05566 122.271 4.06459 126.821 4.07396 131.351 4.25145 113.055 4.25846 117-634 4.26630 122.197 4.27479 126.743 4.28371 131.270 4.47350 113.002 4.48017 117.573 4.48763 122.130 4.49570 126.671 4.50417 131.196 i-SS 1.6 1-65 3.69515 136.049 3.70572 140.533 3.71600 I44-992 3.72574 149-427 3.73470 153.840 3.88448 135-950 3.89453 140.435 3.90432 144.898 3-9I359 149-340 3.92213 153.761 4.08355 135.859 4.09311 140.347 4.10243 144.813 4.11125 149.260 4.11938 153.689 4.29282 135.778 4.30193 140.266 4.31079 144-736 4.31918 149.188 4.32693 153.624 4.51285 135.703 4.52150 140.193 4.52993 144.666 4-53793 149-123 4.54529 153.565 1.8 1.85 1.9 1-95 3.74268 158.232 3.74948 162.608 3.75495 166.969 3.75894 171-319 3.76137 i75-66i 3.92972 158.164 3.93620 162.551 3.94142 166.925 3.94521 171.289 3-94753 I75-646 4.12661 158.101 4.13278 162.499 4.13774 166.885 4.14136 171.262 4.14357 175.632 4.33381 158.044 4.33968 162.452 4.34440 166.849 4.34785 171.237 4.34996 175-620 4.55184 157-993 4.55744 162.409 4.56167 166.816 4.56523 171.215 4.56723 175.609 2.0 3.76220 180.000 3.94832 180.000 4.14431 180.000 4-35067 180.000 4.56791 180.000 Example, cosh (2.0 + i 0.5) = 3.69515 = 3-695*5 /43-57'.04 g - C"4] TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = 7/7. CONTINUED X = 2.25 x = 2-3 x = 2-35 x = 2.4 T ST 2-45 q r y r y r y r y r y o o o 4-79 6 57 O.OOO 5.03722 O.OOO 5- 29047 O.OOO 5-55695 O.OOO S-83732 0.000 0.05 4-79593 4.401 5.03661 4411 5- 28989 4419 5-55639 4.427 5.83680 4-434 O.I 4.79402 8.805 5A3479 8.824 5- 28816 8.840 5-55474 8.856 5-83522 8.869 0.15 4.79088 13.214 5.03181 13.241 5. 28531 I3-265 5-55204 13.288 5-83265 13-308 O.2 4.78661 17.629 5-02773 17.664 5- 28143 17.696 5-54834 17-725 5.82913 I7.75I 0.25 4.78127 22.053 5.02265 22.096 5' ,27662 22.134 5-54375 22.169 5.82476 22.200 -3 4-77503 26.488 5.01672 26-537 5' ,27096 26.581 5-53837 26.620 5.81964 26.656 o-3S 4.76802 30.936 5.01005 30.989 5' ,26461 3I-038 5-53233 31.081 5-81389 31.121 0.4 4.76042 35-397 5.00281 35-454 5- 25772 35-506 5-52578 35-553 5.80765 35-595 0-45 4.75240 39-873 4.99518 39-932 5- 25046 39.986 5-51887 40-035 5.80108 40.079 o-5 4-744I5 44-364 4-98734 44.424 5' 23728 44-479 5-5II77 44-529 5-79434 44-573 o-55 4-73592 48.870 4-97950 48.930 5' 23553 48.984 5.50468 49.034 5.78758 49.078 0.6 4.72784 53-392 4.97183 53-451 5 ,22825 53-503 5-49775 53-551 5-78099 53-593 0.65 4.72017 57-930 4.96454 57-984 5' 22131 58-034 5-49II5 58.078 5-77472 58.119 0.7 4.71308 62.482 4.95780 62.531 5' ,21490 62.576 5-48505 62.617 5.76892 62.653 0-75 4.70675 67.046 4-95I77 67.090 5 .20917 67.129 5.47961 67.165 5-76375 67.197 0.8 4.70134 71.622 4.94662 7I-659 5 .20428 71.691 5-47495 71.721 5-75932 71.748 0.85 4.69697 76.208 4.94249 76-236 5 .20035 76.261 5.47122 76.284 5^5576 76-305 0.9 4-69377 80.801 4-93944 80.820 5 .19745 80.837 5.46847 80.853 5-753i6 80.867 o-9S 4.69182 85-399 4-93759 85.409 5 .19569 85.418 5.46679 85.426 5.75I56 85433 I.O 4.69117 90.000 4.93696 90.000 5 .19510 90.000 5.46623 90.000 5-75103 90.000 1.05 4.69182 94.601 4-93759 94.591 5 .19569 94.582 5.46679 94-574 5.75I56 94-567 i.i 4-69377 99.199 4-93944 99.180 5 .19745 99.163 5.46847 99.147 5.753i6 99-133 i-iS 4.69697 103.792 4.94249 103.764 5 -20035 103-739 5.47122 103.716 5^5576 103.695 1.2 4-70I34 108.378 4.94662 108.341 5 .20428 108.309 5-47495 108.279 5-75932 108.252 1-25 4.70675 112.954 4-95I77 112.910 5 .20917 112.871 5.47961 112.835 5.76375 112.803 J -3 4.71308 117.518 4-9578o 117.469 5 .21490 117.424 5-48505 117-383 5.76892 "7-347 I -3S 4.72017 122.070 4.96454 122.016 5 .22131 121.966 5-49"5 121.922 5-77472 121.881 1.4 4.72784 126.608 4.97183 126.549 5 .22825 126.497 5-49775 126.449 5.78099 126.407 i-4S 4-73592 131.130 4-97950 131.070 5 23553 131.016 5.50468 130.966 5.78758 130.922 i-5 4-744IS I35-636 4-98734 I35-576 5 .23728 I35-52I 5-5II77 I35-47I 5-79434 135427 i-SS 4.75240 140.127 4.99518 140.068 5 .25046 140.014 5-51887 I39-965 5.80108 139.921 1.6 4.76042 144.603 5.00281 144.546 5 .25772 144.494 5-52578 144.447 5-80765 144.405 1-65 4.76802 149.064 5.01005 149.011 5 .26461 148.962 5-53233 148.919 5-81389 148.879 i-7 4-77503 I53-5I2 5.01672 I53-463 5 .27096 I53-4I9 5-53837 i53-38o 5.81964 153-344 i-7S 4.78127 157-947 5.02265 157.904 5 .27662 157-866 5-54375 157-831 5.82476 157.800 1.8 4.78661 162.371 5-02773 162.336 5 .28143 162.304 5-54834 162.275 5-82913 162.249 1.85 4.79088 166.786 5.03181 166.759 5 .28531 166.735 5-55204 166.712 5-83265 166.692 1.9 4.79402 I7I-I95 5-03479 171.176 5 .28816 171.160 5-55474 171.144 5-83522 171.131 i-95 4-79593 175-599 5.03661 I75-589 5 .28989 I75-58I 5-55639 175-573 5-83680 I75.566 2.0 4-79657 180.000 5.03722 180.000 5 .29047 180.000 5.55695 180.000 5.83732 180.000 Example. cosh (2.40 + i 2.0) = 5.55695 /i8o = 5.55695 \i8o. ["Si TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED * = 2.5 a = 2.55 * = 2.6 x = 2.65 * = 2.7 9 r 7 r V r 7 r 7 r V o o e o 6.13229 o.ooo 6.44259 o.ooo 6.76901 o.ooo 7.11234 o.ooo 7-47347 o.ooo 0.05 6.13179 4-440 6.44212 4.446 6.76855 4.451 7.11191 4.456 7.47306 4.460 O.I 6.13030 8.881 6.44069 8.893 6.76720 8.903 7.11062 8.912 7.47183 8.920 0.15 6.12785 13.326 6.43836 13.342 6.76499 13.357 7.10851 13.371 7.46982 13.383 O.2 6.12450 17-774 6.43518 17-796 6.76195 17.815 7.10563 17.833 7.46708 17.848 0.2S 6.12034 22.228 6.43121 22.254 6.75818 22.278 7.10204 22.299 7.46366 22.318 -3 6.11547 26.689 6.42658 26.718 6.75376 26.745 7.09784 26.770 7.45966 26.791 o-35 6.10999 31.157 6.42137 31.190 6.74881 31.219 7.09313 31.246 7-455i8 31.270 0.4 6.10406 35-634 6.41572 35.668 6-74343 35-700 7.08802 35.728 7.45032 35.754 0-45 6.09781 40.119 6.40977 40.155 6.73778 40.188 7.08264 40.218 7.44519 40.245 o-5 6.09139 44.614 6.40367 44.651 6.73197 44-684 7.07711 44-713 7-43994 44-741 0-55 6.08496 49.118 6.39755 49-155 6.72616 49.187 7.07158 49-217 7.43468 49.244 0.6 6.07869 53.632 6.39160 53-667 6.72048 53-699 7.06618 53.727 742955 53-754 0.65 6.07273 58.155 6.38592 58.188 6.71509 58.218 7.06105 58.244 7.42467 58.269 0.7 6.06722 62.686 6.38068 62.716 6.71011 62.743 7.05631 62.768 7.42017 62.790 0-75 6.06229 67.226 6.37601 67.252 6.70565 67.275 7.05208 67.297 7.41614 67.316 0.8 6.05808 71.772 6.37201 71-794 6.70185 71.813 7.04847 71.831 7.41271 71.847 0.85 6.05471 76-324 6.36879 76.340 6.69880 76.356 7.04556 76.369 7.40994 76.382 0.9 6.05223 80.880 6.36644 80.891 6.69656 80.902 7.04343 80.911 7.40791 80.920 o-95 6.05071 85.439 6.36499 85.445 6.69518 85.450 7.04213 85.455 7.40668 85.460 I.O 6.05020 90.000 6.36451 90.000 6.69473 90.000 7.04169 90.000 7.40626 90.000 1.05 6.05071 94-561 6.36499 94-555 6.69518 94-55 7.04213 94.545 7.40668 94.540 i.i 6.05223 99.120 6.36644 99.109 6.69656 99.098 7.04343 99.089 7.40791 99.080 i-iS 6.05471 103.676 6.36879 103.660 6.69880 103.644 7.04556 103.631 7.40994 103.618 1.2 6.05808 108.228 6.37201 108.206 6.70185 108.187 7.04847 108.169 7.41271 108.153 1.25 6.06229 112.774 6.37601 112.748 6.70565 112.725 7.05208 112.703 7.41614 112.684 i-3 6.06722 117.314 6.38068 117.284 6.71011 117.257 7.05631 117.232 7.42017 117.210 i-3S 6.07273 121.845 6.38592 121.812 6.71509 121.782 7.06105 121.756 7.42467 121.731 1.4 6.07869 126.368 6.39160 126.333 6.72048 126.301 7.06618 126.273 7.42955 126.246 i-45 6.08496 130.882 6-39755 130-845 6.72616 130.813 7.07158 130.783 7.43468 130.756 i-5 6.09139 135.386 6.40367 135-35 6-73 J 97 i35-3i6 7.07711 135.286 7-43994 I35-259 i-55 6.09781 139.881 6.40977 139-845 6.73778 139.812 7.08264 139.782 7-445I9 139-755 1X3 6.10406 144.366 6.41572 144-332 6.74343 144.300 7.08802 144.272 7.45032 144.246 1.65 6.10999 148.843 6.42137 148.810 6.74881 148.781 7.09313 148.754 7.45518 148.730 i-7 6.11547 153.3" 6.42658 153.282 6.75376 I53-255 7.09784 153.230 7.45966 153.209 i-75 6.12034 157.772 6.43121 157-746 6.75818 157-722 7.10204 157.701 7.46366 157.682 1.8 6.12450 162.226 6.43518 162.204 6.76195 162.185 7.10563 162.167 7.46708 162.152 1.85 6.12785 166.674 6.43836 166.658 6.76499 166.643 7.10851 166.629 7.46982 166.617 i. g 6.13030 171.119 6.44069 171.107 6.76720 171.097 7.11062 171.088 7.47183 171.080 1-95 6.13179 175-560 6.44212 175.554 6.76855 175-549 7.11191 175.544 7.47306 175.540 2.0 6.13229 180.000 6.44259 180.000 6.76901 180.000 7.11234 180.000 7-47347 180.000 Example. cosh (2.65 + i 0.75) = 7.05208 /67.297 = 7.05208 /67.i7' .49'. TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /T. CONTINUED x = 2.75 x =.2.85 * = 2.9 x = 2.95 9 r 7 r 7 r 7 r 7 r 7 o o o o o o 7-85328 O.OOO 8.25273 O.OOO 8.67281 O.OOO 9.11458 O.OOO 9.57915 O.OOO 0.05 7.85288 4.464 8-25235 4.467 8.67246 4.470 9.11424 4-473 9-57882 4.476 O.I 7.85172 8.928 8.25124 8-935 8.67140 8.941 9.11324 8-947 9.57787 8.952 0.15 7.84980 13-394 8.24942 13.404 8.66967 13.413 9.11160 13.421 9-57630 13-429 0.2 7.84720 17.863 8.24694 17.876 8.66731 17.888 9.10934 17.898 9.57416 17.908 0.25 7-84395 22-335 8.24385 22.351 8.66436 22.364 9.10655 22.378 9-57150 22.389 o-3 7.84015 26.8II 8.24024 26.829 8.66092 26.845 9.10327 26.860 9.56838 26.873 035 7.83588 31.292 8.23617 3I.3I2 8.65706 3 J .330 9.09959 3I-346 9.56488 31-360 0.4 7-83125 35-778 8.23177 35-799 8.65287 35-8i8 9.09561 35-83S 9.56109 35-851 o-4S 7.82638 40.269 8.22713 40.291 8.64846 40.311 9.09142 40-329 9-557II 40.345 o-S 7.82138 44.766 8.22238 44-788 8.64395 44.808 9.08711 44.827 9-55301 44.843 o-55 7.81637 49.269 8.21762 49.290 8.63941 49.310 9.08281 49-329 9.54892 49-345 0.6 7.81149 53-777 8.21298 53-798 8.63500 53-817 9.07861 53-835 9.54492 53-850 0.65 7.80685 58.291 8.20857 58-311 8.63080 58-329 9.07461 58.345 9-54113 58-360 0.7 7.80257 62.810 8.20449 62.828 8.62691 62.845 9.07093 62.859 9.53762 62.873 0-75 7-79875 67-334 8.20085 67-350 8.62347 67.364 9.06764 67-377 9-53450 ' 67-389 0.8 7-79547 71.862 8.19774 7I-875 8.62051 71.887 9.06483 71.898 9-53182 71.907 0.85 7-79285 76.393 8.19524 76.404 8.61813 76.413 9-06257 76.421 9.52967 76.429 0.9 7.79092 80.927 8.19341 80.934 8.61639 80.940 9.06091 80.946 9.52809 80.951 o-95 7-78975 85-463 8.19230 85.467 8.61532 85-470 9.05990 85-473 9-52713 85475 I.O 7-78935 90.000 8.19192 90.000 8.61497 90.000 9.05956 90.000 9.52681 90.000 1.05 7-78975 94-537 8.19230 94-533 8.61532 94-530 9.05990 94.527 9-52713 94.525 i.i 7.79092 99.073 8.19341 99.066 8.61639 99.060 9.06091 99.054 9.52809 99.049 i-iS 7.79285 103.607 8.19524 103.596 8.61813 103.587 9.06257 103.579 9.52967 103.571 1.2 7-79547 108.138 8.19774 108.125 8.62051 108.113 9.06483 108.102 9-53182 108.093 ".S 7-79875 112.666 8.20085 112.650 8.62347 112.636 9.06764 112.623 9-53450 112.611 i-3 7.80257 117.190 8.20449 117.172 8.62691 "7-155 9.07093 117.141 9-53762 117.127 35 7.80685 121.709 8.20857 121.689 8.63080 121.671 9.07461 121.655 9.54ii3 121.640 4 7.81149 126.223 8.21298 126.202 8.63500 126.183 9.07861 126.165 9-54492 126.150 45 7.81637 I30-73I 8.21762 130.710 8.63941 130.690 9.08281 130.671 9.54892 130.655 5 7-82138 I35-234 8.22238 135.212 8.64395 I35-I92 9.08711 I35.I73 9-55301 135.157 55 7.82638 I39-73I 8.22713 139.709 8.64846 139.689 9.09142 139.671 9-557II I39.655 .6 7-83125 144.222 8-23177 144.201 8.65287 144.182 9.09561 144.165 9.56109 144.149 6S 7-83588 148.708 8.23617 148.688 8.65706 148.670 9.09959 148.654 9-56488 148.640 7 7.84015 153-189 8.24024 i53- I 7i 8.66092 I53-I55 9.10327 153-140 9.56838 I53""? 75 7.84395 I57-665 8.24385 157.649 8.66436 I57-636 9.10655 157.622 9-57I50 157-611 .8 7.84720 162.137 8.24694 162.124 8.66731 162.112 9.10934 162.102 9.57416 162.092 85 7.84980 166.606 8.24942 166.596 8.66967 166.587 9.11160 166.579 9-57630 166.571 9 7.85172 171.072 8.25124 171.065 8.67140 171.059 9.11324 171-053 9.57787 171.048 95 7.85288 I75-536 8-25235 175.533 8.67246 175-530 9.11424 I75.527 9-57882 I75-524 2.O 7-85328 180.000 8.25273 180.000 8.67281 180.000 9.11458 180.000 9.579I5 180.000 Example. cosh (2.90 + i 0.9) = 9.06091 /8o.946 = 0.96091 /8o.<;6'.46'. [117] TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED x = 3-o x = 3.05 * = 3-i x = 3.15 x = 3-2 <7 r T r 7 r y r 7 r y o o 10.06766 o.ooo 10.58135 O.OOO 11.12150 O.OOO 11.68946 o.ooo 12.28665 O.OOO 0.05 10.06737 4.478 10.58110 4.480 II.I2I2O 4.482 11.68920 4.484 12.28640 4-485 O.I 10.06645 8.956 10.58020 8.960 II.I2O4O 8.964 11.68840 8.968 12.28560 8.971 0.15 10.06500 13-436 10.57880 13.442 II.II9OO 13-447 11.68710 13-452 12.28440 13.457 O.2 10.06292 17.917 10.57680 I7-925 II.II72O 17.932 11.68540 17.938 12.28280 17.944 0.25 10.06040 22.400 10.57440 22.409 II.II49O 22.418 11.68320 22.426 12.28070 22.433 o-3 10.05743 26.886 10.57160 26.896 II.II22O 26.906 11.68060 26.915 12.27820 26.923 . 0-35 10.05410 3 J -374 10.56840 3I-386 II.IO92O 31.396 11.67780 31.406 12.27550 3!-4i5 0.4 10.05050 35-865 10.56500 35.878 II.IO6OO 35.890 11.67470 35-900 12.27260 35.910 0-45 10.04670 40.360 10.56140 40-373 II.IO25O 40.385 11.67140 40.396 12.26950 40.406 o-S 10.04280 44.858 10-55770 44.871 II.O99OO 44.884 11.66810 44.895 12.26630 44.905 10.03890 49.360 10.55400 49-373 11.09540 49.385 11.66470 49.396 12.26310 49.406 0.6 10.03510 53-865 10.55040 53-878 II.0920O 53-889 11.66140 53.900 12.26000 53-909 0.65 10.03150 58.373 10.54690 58-385 II.08880 58.396 11.65830 58.406 12.25700 58-415 0.7 10.02820 62.885 10.54380 62.896 11.08580 62.906 11.65540 62.915 12.25430 62.923 0-75 10.62520 67-399 10.54090 67.409 II.O83IO 67.417 11.65290 67-425 12.25180 67.432 0.8 10.02260 71.916 10.53850 71.924 II.o8o8o 71.932 11.65070 71-938 12.24980 71.944 0.85 10.02060 76.435 10.53660 76.441 11.07890 76.447 11.64890 76.452 12.24810 76.457 0.9 10.01910 80.956 10.53520 80.960 11.07750 80.964 11.64770 80.967 12.24690 80.970 o-95 10.01820 85.478 10.53430 85.480 II.O767O 85.482 11.64690 85.484 12.24610 85-485 I.O 10.01787 90.000 10-53399 90.000 11.07645 9O.OOO 11.64661 90.000 12.24588 90.000 1.05 10.01820 94.522 10.53430 94.520 II.O767O 94.518 11.64690 94.516 12.24610 94.5I5 i.i 10.01910 99.044 10.53520 99.040 11.07750 99.036 11.64770 99-033 12.24690 99.030 I - I 5 10.02060 103-565 10.53660 103.559 11.07890 103-553 11.64890 103.548 12.24810 103-543 1.2 10.02260 108.084 10.53850 108.076 II.o8o8o 108.068 11.65070 108.062 12.24980 108,056 !.2 5 10.02520 112.601 10.54090 112.591 II.O83IO 112.583 11.65290 112.575 12.25180 112.568 1-3 10.02820 117.115 10.54380 117.104 11.08580 117.094 11.65540 117.085 12.25430 117.077 i-35 10.03150 121.627 10.54690 121.615 II.08880 I2I.6O4 11.65830 121.594 12.25700 121.585 1.4 10.03510 126.135 10.55040 126.122 II.O92OO I26.III 11.66140 126.100 12.26000 126.091 i-45 10.03890 130.640 10.55400 130.627 11.09540 130.615 11.66470 130.604 12.26310 130.594 i-5 10.04280 I35.I42 10.55770 I35-I29 II.O9900 I35.II6 11.66810 135-105 12.26630 I35.095 i-55 10.04670 139.640 10.56140 139.627 II.IO250 139.615 11.67140 139.604 12.26950 139-594 1.6 10.05050 I44.I35 10.56500 144.122 II.IO6OO I44.IIO 11.67470 144.100 12.27260 144.090 1-65 10.05410 148.626 10.56840 148.614 11.10920 148.604 11.67780 148.594 12.27550 148.585 10.05743 I53-II4 10.57160 I53-I04 II.II22O I53-094 11.68060 153-085 12.27820 I53-077 i-75 10.06040 157.600 10.57440 I57-59I II.II490 I57-582 11.68320 157-574 12.28070 I57-567 1.8 10.06292 162.083 10.57680 162.075 II.II72O 162.068 11.68540 162.062 12.28280 162.056 1-85 10.06500 166.564 10.57880 166.558 n.II90O 166.553 11.68710 166.548 12.28440 166.543 1.9 10.06645 171.044 10.58020 171.040 11.12040 171.036 11.68840 171.032 12.28560 171.029 I-9S 10.06737 I75-522 10.58110 I75-520 II.I2I20 I75-5I8 11.68920 175-516 12.28640 I75-5I5 2.0 10.06766 180.000 10.58135 180.000 II.I2I50 iSo.OOO 11.68946 180.000 12.28665 iSo.OOO Example. cosh (3.15 + 2Q.I5) = 11.68710 /i3.452 = 11.68710 /i3.27'.07*. TABLE XI. HYPERBOLIC COSINES, cosh (* + iq) = r y. CONTINUED = 3-25 *' = 3-3 = 3-35 x = 3.4 = 3-45 9 r 7 r V r 7 r 7 r y o o o o o 12.91456 0.000 I3-57476 O.OOO 14.26891 O.OOO 14.99874 o.ooo 15.76607 O.OOO 0-05 12.91430 4.486 13-57455 4.488 14.26870 4.489 4.490 15-76587 4491 O.I 12.91360 8-973 I3-57387 8.976 14.26805 8.978 14.99790 8.980 15-76530 8.982 0.15 12.91240 13.461 I3-57275 13-465 14.26700 13.468 14.99692 13-471 iS-76433 13-474 O.2 12.91085 17.949 i3-57!23 17.954 14.26556 17-959 14-99555 17.963 15-76303 17.966 0.25 12.90888 22.439 13-56936 22.445 14.26377 22.450 14.99385 22.455 15.76144 22.459 o-3 12.90658 26.930 13.56718 26.937 14.26167 26.943 14.99186 26.948 I5-75950 26.953 o-35 12.90398 31.423 13.56470 31.431 14.25933 31-437 14.98960 3L443 I5-75740 31-449 0.4 12.90117 35-9 x 8 13.56203 35-926 14.25614 35-933 14.98721 35-939 i5-755io 35-945 o-45 12.89820 40.415 I3-559 I 8 40.423 14.25412 40.430 14.98466 40-437 15.75270 40.443 0-5 12.89518 44.914 I3-55633 44.922 14-25137 44-929 14.98205 44.936 15.75020 44.942 0-55 12.89215 49-4I5 I3-55346 49.423 14.24863 49.430 14-97945 49-437 I5-74772 49-443 0.6 12.88910 53.918 13.55062 53.926 14.24595 53-933 14.97690 53-939 I5-74529 53-945 0.65 12.88637 58-423 13.54796 58.431 14.24339 58.437 14.97448 58.443 15.74300 58.449 o-7 12.88380 62.930 I3-54550 62.937 14.24106 62.943 14.97225 62.948 15.74088 62.953 0-75 12.88146 67439 I3-54322 67.445 14.23897 67450 14.97023 67455 15-73898 67459 0.8 12.87949 71.949 I3-54I4I 71-954 14.23718 71-959 14.96855 71.963 I5-73736 71.966 0.85 12.87790 76.461 I3-5399I 76.465 I4-23573 76.468 14.96716 76.471 15-73603 76.474 0.9 12.87670 80.973 13-53878 80.976 14.23470 80.978 14.96616 80.980 i5-735io 80.982 0-95 12.87600 85.486 13-53810 85.488 14.23403 85-489 14.96556 85.490 I5-73450 85.491 I.O 12.87578 90.000 13.53788 90.000 14.23382 90.000 14.96536 90.000 I5-73432 90.000 1.05 12.87600 94-5H 13-53810 94.512 14.23403 94-5" 14.96556 94.510 I5-73450 94.509 i.i 12.87670 99.027 I3-53878 99.024 14.23470 99.022 14.96616 99.020 i5-735io 99.018 1.15 12.87790 103.539 I3-5399I 103.535 I4-23573 103.532 14.96716 103.529 15-73603 103.526 1.2 12.87949 108.051 I3-54I4I 108.046 14.23718 108.041 14.96855 108.037 I5-73736 108.034 T .25 12.88146 112.561 I3-54322 "2.555 14.23897 "2.550 14.97023 "2-545 15-73898 112.541 1.3 12.88380 117.070 I3.54550 117.063 14.24106 117.057 14.97225 117.052 15.74088 117.047 1.35 12.88637 121.577 I3-54796 121.569 14.24339 121.563 14.97448 121-557 15.74300 121-551 1.4 12.88910 126.082 13.55062 126.074 14.24595 126.067 14.97690 126.061 I5-74529 126.055 i-45 12.89215 130-585 I3-55346 130.577 14.24863 130.570 14.97945 130.563 I5-74772 130.557 i-S 12.89518 135.086 I3-55633 135-078 14-25137 I35-07I 14.98205 135.064 15.75020 I35-058 i-55 12.89820 139-585 I3-559I8 139-577 14.25412 I39-570 14.98466 139-563 15-75270 139-557 1.6 12.90117 144.082 13.56203 144.074 14.25614 144.067 14.98721 144.061 i5-755io I44.055 1.65 12.90398 148.577 13.56470 148.569 14-25933 148.563 14.98960 148.557 I5-75740 148.551 12.90658 I53-070 13.56718 153-063 14.26167 I53.057 14.99186 153-052 I5-75950 I53-047 1-75 12.90888 I57-56I 13-56936 157-555 14.26377 IS7.550 14-99385 157-545 15.76144 I57.54I 1.8 12.91085 162.051 13-57123 162.046 14.26556 162.041 14-99555 162.037 15-76303 162.034 1.85 12.91240 166.539 I3-57275 166.535 14.26700 166.532 14.99692 166.529 I5-76433 166.526 1.9 12.91360 171.027 I3-57387 171.024 14.26805 171.022 14.99790 171.020 15-76530 171.018 i-95 12.91430 I75-5I4 13-57455 175-512 14.26870 175-5" 14.99853 i75-5io 15-76587 I75.509 2.0 12.91456 180.000 I3-57476 180.000 14.26891 180.000 14.99874 180.000 15.76607 180.000 Example. cosh (3.4 + to. 75) = 14.97023 = 14-97023 /67.27'.i8. ["9] TABLE XI. HYPERBOLIC COSINES, cosh (x + iq) = r /y. CONTINUED x = 3-5 x = 3-55 x = 3.6 x = 3-65 * = 3-7 9 r y r y r y r y r y o o o o 16.57282 o.ooo 17.42102 o.ooo 18.31278 o.ooo 19.25033 o.ooo 20.23601 o.ooo 0.05 16.57260 4.492 17.42083 4.493 18.31260 4.493 19.25015 4.494 20.23585 4.495 O.I 16.57210 8.984 17.42030 8.985 18.31210 8.987 19.24970 8.988 20.23540 8.989 0.15 16.57120 13.476 17.41945 13.479 18.31130 13.481 19.24890 13.483 20.23460 13.484 O.2 16.56996 17.969 17.41830 17-972 18.31020 17.975 19.24790 17.977 20.23365 17.979 O.25 16.56840 22.463 17.41680 22.467 18.30880 22.470 19.24650 22.473 20.23240 22.475 O-3 16.56660 26.958 17.41510 26.962 18.30715 26.965 19.24496 26.969 20.23090 26.972 0-35 16.56460 31.454 17.41310 31.458 18.30530 31.462 19.24323 31.466 20.22930 31.469 0.4 16.56240 35.950 17.41110 35.955 18-30335 35-959 19.24134 35.963 20.22750 35.967 0.45 16.56010 40.448 17.40900 40.453 18.30126 40.458 19.23940 40.462 20.22560 40.465 o-5 16.55770 44-948 17.40665 44-953 18.29910 44-957 19.23732 44.961 20.22365 44-965 0-55 16.55538 49-448 17.40441 49-453 18.29699 49.458 19.23530 49.462 20.22170 49.465 0.6 16.55307 53-950 17.40222 53.955 18.29490 53-959 19-23331 53-963 20.21981 53.967 0.65 16.55087 58.454 17.40015 58-458 18.29292 58.462 19.23142 58.465 20.21804 58.469 0.7 16.54885 62.958 17.39822 62.962 18.29109 62.965 19.22968 62.969 20.21639 62.972 0-75 16.54702 67.463 17.39650 67.466 18.28942 67.470 19.22813 67.473 20.21490 67.475 0.8 16.54550 71-969 17.39504 71-972 18.28805 71-975 19.22681 71.977 20.21365 71-979 0.85 16.54428 76.476 17.39386 76.479 18.28694 76.481 19.22577 76.483 20.21268 76.484 0.9 16.54337 80.984 17.39300 80.985 18.28611 80.987 19.22496 80.988 20.21189 80.989 0-95 16.54281 85.492 17.39248 85.493 18.28560 85.493 19.22448 85.494 20.21140 85.495 I.O 16.54263 90.000 17.39230 90.000 18.28546 90.000 19.22434 90.000 20.21129 90.000 1-05 16.54281 94.508 17.39248 94.507 18.28560 94.507 19.22448 94.506 20.21140 94.506 i.i 16.54337 99.016 17.39300 99.015 18.28611 99.013 19.22496 99.012 20.21189 99.011 1.15 16.54428 103.524 17.39386 103.522 18.28694 103.519 19.22577 103.518 20.21268 103.516 1.2 16.54550 108.031 I 7-395 O 4 108.028 18.28805 108.025 19.22681 108.023 20.21365 108.021 25 16.54702 112.537 17.39650 112.534 18.28942 112.530 19.22813 112.527 20.21490 112.525 3 16.54885 117.042 17.39822 117.038 18.29109 117-035 19.22968 117.031 20.21639 117.028 35 16.55087 121.547 17.40015 121.542 18.29292 121.538 19.23142 121.535 20.21804 121.531 4 16.55307 126.050 17.40222 126.045 18.29490 126.041 19.23331 126.037 20.21981 126.033 45 16.55538 130.552 17.40441 130.547 18.29699 130.542 19.23530 130.538 20.22170 130.535 5 16.55770 135.052 17.40665 135.047 18.29910 135.043 19.23732 135.039 20.22365 135.035 55 16.56010 139.552 17.40900 139.547 18.30126 139.542 19.23940 139.538 20.22560 139.535 .6 16.56240 144.050 17.41110 144-045 18.30335 144.041 19.24134 144.037 20.22750 144.033 65 16.56460 148.546 17.41310 148.542 18.30530 148.538 19.24323 148.534 20.22930 148.531 7 16.56660 153.042 17.41510 153.038 18.30715 153.035 19.24496 153.031 20.23090 153.028 75 16.56840 157-537' 17.41680 157-533 18.30880 157.530 19.24650 157-527 20.23240 157.525 .8 16.56996 162.031 17.41830 162.028 18.31020 162.025 19.24790 162.023 20.23365 162.021 1.85 16.57120 166.524 17.41945 166.521 18.31130 166.519 19.24890 166.517 20.23460 166.516 1.9 16.57210 171.016 17.42030 171.015 18.31210 171.013 19.24970 171.012 20.23540 171.011 i-95 16.57260 175.508 17.42083 175-507 18.31260 175-507 19.25015 175.506 20.23585 175-505 2.O 16.57282 180.000 17.42102 180.000 18.31278 180.000 19-25033 180.000 20.23601 180.000 Example. cosh (3.65 + i 0.05) = 19.25015 /4.494 = 19.25015 /4.29 / .38*. [120] TABLE XL HYPERBOLIC COSINES, cosh (x + iq) = r /r CONTINUED x = 3-75 * = 3- = 3-8S = 3-95 21.27230 o.ooo 22.36178 o.ooo 23-50717 o.ooo 24.71135 o.ooo 25-9773I o.ooo 0.05 21.27212 4.495 22.36163 4.495 23.50702 4.496 24.71120 4.496 25-97720 4.497 O.I 2I.27I7O 8.990 22.36122 8.991 23.50664 8.992 24.71090 8.993 25.97680 8.993 0.15 2I.27IOO 13.486 22.36055 13.487 23.50600 13.488 24.71024 13.489 25-97630 13.490 0.2 2I.270OO 17.981 22.35962 17-983 23.50512 17.985 24.70940 17.986 25-97550 I7-988 0.2S 21.26885 22.478 22.35850 22.480 23.50404 22.482 24.70840 22.483 25-97450 22.485 o-3 21.26745 26.974 22.35716 26.977 23.50277 26.979 24.70720 26.981 . 25.97337 26.983 o-3S 21.26586 31.472 22.35565 31-474 23-50136 31-477 24.70580 31.479 25.97206 31.481 0.4 21.26415 35.970 22.35403 35-973 23.49980 35.975 24.70440 35.978 25.97066 35.980 o-4S 21.26236 40.469 22.35230 40.472 23.49820 40.474 24.70280 40.477 25.96920 40.479 o-5 2I.26O52 44.968 22.35060 44-971 23.49650 44-974 24.70120 44.977 25.96770 44-979 o-55 21.25869 49.469 22.34883 49-472 23.49486 49.474 24.69964 49.477 25.96618 49.479 0.6 21.25685 53.970 22.34712 53.973 23.49322 53.975 24.69809 53.978 25.96471 53.980 0.65 21.25520 58.472 22.34550 58.474 23-49II5 58.477 24.69662 58.479 25-96333 58-481 0.7 21.25362 62.974 22.34401 62.977 23.49028 62.979 24.69528 62.981 25.96205 62.983 0-75 21.25221 67.478 22.34270 67.480 23.48900 67.482 24.69406 67.483 25.96090 67.485 0.8 21.25102 71.981 22.34153 71-983 23.48791 71.985 24.69302 71.986 25.95991 71.988 0.85 2I.250O6 76.486 22.34061 76.487 23.48704 76.488 24.69223 76.489 25.95911 76.490 0.9 21.24935 80.990 22.33995 80.991 23.48640 80.992 24.69159 80.993 25-95854 80.993 o-95 21.24891 85.495 22.33952 85.496 23.48601 85.496 24.69120 85.496 25.95820 85.497 I.O 21.24878 90.000 22.33941 90.000 23.48589 90.000 24.69110 90.000 25.95806 90.000 1.05 21.24891 94-50S 22.33952 94-504 23.48601 94.504 24.69120 94.504 25.95820 94-503 i.i 21.24935 99-010 22-33995 99-009 23.48640 99.008 24.69159 99.007 25-95854 99-007 i-iS 21.25006 103.514 22.34061 103.513 23.48704 103.512 24.69223 103.511 25.95911 103.510 1.2 2I.25IO2 IO8.OI9 22.34153 108.017 23.48791 108.015 24.69302 108.014 25-9599I 108.013 I-2S 21.25221 112.522 22.34270 112.520 23.48900 112.518 24.69406 112.517 25.96000 112.515 i-3 21.25362 II7.O26 22.34401 117.023 23.49028 117.021 24.69528 117.019 25.96205 117.017 i-3S 2I.2552O 121.528 22.34550 121.526 23.49115 121.523 24.69662 121.521 25-96333 121.519 1.4 21.25685 I26.O3O 22.34712 126.027 23.49322 126.025 24.69809 126.022 25.96471 126.020 i. 45 21.25869 130.531 22.34883 130.528 23.49486 130.525 24.69964 130.523 25.96618 130.521 i-S 21.26052 135.032 22.35060 135.029 23.49650 135.026 24.70120 135.023 25.96770 135.021 i-55 21.26236 I39-53 1 22.35230 139.528 23.49820 139.526 24.70280 139.523 25.96920 139.521 1.6 21.26415 144.030 22.35403 144.027 23.49980 144-025 24.70440 144.022 25.97066 144.020 1.65 21.26586 148.528 22.355 6 5 148.526 23.50136 148.523 24.70580 148.521 25.97206 148.519 i-7 21.26745 153.026 22.35716 153.023 23-50277 153-021 24.70720 153.019 25-97337 I53-OI7 i-75 21.26885 I57-522 22.35850 157.520 23.50404 157-518 24.70840 157.517 25-9745 I57-5I5 1.8 21.27000 I62.OI9 22.35962 162.017 23.50512 162.015 24.70940 162.014 25.97550 162.012 1.85 2I.27IOO 166.514 22.36055 166.513 23.50600 166.512 24.71024 166.511 25.97630 166.510 1.9 2I.27I7O I7I.OIO 22.36122 171.009 23.50664 171.008 24.71090 171.007 25.97680 171.007 i-9S 21.27212 I75-505 22.36163 175-505 23.50702 175-504 24.71120 i75-54 25.97720 175.503 2.0 2I.2723O iSo.OOO 22.36178 180.000 23.50717 180.000 24.71135 180.000 25-9773I 180.000 Example. cosh (3.85 + i 1.05) = 23.48601 = 23.48601 /94.3o / .i4 > . [121] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r * = X = 0.05 * = O.I * = 3.15 X = O.2 9 r 7 r 7 f 7 r 7 r 7 o o 0.00 90 0.04996 o.oo 0.09967 O.OO 0.14889 O.OO 0.19738 O.OO 0.05 0.07870 9P 0.09322 57.368 0.12699 37.846 0.16840 27.190 0.21246 20.849 O.I 0.15838 90 0.16607 72.040 O.I87II 56.914 0.21732 45.420 0.25294 36.955 0.15 0.24008 90 0.24520 77.558 0.25987 66.084 0.28232 56.148 0.31045 47.863 0.2 0.32492 90 0.32870 80.329 0.33969 71.092 0-35699 62.612 0-37939 55.054 0.25 0.41421 90 0.41727 81.937 0.42568 74.107 0-43932 66.700 0.45731 59.848 0-3 0.50953 9 0.51180 82.942 0.51851 76.025 0.52931 69.373 0.54368 63.082 o-3S 0.61280 90 0.61455 83.585 0.61970 77.267 0.62802 71.131 0.63915 65.250 0.4 0.72654 90 0.72778 83.987 o.73 J 43 78.047 0-73734 72.245 0.74525 66.641 0-45 0.85408 90 0.85476 84.209 0.85678 78.478 0.86004 72.864 0.86439 67.419 0-5 I.OOOOO 90 I.OOOOO 84.279 I.OOOOO 78.616 I.OOOOO 73.064 I.OOOOO 67.670 0-5S 1.17085 90 1.16991 84.209 1.16717 78.478 1.16274 72.864 1.15688 67.419 0.6 1.37638 90 1.37404 83.987 1.36718 78.047 1.35623 72.245 1-34183 66.641 0.65 1.63185 90 1.62722 83.585 1.61369 77.267 1.59231 71.131 1.56459 65.250 0.7 1.96261 90 1.95388 82.942 1.92859 76.025 1.88925 69.373 I.83933 63.082 0-75 2.41421 90 2.39735 81.937 2.34919 74.107 2.27623 66.700 2.18669 59.848 0.8 3.07768 90 3-04234 80.329 2.94391 71.092 2.80120 62.612 2.63581 55-054 0.85 4.16530 90 4.07824 77.558 3.84810 66.084 3.54212 56.148 3-22115 47-863 0.9 6.31375 90 6.02149 72.040 5.34442 56.914 4.60155 45.420 3-95347 36.955 o-9S 12.70620 90 10.72750 57.368 7.87464 37.846 5-93842 27.190 4.70673 20.849 I.O 00 90 20.01667 G.OO 10.03331 O.OO 6.71659 O.OO 5.06649 O.OO 1.05 12.70620 90 10.72750 57-368 7.87464 37.846 5-93842 27.190 4.70673 20.849 .1 6.31375 90 6.02149 72.O4O 5.34442 36.914 4.60155 45.420 3-95347 36.955 IS 4.16530 90 4.07824 77.558 3.84810 66.084 3-54212 56.148 3-22115 47.863 .2 3-07768 90 3-04234 80.329 2.94391 71.092 2.80120 62.612 2.63581 55.054 25 2.41421 90 2-39735 81.937 2.34919 74.107 2.27623 66.700 2.18669 59.848 3 1.96261 90 1.95388 82.942 1.92859 76.025 1.88925 69.373 I-83933 63.082 35 1.63185 90 1.62722 83.585 1.61369 77.267 1-59231 71.131 1.56459 65.250 1.4 1-37638 90 1.37404 83.987 1.36718 78.047 1.35623 72.245 1-34183 66.641 1-45 1.17085 90 1.16991 84.209 1.16717 78.478 1.16274 72.864 1.15688 67.419 5 I.OOOOO 90 I.OOOOO 84.279 I.OOOOO 78.616 I.OOOOO 73.064 1.09000 67.670 55 0.85408 90 0.85476 84.209 0.85678 78.478 0.86004 72.864 0.86439 67.419 .6 0.72654 90 0.72778 83-987 0.73*43 78.047 0.73734 72.245 0.74525 66.641 65 0.61280 90 0.61455 83.585 0.61970 77.267 0.62802 71.131 0.63915 65.250 7 0.50953 90 0.51180 82.942 0.51851 76.023 0.52931 69.373 0.54368 63.082 75 0.41421 90 0.41727 81.937 0.42568 74.107 0.43932 66.700 0.45731 59.848 .8 0.32492 90 0.32870 80.329 0.33969 71.092 0.35699 62.612 0-37939 55.054 85 0.24008 90 0.24520 77.558 0.25987 66.084 0.28232 56.148 0.31045 47.863 9 0.15838 9P 0.16607 72.O4O 0.18711 56.914 0.21732 45.420 0.25294 36.953 95 0.07870 90 0.09322 57.368 0.12699 37.846 0.16840 27.190 0.21246 20.849 2.0 o.oo 90 0.04996 o.oo 0.09967 o.oo 0.14889 o.oo 0.19738 o.oo Note. Negative quantities are in heavy type. Examples, tanh (o.i + i 0.25) = 0.42568 /74. 107 = 0.42568 /74.o6'.2$". m tanh (o.i + i 1.2) = 2.94391 \7i.o92 = 2.94391 \7i.o5'.3i*. [122] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r * = 0.25 x = 0.3 = o-3S x = 0.4 CONTINUED = 0.45 q r V r 7 r 7 f 7 r 7 o o o o o 0.24492 o.oo 0.29131 O.OO 0.33638 o.oo 0.37995 o.oo 0.42190 0.00 0.05 0.25721 16.710 0.30168 13-805 0-34534 11.652 0.38784 9.990 0.42894 8.665 O.I 0.29145 30.669 0.33123 25.891 0.37127 22.164 0.41000 19.185 0.44965 16.753 0.15 0.34237 41.063 0.37657 35-492 0.41192 30.900 0.44758 27.076 0.48295 23.858 O.2 0.40561 48.442 0-43445 42.715 0.46491 37-770 0.49617 33-498 0.52758 29.796 0.2S 0.47875 53-612 0.50275 48.001 0.52849 42.989 0.55525 38.527 0.58242 34-560 0-3 0.56098 57-214 0.58056 51-799 0.60177 46.843 0.62401 42.331 0.64675 38.242 0-35 0.65262 59-679 0.66796 54-453 0.68466 49-590 0.70225 45-093 0.72031 40.958 0.4 0.75486 61.281 0.76580 56.200 0.77774 51-423 0.79033 46.961 0.80327 42.815 0-45 0.86968 62.184 0.87570 57-194 0.88225 52.474 0.88914 48.039 0.89620 43.896 0-5 I.OOOOO 62.476 I.OOOOO 57.518 I.OOOOO 52.817 I.OOOOO 48.392 I.OOOOO 44.250 o-55 1.14985 62.184 1.14195 57-194 1-13347 52.474 1.12469 48.039 1.11583 43.896 0.6 1.32476 61.281 1.30582 56.200 1.28577 51-423 1.26529 46.961 1.24492 42.815 0.65 1.53228 59-679 1.49710 54-453 1.46059 49.590 1.42400 45.093 1.38830 40.958 0.7 1.78259 57-214 1.72246 51-799 1.66176 46.843 1.60255 42.331 1.54620 38.242 0-75 2.08878 53-612 1.98907 48.001 1.89219 42.989 1.80100 38.527 1.71698 34-560 0.8 2.46545 48.442 2.30177 42.715 2.15096 37-770 2.01545 33-498 1.89545 29.796 0.85 2.92081 41.063 2-65553 35-492 2.42764 30.900 2.23422 27.076 2.07060 23.858 0.9 3-43113 30.669 3.01903 25.891 2.69344 22.164 2.43370 19.185 2.22397 16.753 0-95 3.88795 16.710 3.31480 13-805 2.89571 11.652 2.57838 9.990 2.33132 8.665 I.O 4.08299 0.00 3-43274 o.oo 2.97287 o.oo 2.63193 o.oo 2.37024 o.oo 1.05 3.88795 16.710 3.31480 13.805 2.89571 11.652 2.57838 9.990 2.33132 8.665 i.i 3-43113 30.669 3.01903 25.891 2.69344 22.164 2.43370 19-185 2.22397 16.753 i-iS 2.92081 41.063 2.65553 35.492 2.42764 30.900 2.23422 27.076 2.07060 23.858 1.2 2.46545 48.442 2.30177 42.715 2.15096 37.770 2.01545 33-498 1.89545 29.796 1.25 2.08878 53.612 1.98907 48.001 1.89219 42.989 1.80100 38.527 1.71698 34.560 i-3 1.78259 57.214 1.72246 51-799 1.66176 46.843 1.60255 42.331 1.54620 38.242 i-3S 1.53228 59.679 1.49710 54-453 1.46059 49.590 1.42400 45.093 1.38830 40.958 1.4 1.32476 61.281 1.30582 56.200 1.28577 51-423 1.26529 46.961 1.24492 42.815 i-4S 1.14985 62.184 1.14195 57-194 1-13347 52.474 1.12469 48.039 1.11583 43.896 i-5 I.OOOOO 62.476 I.OOOOO 57.518 I.OOOOO 52.817 I.OOOOO 48.392 I.OOOOO 44.250 i-55 0.86968 62.184 0.87570 57-194 0.88225 52.474 0.88914 48.039 0.89620 43.896 1.6 0.75486 61.281 0.76580 56.200 0.77774 51-423 0.79033 46.961 0.80327 42.815 1.65 0.65262 59-679 0.66796 54-453 0.68466 49.590 0.70225 45.093 0.72031 40.958 i-7 0.56098 57-214 0.58056 51-799 0.60177 46.843 0.62401 42.331 0.64675 38.242 i-7S 0.47875 53-612 0.50275 48.001 0.52849 42.989 0.55525 38.527 0.58242 34.560 1.8 0.40561 48.442 0.43445 42.715 0.46491 37.770 0.49617 33.498 0.52758 29.796 1.85 0.34237 41.063 0.37657 35-492 0.41192 30.900 0.44758 27.076 0.48295 23.858 1.9 0.29145 30.669 0.33123 25.891 0.37127 22.164 0.41090 19.185 0.44965 16.753 i. 95 0.25721 16.710 0.30168 13-805 0.34534 11.652 0.38784 9.990 0.42894 8.665 2.0 0.24492 o.oo 0.29131 o.oo 0.33638 o.oo 0.37995 o.oo 0.42190 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.4 + i o) = 0.37995 /o tanh (0.45 + * i.i) = 2.22397 \i6.753 = 2.22397 \i6.45'.ii*. [123] TABLE XII. HYPERBOLIC TANGENTS, tanh (* + *?) = r /y. CONTINUED * = o.s * = 0.55 * = 0.6 x = 0.65 * = 0.7 q r y r 7 r 7 r 7 r 7 o o o o 0.46211 o.oo 0.50052 o.oo 0.53704 o.oo 0.57167 o.oo 0.60437 o.oo 0.05 0.46846 7.582 0.50628 6.680 0.54230 5.917 0.57648 5.263 0.60878 4.696 O.I 0.48720 14-732 0.52334 13.027 0.55790 11.570 0.59079 10.312 0.62194 9.217 0.15 0.51758 21.122 0.55115 18.773 0.58344 16.739 0.61428 14.966 0.64357 13.409 0.2 0.55865 26.572 0.58900 23.753 0.61835 21.276 0.64650 19.090 0.67331 17.153 0.2S 0.60952 31.035 0.63616 27.897 0.66205 25.101 "0.68696 22.604 0.71076 20.372 0-3 0.66956 34-544 0.69209 31.204 0.71405 28.190 0.73523 25.471 0.75548 23.017 0-35 0.73847 37.I69 0.75645 33.707 0.77399 30.553 0.79092 27.683 0.80711 25.074 0.4 0.8l62* 38.983 0.82914 35.453 0.84168 32.214 0.85377 29.248 0.86531 26.539 0-45 0.90328 40.046 0.91025 36.482 0.91703 33. J 98 .9 2 354 3. l8 o 0.92973 27.414 0-5 1. 00000 40.395 i.ooooo 36.822 i.ooooo 33.524 i.ooooo 30.489 i.ooooo 27.705 0-55 1.10708 40.046 1.09860 36.482 1.09048 33.198 1.08279 30.180 1.07558 27.414 0.6 1.22508 38.983 1.20607 35-453 1. 18810 32.214 1.17128 29.248 1.15566 26.539 0.65 1-35414 37.169 1-32197 33.707 1.29201 30.553 1.26434 27.683 1.23898 25.074 0.7 1-49352 34-544 1.44490 31.204 1.40047 28.190 1.36012 25.471 1.32366 23.017 0-75 1.64064 31.035 1.57193 27.897 1.51047 25.101 1.45568 22.604 1.40695 20.372 0.8 1.79004 26.572 1.69780 23.753 1.61722 21.276 1.54680 19.090 1.48519 17.153 0.85 1.93206 21.122 1.81438 18.773 1.71398 16.739 1.62793 14.966 1.55384 13.409 0.9 2.05254 14.732 1.91081 13.027 1.79243 11.570 1.69266 10.312 1.60788 9.217 o-9S 2.13465 7.582 1.97520 6.680 1.84400 5.917 1.73467 5.263 1.64262 4.696 1.0 2.16395 o.oo 1.99792 o.oo 1.86202 o.oo 1.74926 o.oo 1.65462 o.oo 1.05 2.13465 7.582 1.97520 6.680 1.84400 5.917 1.73467 5.263 1.64262 4.696 i.i 2.05254 14-732 1.91081 13.027 1.79243 11.570 1.69266 10.312 1.60788 9.217 i.iS 1.93206 21.122 1.81438 18.773 1.71398 16.739 1.62793 14.966 1.55384 13.409 1.2 1.79004 26.572 1.69780 23.753 1.61722 21.276 1.54680 19.090 1.48519 17.153 I-2S 1.64064 31.035 1.57193 27-897 1.51047 25.101 1.45568 22.604 1.40695 20.372 i-3 1.49352 34-544 1.44490 31.204 1.40047 28.190 1.36012 25.471 1.32366 23.017 i-3S 1.35414 37-169 1.32197 33-707 1.29201 30.553 1.26434 27.683 1.23898 25.074 1.4 1.22508 38.983 1.20607 35-453 I. 18810 32.214 1.17128 29.248 1.15566 26.539 1-45 1.10708 40.046 1.09860 36.482 1.09048 33.198 1.08279 30.180 1.07558 27.414 i-S i.ooooo 40.395 i.ooooo 36.822 i.ooooo 33.524 i.ooooo 30.489 i.ooooo 27.705 i-SS 0.90328 40.046 0.91025 36.482 0.91703 33-198 0.92354 30.180 0.92973 27.414 1.6 0.81628 38.983 0.82914 35.453 0.84168 32.214 0.85377 29.248 0.86531 26.539 1.65 0.73847 37-i69 0.75645 33.707 0.77399 30.553 0.79092 27.683 0.80711 25.074 i-7 0.66956 34-544 0.69209 31.204 0.71405 28.190 0.73523 25.471 0.75548 23.017 i-7S 0.60952 31-035 0.63616 27.897 0.66205 25.101 0.68696 22.604 0.71076 20.372 1.8 0.55865 26.572 0.58900 23.753 0.61835 21.276 0.64650 19.090 0.67331 17-153 1.85 0.51758 21.122 0.55115 18.773 0.58344 16.739 0.61428 14.966 0.64357 13.409 1.9 0.48720 14.732 0.52334 13.027 0.55790 11.570 0.59079 10.312 0.62194 9.217 1-95 0.46846 7.582 0.50628 6.680 0.54230 5.917 0.57648 5.263 0.60878 4.696 2.0 0.46211 o.oo 0.50052 o.oo 0.53704 o.oo 0.57167 o.oo 0.60437 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.7 + i 0.7) = 1.32366 /23.oi7 = 1.32366 /23.oi > .oi*'. tanh (0.6 + 1.5) = i.oooo \33.524 = i.oooo \33.3i'.26"'. [124] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + ig) = r Ay. CONTINUED x = 0.75 x = 0.85 x = o.g x = 0.95 0.63515 o.oo 0.66403 0.00 0.69107 o.oo 0.71629 o.oo 0.73978 o.oo 0.05 0.63921 4.202 0.66777 3.767 0.69451 3.384 0.71947 3-043 0.74269 2.741 O.I 0.65131 8.257 0.67892 7.411 0.70478 6.662 0.72892 5-995 0.75141 5.401 0.15 0.67125 12.036 0.69730 10.819 0.72172 9-737 0-74453 8.771 0.76578 7-909 O.2 0.69871 15-432 0.72264 13.898 0.74509 12.526 0.76607 11.297 0.78561 10.196 O.2S 0-73333 18.371 0.75461 16.576 0-77459 14.964 0.79326 I3-5I3 0.81065 12.208 0-3 0.77471 20.804 0.79285 18.807 0.80986 17.004 0.82576 15-375 0.84054 13-904 0.35 0.82247 22.707 0.83695 20.559 0.85051 18.613 0.86316 16.848 0.87492 15-250 0.4 0.87623 24.068 0.88650 21.819 0.89611 19-773 0.90504 17.914 0.91333 16.226 0.45 0-93557 24.885 0.94104 22.576 0.94614 20.472 0.95086 18.557 0.95523 16.816 0.5 I.OOOOO 25-157 .00000 22.828 I.OOOOO 20.706 I.OOOOO 18.772 I.OOOOO 17.013 0.55 1.06887 24.885 .06265 22.576 1.05693 20.472 1.05168 18.557 1.04687 16.816 0.6 1.14125 24.068 .12803 21.819 1.11594 19-773 1.10492 17.914 1.09490 16.226 0.65 1.21585 22.707 .19482 20.559 1.17576 18.613 1.15853 16.848 1.14297 15-250 0.7 1.29081 20.804 .26128 18.807 1.23478 17.004 I.2IIOI 15-375 1.18971 13.904 0-75 1.36365 18.371 1-32519 16.576 1.29101 14.964 I.26O62 I3-5I3 1.23358 12.208 0.8 1.43121 15-432 1.38382 13.898 1.34212 12.526 1.30536 11.297 1.27289 10.196 0.85 1.48976 12.036 1.43411 10.819 L38559 9-737 I-343I3 8.771 1.30586 7.909 0.9 1-53537 8.257 1.47293 7.411 1.41889 6.662 I.37I89 5-995 1.33083 5.401 o-9S 1.56444 4.202 I-4975I 3-767 1.43986 3-384 1.38992 3-043 1.34645 2.741 I.O 1-57443 o.oo I.50594 o.oo 1.44703 o.oo 1.39606 o.oo 1.35175 o.oo 1.05 1.56444 4.202 J-4975 1 3.767 1.43986 3.384 1.38992 3.043 1.34645 2.741 i.i 1-53537 8.257 1.47293 7.411 1.41889 6.662 I.37I89 5-995 1.33083 5.401 i-i5 1.48976 12.036 1.43411 10.819 I.38559 9-737 L343I3 8.771 1.30586 7.909 1.2 1.43121 15.432 1.38382 13.898 1.34212 12.526 1.30536 11.297 1.27289 10.196 1-25 1-36365 18.371 1-325*9 16.576 .29101 14.964 1.26062 I3.5U 1-23358 12.208 i-3 1.29081 20.804 1.26128 18.807 23478 17.004 I.2IIOI 15-375 1.18971 13.904 1-35 1-21585 22.707 1.19482 20.559 17576 18.613 I.I5853 16.848 1.14297 15.250 1.4 1.14125 24.068 1.12803 21.819 -II594 19-773 I.I0492 17.914 1.09490 16.226 i -45 1.06887 24.885 1.06265 22.576 05693 20.472 I.05I68 18.557 1.04687 16.816 i-5 I.OOOOO 25.157 I.OOOOO 22.828 I.OOOOO 20.706 I.OOOOO 18.772 I.OOOOO 17.013 1-55 0-93557 24.885 0.94104 22.576 0.94614 20.472 0.95086 iS-557 0.95523 16.816 1.6 0.87623 24.068 0.88650 21.819 0.89611 19-773 0.90504 17.914 0.91333 16.226 1.65 0.82247 22.707 0.83695 20.559 0.85051 18.613 0.86316 16.848 0.87492 15-250 i.7 0.77471 20.804 0.79285 18.807 0.80986 17.004 0.82576 15-375 0.84054 13.904 i-75 0-73333 18.371 0.75461 16.576 0-77459 14.964 0.79326 13-513 0.81065 12.208 1.8 0.69871 15.432 0.72264 13.898 0-74509 12.526 0.76607 11.297 0.78561 10.196 1.85 0.67125 12.036 0.69730 10.819 0.72172 9-737 0.74453 8.771 0.76578 7.909 1.9 0.65131 8.257 0.67892 7.411 0.70478 6.662 0.72892 5-995 0.75141 5.401 i-95 0.63921 4.202 0.66777 3.767 0.69451 3-384 0.71947 3.043 0.74269 2.741 2.0 0.63515 o.oo 0.66403 o.oo 0.69107 o.oo 0.71629 o.oo 0.73978 o.oo Note. Negative quantities are in heavy type. Examples, tanh (0.9 + i i.o) = 1.39606 /o. tanh (0.95 + i 1.55) = 0.95523 \i6.8i6 = 0.95523 \i6.48'.s8'. TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = x = 1.05 * = 1.15 1 0.76159 o.oo 0.78181 o.oo 0.80050 o.oo 0.81775 -oo 0.83365 o.oo 5.05 0.76428 2.470 0.78427 2.228 0.80277 2.OIO 0.81984 1.815 0.83557 1-639 3.1 0.77229 4.870 0.79164 4.394 0.80954 3.966 0.82605 3-582 0.84127 3.235 'IS 0.78552 7.134 0.80380 6.440 0.82071 5.815 0.83630 5.254 0.85067 4.747 X2 0.80376 9.206 0.82058 8.315 0.83611 7.513 0.85043 6.790 0.86361 6.138 5.2S 0.82678 11.032 0.84172 9.971 0.85551 9.015 0.86822 8.151 0.87991 7.371 5-3 0.85425 12.575 0.86693 11.373 0.87862 10.288 0.88939 9.306 0.89927 8.419 5-35 0.88580 13.802 0.89585 12.492 0.90509 11.305 0.91359 10.231 0.92138 9.258 3.4 0.92098 14.694 0.92802 13.305 0.93449 12.045 0.94042 10.904 0.94585 9.870 3-45 0.95925 15.233 0.96294 13.798 0.96632 12.495 0.96941 11.313 0.97223 10.242 3-5 i.ooooo 15.414 i.ooooo 13.963 i.ooooo 12.646 i.ooooo 11.451 i.ooooo 10.368 ^55 1.04248 15.233 1.03849 13.798 1.03486 12.495 1.03155 11.313 1.02854 10.242 3.6 1.08581 14.694 1.07756 13.305 1.07010 12.045 1.06335 10.904 1.05725 9.870 x6 S 1.12892 13.802 1.11626 12.492 1.10486 11.305 1.09458 10.231 1.08533 9-258 3-7 1.17061 12.575 1.15349 11.373 1.13815 10.288 1.12437 9.306 I.II20I 8.419 3-75 1.20951 11.032 1.18804 9.971 1.16889 9- OI 5 1.15179 8.151 1.13649 7.371 D.8 1.24415 9.206 1.21866 8.315 1.19602 7.513 1.17587 6.790 1.15793 6.138 3.85 1-27305 7-134 1.24409 6.440 1.21846 5.815 1.19574 5.254 I-I7555 4-747 D.9 1.29485 4.870 1.26320 4-394 i-235 2 7 3-966 1.21058 3.582 1.18868 3.235 3-95 1.30843 2.470 1.27506 2.228 1.24569 2.OIO 1.21976 1.815 1.19680 1.639 i.o 1.31304 o.oo 1.27008 o.oo 1.24922 o.oo 1.22286 o.oo 1.19954 o.oo i-S 1.30843 2.470 1.27506 2.228 1.24569 2.0IO 1.21976 1.815 1.19680 1.639 i.i 1.29485 4.870 1.26320 4.394 1.23527 3.966 1.21058 3.582 1.18868 3.235 IS 1-27305 7.134 1.24409 6.440 1.21846 5.815 1.19574 5.254 I-I7555 4-747 .2 1.24415 9.206 1.21866 8.315 1.19602 7.513 1.17587 6.790 1.15793 6.138 25 1.20951 11.032 1.18804 9.971 1.16889 9' 01 5 1.15179 8.151 1.13649 7.371 3 1.17061 12.575 1.15349 11.373 1.13815 10.288 1.12437 9.306 I.II20I 8.419 35 1.12892 13.802 1.11626 12.492 1.10486 11.305 1.09458 10.231 I-08533 9.258 4 1.08581 14.694 1.07756 13.305 1.07010 12.045 1.06335 10.904 I.O5725 9.870 45 1.04248 15.233 1.03849 13.798 1.03486 12.495 1.03155 11.313 I.O2854 IO.242 5 i.ooooo 15.414 i.ooooo 13.963 i.ooooo 12.646 i.ooooo 11.451 i.ooooo 10.368 55 0.95925 15.233 0.96294 13.798 0.96632 12.495 0.96941 11.313 0.97223 10.242 .6 0.92098 14.694 0.92802 13.305 0.93449 12.045 0.94042 10.904 0.94585 9.870 65 0.88580 13.802 0.89585 12.492 0.90509 11.305 0.91359 10.231 0.92138 9.258 7 0.85425 12.575 0.86693 "-373 0.87862 10.288 0.88939 9.306 0.89927 8.419 75 0.82678 11.032 0.84172 9.971 0.85551 9.015 0.86822 8.151 0.87991 7.371 .8 0.80376 9.206 0.82058 8.315 0.83611 7.513 0.85043 6.790 0.86361 6.138 85 0.78552 7.134 0.80380 6.440 0.82071 5.815 0.83630 5.254 0.85067 4.747 0.77229 4.870 0.79164 4.394 0.80954 3.966 0.82605 3.582 0.84127 3.235 1-95 0.76428 2.470 0.78427 2.228 O.80277 2.OIO 0.81984 1.815 0.83557 -39 2.0 0.76159 o.oo 0.78181 o.oo 0.80050 o.oo 0.81775 o.oo 0.83365 o.oo Note. Negative quantities are in heavy type. Examples, tanh (i.i + i 0.7) = 1.13815 /io.288 = 1.13815 /io.i7 / .i7*. tatth (1.2 + f 0.89927 \8.4i9 = 0.89927 \8.2s'.o8*. [126] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r CONTINUED * = 1.25 * = 1.3 * = i-35 x = 1.4 x = 1-45 9 f t r y f y f 7 r 7 o o o o 0.84828 o.oo 0.86172 o.oo 0.87405 o.oo 0.88535 o.oo 0.89569 o.oo o.os 0.85004 1.481 0.86333 1.338 0.87552 1. 210 0.88669 I-O94 0.89692 0.989 O.I 0.85526 2.924 0.86811 2.643 0.87990 2.389 0.89069 2.161 0.90057 1.953 0.15 0.86387 4.291 0-87599 3-879 O.887II 3.508 0.89728 3.172 0.90659 2.869 O.2 0.87573 5-549 0.88684 5-017 0.89702 4.538 0.90634 4.104 0.91485 3.712 0.25 0.89063 6.666 0.90047 6.029 0.90947 5.454 0.91769 4.934 0.92521 4.463 0-3 0.90833 7.616 0.91663 6.890 O.9242I 6.234 0.93114 5.640 0.93746 5.103 0-35 0.92852 8.378 0.93504 7.581 0.94099 6.86O 0.94642 6.207 0-95*37 5-6i7 0.4 0.95082 8.933 0.95534 8.085 0-95947 7-3I7 0.96323 6.622 0.96665 5.993 0-45 0.97481 9.272 0.97715 8.393 0.97928 7.596 0.98122 6.875 0.98298 6.222 o-S i.ooooo 9.385 i.ooooo 8.496 i.ooooo 7.689 i.ooooo 6.960 i.ooooo 6.299 o-SS 1.02584 9.272 1.02338 8.393 1.02116 7.596 1.01914 6.875 I.OI733 6.222 0.6 1.05173 8.933 1.04674 8.085 1.04224 7.317 1.03817 6.622 1.03450 5.993 0.65 1.07699 8.378 1.06948 7.581 1.06271 6.860 1.05661 6.207 1.05112 5.617 0.7 1.10092 7.616 1.09096 6.890 1.08200 6.234 1.07395 5-640 1.06671 5.103 0-75 1.12280 6.666 1.11054 6.029 1.09955 5-454 1.08963 4.934 1.08084 4-463 0.8 1.14192 5.549 1.12760 5.017 1.11483 4.538 1.10335 4.104 1.09308 3.712 0.85 1.15758 4-291 1.14156 3.879 1.12726 3.508 1.11448 3.172 1.10304 2.869 0.9 1.16924 2.924 1.15193 2.643 1.13650 2.389 1.12272 2.161 1.11041 1.953 0-95 1.17642 1.481 1.15831 1.338 I.I42I8 1. 210 1.12779 1-094 1.11493 0.989 I.O .17885 o.oo 1.16047 o.oo 1.14410 o.oo 1.12950 o.oo .11646 o.oo 1.05 .17642 1.481 1.15831 1.338 I.I42I8 I.2IO 1.12779 1.094 .11493 0.989 .1 .16924 2.924 1.15193 2.643 I.I3650 2.389 1.12272 2.161 .11041 1.953 IS .15758 4.291 1.14156 3.879 I.I2726 3.508 1.11448 3.172 .10304 2.869 .2 .14192 5.549 1.12760 5.017 I.II483 4.538 1.10335 4.104 .09308 3.712 25 .12280 6.666 1.11054 6.029 1.09955 5-454 1.08963 4.934 1.08084 4.463 3 .10092 7.616 1.09096 6.890 1.08200 6.234 1.07395 5.640 1.06671 5.103 35 .07699 8.378 1.06948 7.581 I.O627I 6.86O 1.05661 6.207 1.05112 5.617 1.4 1.05173 8.933 1.04674 8.085 1.04224 7.317 1.03817 6.622 1.03450 5.993 i-45 1.02584 9.272 1.02338 8.393 1.02116 7.596 1.01914 6.875 I.OI733 6.222 -5 i.ooooo 9.385 i.ooooo 8.496 i.ooooo 7.689 i.ooooo 6.960 i.ooooo 6.299 55 0.97481 9.272 0.97715 8.393 0.97928 7.596 0.98122 6.875 0.98298 6.222 .6 0.95082 8.933 0-95534 8.085 0.95947 7.317 0.96323 6.622 0.96665 5-993 65 0.92852 8.378 0.93504 7-58i 0.94099 6.860 0.94642 6.207 0.95137 5-617 -7 0.90833 7-616 0.91663 6.890 0.92421 6.234 0.93114 5.640 0.93746 5.103 -75 0.89063 6.666 0.90047 6.029 0.90947 5-454 0.91769 4.934 0.92521 4-463 .8 0-87573 5-549 0.88684 5.017 0.89702 4.538 0.90634 4.104 0.91485 3.712 85 0.86387 4.291 0-87599 3.879 0.88711 3.508 0.89728 3.172 0.90659 2.869 9 0.85526 2.924 0.868 1 r 2.643 0.87990 2.389 0.89069 2.161 0.90057 1.953 95 0.85004 1.481 0.86333 1.338 0.87552 I.2IO 0.88669 1.094 0.89692 0.989 2.0 0.84828 o.oo 0.86172 o.oo 0.87405 o.oo 0.88535 o-oo 0.89569 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.25 + to. 25) = 0.89063 /6.666 = 0.89063 /6.3g '58'. tanh (1.25 + i 1.25) = 1.12280 \6.666 = 1. 12280 \6.39'.58*. [127] TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r /y. CONTINUED *= i-S x = i-5S x 1.6 x = 1.65 x = 1.7 q r 7 r y r 7 r 7 r 7 o o o o o 0.90515 o.oo 0.91379 o.oo 0.92167 o.oo 0.92886 o.oo 0.93541 o.oo 0.05 0.90627 0.894 0.91481 0.809 0.92260 0.732 0.92970 O.662 0.93618 0.599 O.I 0.90960 1.767 0.91785 1.598 0.92537 1.446 0.93223 1.308 0.93848 1.183 0.15 0.91509 2.595 0.92285 2.347 0.92993 2.123 0.93638 I.92I 0.94226 1.738 0.2 0.92263 3.357 0.92972 3.038 0.93619 2.748 0.94207 2.486 0.94744 2.249 0.2S 0.93207 4-038 0.93832 3-653 0.94401 3.305 O.9492O 2.990 0.95392 2.705 o-3 0.94323 4.617 0.94847 4.177 0.95325 3-780 0.95760 3.420 0.96155 3.095 o-35 0.95588 5.083 0.95998 4.599 0.96371 4.161 0.96710 3.766 0.97018 3.407 0.4 0.96976 5.423 0.97259 4.908 0.97516 4.440 0.97749 4.019 0.97961 3.636 0-45 0.98458 5.631 0.98603 5.096 0-98735 4-6n 0.98854 4.173 0.98962 3.776 o-S i.ooooo 5.700 i.ooooo 5.159 i.ooooo 4.668 i.ooooo 4.225 i.ooooo 3.822 o-5S 1.01566 5.631 1.01417 5.096 1.01281 4.611 1.01159 4-173 1.01049 3-776 0.6 1.03118 5.423 1.02818 4.908 1.02548 4.440 1.02303 4.019 1.02082 3.636 0.65 1.04616 5.083 1.04169 4.599 1.03766 4.161 1.03402 3.766 1.03074 3.407 0.7 1.06019 4.617 1-05433 4-177 1.04904 3.780 1.04428 3.420 1-03999 3-095 0-75 1.07289 4.038 1.06574 3.653 1-05931 3-305 1-05353 2.990 1.04831 2.705 0.8 1.08386 3.357 1-07559 3-038 1.06817 2.748 1.06149 2.486 1.05548 2.249 0.85 1.09279 2.595 1.08360 2.347 1-07535 2.123 1.06795 1-921 1.06128 1.738 0.9 1.09938 1.767 1.08951 1.598 1.08065 1.446 1.07270 1.308 1.06556 1.183 o-95 1.10343 0.894 1.09313 0.809 1.08390 0.732 1.07562 0.662 1.06817 0.599 I.O 1.10479 o.oo 1.09436 o.oo 1.08500 o.oo 1.07659 o.oo 1.06906 o.oo 1.05 1.10343 0.894 1.09313 0.809 1.08390 0.732 1.07562 0.662 1.06817 0.599 i.i 1.09938 1.767 1.08951 1.598 1.08065 1.446 1.07270 1.308 1.06556 1.183 i-iS 1.09279 2.595 1.08360 2.347 1-07535 2.123 1.06795 1-921 1.06128 1.738 1.2 1.08386 3.357 1-07559 3.038 1.06817 2.748 1.06149 2.486 1.05548 2.249 1.25 1.07289 4.038 1.06574 3-653 1.05931 3.305 1.05353 2.990 1.04831 2.705 i-3 1.06019 4.617 1.05433 4.177 1.04904 3.780 1.04428 3.420 1.03999 3-095 1-35 1.04616 5.083 1.04169 4.599 1.03766 4.161 1.03402 3.766 1.03074 3.407 1.4 1.03118 5.423 1.02818 4.908 1.02548 4.440 1.02303 4.019 1.02082 3.636 I-4S 1.01566 5.631 1.01417 5.096 1.01281 4.611 1.01159 4.173 1.01049 3.776 i-S i.ooooo 5.700 i.ooooo 5.159 i.ooooo 4.668 i.ooooo 4.225 i.ooooo 3.822 1-55 0.98458 5.631 0.98603 5.096 0.98735 4.611 0.98854 4.173 0.98962 3.776 1.6 0.96976 5-423 0.97259 4.908 0.97516 4.440 0.97749 4.019 0.97961 3.636 1.65 0.95588 5.083 0.95998 4.599 0.96371 4.161 0.96710 3.766 0.97018 3.407 i-7 0.94323 4.617 0.94847 4.177 0.95325 3.780 0.95760 3.420 0.96155 3.095 i-75 0.93207 4.038 0.93832 3.653 0.94401 3.305 0.94920 2.990 0.95392 2.705 1.8 0.92263 3-357 0.92972 3.038 0.93619 2.748 0.94207 2.486 0.94744 2.249 1.85 0.91509 2.595 0.92285 2.347 0.92993 2.123 0.93638 1.921 0.94226 1.738 1.9 0.90960 1.767 0.91785 1.598 0.92537 1.446 0.93223 1.308 0.93848 1.183 1-95 0.90627 0.894 0.91481 0.809 0.92260 0.732 0.92970 0.662 0.93618 0.599 2.0 0.90515 o.oo 0.91379 o.oo 0.92167 o.oo 0.92886 o.oo 0.93541 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.7 + *' Q-7) = 1-03999 /3-95 = 1-03999 /3-5 > -42 y . tanh (1.6 + u) = 0.95325 \3.78o = 0.95325 \346'.48*. [128] TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r fa. CONTINUED = 1.75 x = 1.8 * = 1.85 * = 1.9 x = 1.95 9 r y r y r y r y r 7 o o o o o o 0.94138 o.oo 0.94681 o.oo 0.95175 o.oo 0.95624 o.oo 0.9603 2 O.OO 0.05 0.94208 0.542 0.94745 0.490 0.95232 0.443 0.95677 0.401 0.96080 0.363 O.I 0.94417 1.070 0.94935 0.968 0.95406 0.876 0.95834 0.793 O.96222 O.7I7 0.15 0.94761 1.572 0.95247 1.422 0.95690 1.287 0.96092 1.165 0.96457 1.053 O.2 0.95232 2.035 0.95676 1.841 0.96079 1.666 0.96445 1.507 0.96778 1.364 0.25 0.95821 2.448 0.96211 2.215 0.96565 2.004 0.96886 1.813 0.97178 1.640 -3 0.96514 2.800 0.96840 2.533 0.97136 2.292 0.97405 2.074 0.97649 1.877 o-3S 0.97297 3.083 0.97551 2.789 0.97781 2.524 0.97990 2.285 0.98179 2.067 0.4 0.98153 3.291 0.98327 2.977 0.98485 2.694 0.98628 2.438 0.98757 2.2O6 o-4S 0.99061 3.417 0.99149 3.092 0.99230 2.798 0.99303 2.531 0.99369 2.291 o-S i. ooooo 3.459 i.ooooo 3.130 i.ooooo 2.833 i.ooooo 2.562 i.ooooo 2.319 o-5S 1.00948 3.417 1.00858 3.092 1.00776 2.798 1.00702 2.531 1.00635 2.291 0.6 1.01882 3.291 1.01702 2.977 1.01539 2-694 1.01392 2.438 1.01258 2.206 0.65 1.02778 3.083 1.02511 2.789 1.02269 2.524 1.02051 2.285 1.01854 2.067 0.7 1.03612 2.800 1.03263 2.533 1.02948 2.292 1.02664 2.074 1.02408 1.877 o.7S 1.04362 2.448 1.03939 2.215 1.03558 2.004 1.03214 1.813 1.02904 1.640 0.8 1.05007 2.035 1.04520 1.841 1.04081 1.666 1.03686 1.507 1.03329 1.364 0.85 1.05529 1.572 1.04990 1.422 1.04504 1.287 1.04067 1.165 1.03673 1.053 0.9 1.05913 1.070 1.05336 0.968 1.04816 0.876 1.04347 0.793 1.03926 0.717 0-95 1.06148 0.542 1.05547 0.490 1.05006 0.443 1.04519 0.401 1.04080 0.363 I.O 1.06228 o.oo 1.05619 o.oo 1.05070 o.oo 1.04576 o.oo 1.04131 o.oo 1.05 1.06148 0.542 1.05547 0.490 1.05006 0.443 1.04519 0.401 1.04080 0.363 i.i 1.05913 1.070 1.05336 0.968 1.04816 0.876 1.04347 0.793 1.03926 0.717 i-iS 1.05529 1.572 1.04990 1.422 1.04504 1.287 1.04067 1.165 1.03673 1.053 1.2 1.05007 2.035 1.04520 1.841 1.04081 1.666 1.03686 1.507 1.03329 1.364 1.25 1.04362 2.448 I -3939 2.215 1.03558 2.004 1.03214 1.813 1.02904 1.640 1-3 1.03612 2.800 1.03263 2.533 1.02948 2.292 1.02664 2.074 1.02408 1.877 i-3S 1.02778 3.083 1.02511 2.789 1.02269 2.524 1.02051 2.285 1.01854 2.067 1.4 1.01882 3.291 1.01702 2.977 1.01539 2.694 1.01392 2.438 1.01258 2.206 1-45 1.00948 3.417 1.00858 3.092 1.00776 2.798 1.00702 2.531 1.00635 3 - 2 9i i-5 i.ooooo 3.459 i.ooooo 3.130 i.ooooo 2.833 i.ooooo 2.562 i.ooooo 2.319 i-S5 0.99061 3.417 0.99149 3-092 0.99230 2.798 0.99303 2.531 0.99369 2.291 1.6 0.98153 3.291 0.98327 2.977 0.98485 2.694 0.98628 2.438 0.98757 2.206 1.65 0.97297 3.083 0.97551 2.789 0.97781 2.524 0.97990 2.285 0.98179 2.067 1-7 0.96514 2.800 0.96840 2.533 0.97136 2.292 0.97405 2.074 0.97649 1.877 J-75 0.95821 2.448 0.96211 2.215 0.96565 2.004 0.96886 1.813 0.97178 1.640 1.8 0.95232 2.035 0.95676 1.841 0.96079 1.666 0.96445 1.507 0.96778 1.364 1.85 0.94761 1.572 0.95247 1.422 0.95690 1.287 0.96092 1.165 0.96457 1.053 1.9 0.94417 1.070 0.94935 0.968 0.95406 0.876 0.95834 0.793 0.96222 0.717 1-95 0.94208 0.542 0.94745 0.490 0.95232 0.443 0.95677 0.401 0.96080 0.363 2.O 0.94138 o.oo 0.94681 o.oo o-9S 1 75 - 0.95624 o.oo 0.96032 o.oo Note. Negative quantities are in heavy type. Examples, tanh (1.9 + i 0.05) = 0.95677 /o.40i = 0.95677 /o.24' ,04'. tanh (i 95 + i . i.ooo\2.i9'.o8*. [129] TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) - f CONTINUED x = 2.05 * = 2.15 o.os O.I 0.15 O.2 0.96403 o.oo 0.96446 0.3 28 0.96576 0.649 0.96789 0.953 0.97080 1.234 0.96740 o.oo 0.96779 0.297 0.96897 0.587 0.97090 0.862 0.97354 1.117 0.97045 o.oo 0.97081 0.268 0.97188 0.531 0.97363 0.780 0.97603 i.oio 0.97323 o.oo 0.97355 0.243 0.97452 0.481 0.97611 0.706 0.97829 0.914 0.97574 o.oo 0.97604 O.22O 0.97692 0.435 0.97842 0.639 0.98033 0.827 0.25 0-3 0-35 0.4 0-45 0.97443 1.484 0.97870 1.698 0.98351 1.870 0.98875 1.996 0.99429 2.072 0.97684 1.343 0.98071 1.537 0.98507 1.692 0.98981 1. 806 0.99483 1.876 0.97902 1.215 0.98253 1.390 0.98648 1.531 0.99078 1.634 0.99532 1.698 0.98100 1.099 0.98418 1.258 0.98776 1.385 0.99165 1.479 0.99577 1-536 0.98279 0.995 0.98567 1.138 0.98892 1.253 0.99244 1.337 0.99617 1.389 0-5 0-55 0.6 0.65 0.7 i.ooooo 2.098 1.00574 2.072 1.01138 1.996 1.01676 1.870 1.02176 1.698 i.ooooo 1.899 1.00520 1.876 1.01029 1. 806 1.01516 1.692 1.01967 1.537 i.ooooo 1.718 1.00470 1.698 1.00931 1.634 1.01371 1.531 1.01778 1.390 i.ooooo 1.555 1.00425 1.536 1.00842 1.479 1.01240 1.385 1.01608 1.258 i.ooooo 1.406 1.00385 1.389 1.00762 1.337 I.OII2I 1.253 1.01454 1.138 0-75 0.8 0.85 0.9 0-95 1.02624 1.484 1.03008 1.234 1.03318 0.953 1.03545 0.649 1.03685 0.328 1.02371 1.343 1.02718 1.117 1.02998 0.862 1.03203 0.587 1.03328 0.297 1.02143 1-215 1.02456 i.oio 1.02708 0.780 1.02894 0.531 1.03007 0.268 1.01937 1.099 I.O222O 0.914 1.02447 0.706 1.02614 0.481 1.02717 0.243 1.01751 0.995 1.02006 0.827 1.02206 0.639 1.02363 0.435 I.O2455 O.22O I.O 1.05 i.i 1.2 1.03731 o.oo 1.03685 0.328 1.03545 0.649 1.03318 0.953 1.03008 1.234 1.03370 o.oo 1.03328 0.297 1.03202 0.587 1.02998 0.862 1.02718 1.117 1.03045 o.oo 1.03007 0.268 1.02894 0.531 1.02708 0.780 1.02456 i.oio 1.02751 o.oo 1.02717 0.243 1.02614 0.481 1.02447 0.706 I.O222O O.9I4 1.02486 o.oo 1.02455 0.220 1.02363 0.435 1.02206 0.639 1.02006 0.827 1.25 1-35 1.4 1-45 1.02624 1.484 1.02176 1.698 1.01676 1.870 1.01138 1.996 1.00574 2.072 1.02371 1.343 1.01967 1.537 1.01516 1.692 1.01029 1-806 1.00520 1.876 1.02143 1-215 1.01778 1.390 1.01371 1.531 1.00931 1.634 1.00470 1.698 1.01937 1.099 1.01608 1.258 1.01240 1.385 1.00842 1.479 1.00425 1.536 1.01751 0.995 1.01454 1.138 I.OII2I 1.253 1.00762 1.337 1.00385 1.389 i-S i-SS 1.6 1.65 i-7 i.ooooo 2.098 0.99429 2.072 0.98875 1.996 0.98351 1.870 0.97870 1.698 i.ooooo 1.899 0.99483 1.876 0.98981 1. 806 0.98507 1.692 0.98071 1.537 i.ooooo 1.718 0.99532 1.698 0.99078 1.634 0.98648 1.531 0.98253 1.390 i.ooooo 1.555 0.99577 1-536 0.99165 1.479 0.98776 1.385 0.98418 1.258 i.ooooo 1.406 0.99617 1.389 0.99244 1.337 0.98892 1.253 0.98567 1.138 i.'s 1.85 1.9 1-95 0.97443 1.484 0.97080 1.234 0.96789 0.953 0.96576 0.649 0.96446 0.328 0.97684 1.343 0.97354 1. 117 0.97090 0.862 0.96897 0.587 0.96779 0.297 0.97902 1.215 0.97603 i.oio 0.97363 0.780 0.97188 0.531 0.97081 0.268 0.98100 1.099 0.97829 0.914 0.97611 0.706 0.97452 0.481 0.97355 0.243 0.98279 0.995 0.98033 0.827 0.97842 0.639 0.97692 0.435 0.97604 O.22O 2.0 0.96403 o.oo 0.96740 o.oo 0.97045 o.oo 0.97323 o.oo 0.97574 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.2 + 40) =0.97574 /o. tanh (2.2 + i 1.95) = 0.97604 \o.22o = 0.97604 \o.i3'.i2*. [130] TABLE XII. HYPERBOLIC TANGENTS, tanh (* + iq) = r /y. CONTINUED x = 2.25 x = 2.3 X = 2.35 x = 2.4 x = 2.45 1 r y * y r y r y r y o o o o o 0.97803 o.oo 0.98010 o.oo 0.98197 o.oo 0.98367 o.oo 0.98522 o.oo 0.05 0.97829 0.199 0.98034 0.180 0.98219 0.163 0.98388 0.147 0.98540 0.134 O.I 0.97909 0.394 0.98106 0.356 0.98285 O.322 0.98447 0.291 0.98594 0.264 0.15 0.98040 0.578 0.98225 0.523 0.98393 0.473 0.98544 0.428 0.98682 0.388 0.2 0.98219 0.748 0.98387 0.677 0.98539 0.6l3 0.98677 0.554 0.98802 0.501 0.25 0.98441 0.900 0.98589 0.814 0.98722 0-737 0.98843 0.666 0.98953 0.603 0-3 0.98703 1.030 0.98825 0.932 0.98937 0.843 0.99037 0.763 0.99129 0.690 o-3S 0.98997 1.134 0.99092 1.027 0.99178 0.929 0.99256 0.841 0.99326 0.760 0-4 0.99316 1. 211 0.99381 1.095 0.99440 0.991 0.99493 0.896 0.99541 0.812 o.4S 0.99653 1.257 0.99686 1.138 0.99716 I.O3O 0.99743 0.931 0.99767 0.843 0-5 i.ooooo 1.273 i.ooooo 1.152 i.ooooo 1.042 i.ooooo 0.942 i.ooooo 0.853 o-5S 1.00348 1.257 1.00315 1.138 1.00285 1-030 1.00258 0.931 1.00233 0-843 0.6 1.00689 I-2II 1.00623 I -95 1.00564 0.991 1.00510 0.896 1.00461 0.812 0.65 I.OIOI4 I.I34 1.00917 1.027 1.00829 0.929 1.00750 0.841 1.00678 0.760 0.7 I.OI3I4 1.030 1.01189 0.932 1.01075 0.843 1.00972 0.763 1.00879 0.690 0-75 1.01583 O.gOO 1.01431 0.814 1.01295 0.737 1.01171 0.666 1.01058 0.603 0.8 I.Ol8l4 0.748 1.01640 0.677 1.01482 0.613 1.01340 0.554 I.OI2I2 O.5OI 0.85 I.OI999 0.578 1.01807 0.523 1.01634 0.473 1.01477 0.428 1.01336 0.388 0.9 I.02I36 0.394 1.01930 0.356 1.01745 -3 22 1.01578 0.291 I.OI426 0.264 0-95 I.O22I9 0.199 1.02005 0.180 1.01813 0.163 1.01639 0.147 I.OI482 0.134 I.O I.O2247 O-O 1.02031 o.oo 1.01836 o.oo 1.01659 o.oo 1.01500 o.oo 1.05 I.O22I9 O.IQ9 1.02005 0.180 1.01813 0.163 1.01639 0.147 1.01482 0.134 i.i I.O2I36 O.394 1.01930 0.356 1.01745 0.322 1.01578 0.291 1.01426 0.264 i-iS 1.01999 O.578 1.01807 0.523 1.01634 0.473 1.01477 0.428 1.01336 0.388 1.2 I.Ol8l4 0.748 1.01640 0.677 1.01482 0.613 1.01340 0.554 1 .01 21 2 O.5OZ 25 1.01583 O.90O 1.01431 0.814 1.01295 0.737 1.01171 0.666 I.OIO58 O.6O3 3 I.OI3I4 I.03O 1.01189 0.932 1.01075 0.843 1.00972 0.763 1.00879 0.60X) 35 I.OIOI4 I-I34 1.00917 .027 1.00829 0.929 1.00750 0.841 1.00678 0.760 4 1.00689 I -2 1 1 1.00623 .095 1.00564 0.991 1.00510 0.896 1.00461 0.812 45 J.00348 1.257 1.00315 .138 1.00285 1.030 1.00258 0.931 1.00233 0.843 5 i.ooooo 1.273 i.ooooo .152 i.ooooo 1.042 i.ooooo 0.942 i.ooooo 0.853 55 0.99653 1.257 0.99686 .138 0.99716 1.030 0.09743 -93i 0.99767 0.843 1.6 0.99316 Z.2II 0.99381 .095 0.99440 0.991 0.99493 0.896 0.99541 0.812 1.65 0.98997 1. 134 0.99092 1.027 0.99178 0.929 0.99256 0.841 0.99326 0.760 i-7 0.98703 1.030 0.98825 0.932 0.98937 0.843 0.99037 0.763 0.99129 0.690 1-75 0.98441 O.9OO 0.98589 0.814 0.98722 0.737 0.98843 0.666 0.98953 0.603 1.8 0.98219 0.748 0.98387 0.677 0.98539 0.613 0.98677 0.554 0.98802 0.501 1.85 0.98040 O.578 0.98225 0.523 0.98393 0.473 0.98544 0.428 0.98682 0.388 1.9 0.97909 0.394 0.98106 0.356 0.98285 0.322 0.98447 0.291 0.98594 0.164 1-95 0.97829 0.199 0.98034 0.180 0.98219 0.163 0.98388 0.147 0.98540 0.134 *.o 0.97803 o.oo 0.98010 o.oo 0.98197 o.oo 0.98367 o.oo 0.98522 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.45 + oj) = 1.00879 /o-69Q = 1.00879 /o.4i f -24'- tanh (2.45 + * rj) = 0.99129 \o.6oo = 0.99129 \o.4i'-24 r . [131] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r fa. CONTINUED * = 2.5 x = 2.55 X = 2.6 * = 2.65 x = 2.7 9 r y f 7 r 7 r 7 f Y o o o 0.98661 o.oo 0.98788 o.oo 0.98903 o.oo 0.99007 o.oo 0.99101 o.oo 0.05 0.98678 O.I2I 0.98803 0.109 0.98916 0.099 0.99019 0.089 0.99112 0.081 O.I 0.98727 0.239 0.98847 0.216 0.98956 0.195 0.99055 0.177 0.99145 0.160 0.15 0.98806 0.350 0.98919 0.317 0.99022 0.287 0.99114 0.260 0.99198 0.235 O.2 0.98916 0.454 0.99018 0.411 0.99111 0.372 0.99196 0.336 0.99272 0.304 0.2$ 0.99052 0.546 0.99142 0.494 0.99223 0.447 0.99297 0.405 0-99363 0-366 0-3 O.992II O.625 0.99286 0.565 0.99354 0.512 0.99415 0.463 0.99471 0.419 0-35 0.99390 0.688 0.99448 0.623 0.99500 0.563 0.99548 0.510 0.99591 0.461 0.4 0.99584 0.734 0.99624 0.665 0.99660 0.601 0.99692 0.544 0.99721 0.492 o-4S 0.99789 0.763 0.99809 0.690 0.99828 0.624 0.99844 0.565 0.99859 0.511 o.S i.ooooo 0.772 i.ooooo 0.699 i.ooooo 0.632 i.ooooo 0.573 i.ooooo 0.518 0-55 I.OO2II 0.763 1.00191 0.690 1.00173 0.624 1.00156 0.565 1.00141 0.511 0.6 I.004I7 0.734 1.00377 0.665 1.00342 0.601 1.00309 0.544 1.00280 0.492 0.65 1.00614 0.688 1.00555 0-623 1.00502 0.563 1.00454 0.510 1.00411 0.461 0.7 1.00795 0.625 1.00719 0.565 1.00651 0.512 1.00589 0.463 1.00532 0.419 0-75 1.00958 0.546 1.00866 0.494 1.00783 0.447 1.00708 0.405 1.00641 0.366 0.8 1.01096 0.454 1.00991 0.411 1.00897 0.372 1.00811 0.336 1.00733 0.304 0.85 1.01208 0.350 1.01092 0.317 1.00988 0.287 1.00894 0.260 1.00808 0.235 0.9 1.01290 0.239 1.01166 0.216 1.01055 0.195 1.00954 0.177 1.00863 o.i 60 o-9S 1.01340 o.i 21 I.OI2I2 O.IO9 1.01096 0.099 1.00991 0.089 1.00896 0.081 I.O 1.01357 0.00 1.01227 o.oo I. OHIO O.OO 1.01003 O'O 1.00907 o.oo i. 05 I.OI340 O.I2I I.OI2I2 O.IO9 1.01096 0.099 1.00991 0.089 1.00896 0.081 i.i 1.01290 0.239 1.01166 0.216 1.01055 0-195 1.00954 0.177 1.00863 o.i 60 i.iS 1.01208 0.350 I.OIO92 O.3I7 1.00988 0.287 1.00894 0.260 1.00808 0.235 1.2 1.01096 0.454 I.OO99I O.4II 1.00897 0.372 1. 0081 1 0.336 1.00733 0.304 I.2S 1.00958 0.546 I.OO866 0.494 1.00783 0.447 1.00708 0.405 1.00641 0.366 1-3 1.00795 0.625 I.007I9 0.565 1.00651 0.512 1.00589 0.463 1.00532 0.419 1-35 1.00614 0.688 !-oo555 0.623 1.00502 0.563 1.00454 0.510 1.00411 0.461 1.4 1.00417 0.734 1.00377 0.665 1.00342 0.601 1.00309 0.544 1.00280 0.492 i-45 I.OO2II 0.763 1.00191 0.690 1.00173 0.624 1.00156 0.565 1.00141 0.511 i-5 i.ooooo 0.772 i.ooooo 0.699 i.ooooo 0.632 i.ooooo 0.573 i.ooooo 0.518 i-SS 0.99789 0.763 0.99809 0.690 0.99828 0.624 0.99844 0.565 0.99859 0.511 1.6 0.99584 0.734 0.99624 0.665 0.99660 0.601 0.99692 0.544 0.99721 0.492 1.65 0.99390 0.688 0.99448 0.623 0.99500 0.563 0.99548 0.510 0.99591 0.461 1-7 0.99211 0.625 0.99286 0.565 0.99354 0.512 0.99415 0.463 0.99471 0.419 1-75 0.99052 0.546 0.99142 0.494 0.99223 0.447 0.99297 0.405 0.99363 0.366 1.8 0.98916 0.454 0.99018 0.411 0.99111 0.372 0.99196 0.336 0.99272 0.304 1.85 0.98806 0.350 0.98919 0.317 0.99022 0.287 0.99114 0.260 0.99198 0.235 1.9 0.98727 0.239 0.98847 0.216 0.98956 0.195 0.99055 0.177 0.99145 0.160 i-95 0.98678 O.I2I 0.98803 0.109 0.98916 0.099 0.99019 0.089 0.99112 0.081 2.0 0.98661 o.oo 0.98788 o.oo 0.98903 o.oo 0.99007 o.oo 0.99101 o.oo Note. Negative quantities are in heavy type. Examples, tanh (2.5 + * 0.25) = 0^99052 /o.546 = 0.99052 / o . tanh (2.5 + * 1.75) = 0.99052 \o.546 = 0.99052 \o.32'.46*. TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) - r^y. CONTINUED = 2.75 = 2.8 * = 2.85 x = 2.9 x = 2.95 o 0.05 O.I 0.15 O.2 0.99186 o.oo 0.99196 0.073 0.99226 0.145 0.99275 0.213 0.99341 0.275 0.99263 o.oo 0.99272 0.066 0.99299 0.131 0.99343 0.192. 0.99404 0.249 Q-99333 o.oo 0.99341 0.060 0.99366 0.119 0.99406 0.174 0.99460 0.226 0.99396 o.oo 0.99404 0.054 0.99426 0.107 0.99462 0.158 0.99511 0.204 0.99454 o.oo 0.99460 0.049 0.99480 0.097 0.99513 0.143 0-99558 0.184 O.2S -3 0.35 0.4 0.45 0.99424 0.331 0.99521 0.379 0.99630 0.417 0.99748 0.446 0.99872 0.463 0.99478 0.300 0.99566 0.343 0.99665 0.378 0.99772 0.403 0.99884 0.419 0.99528 0.271 0.99607 0.310 0.99697 0.342 0.99793 0.365 0.99895 0.379 0-99573 0.245 0.99645 0.281 0.99726 0.309 0.99813 0.330 0.99905 0.343 0.99613 O.222 0.99679 0.254 0.99752 O.28O 0.99831 0.299 0.99914 0.310 o-5 o.SS 0.6 0.65 0.7 i .00000 0.468 1.00128 0.463 1.00253 -446 1.00372 0.417 1.00482 0.379 i .00000 0.424 1.00116 0.419 1.00229 -43 1.00336 0.378 1.00436 0.343 i .00000 0.383 1.00105 0.379 1.00207 0.365 1.00304 0.342 1.00394 0.310 i .00000 0.347 1.00095 0.343 1.00187 0.330 1.00275 0.309 1.00356 0.281 i. ooooo 0.313 1.00086 0.310 1.00169 0.209 1.00249 0.280 1.00323 0.254 0-75 0.8 0.85 0.9 o-95 1.00580 0.331 1.00664 0.275 1.00731 0.213 1.00780 0.145 1.00810 0.073 1.00524 0.300 1.00600 0.249 i .0066 1 0.192 1.00706 0.131 1.00733 0.066 1.00474 0.271 1.00543 0.226 1.00598 0.174 1.00638 0.119 1.00663 0.060 1.00429 0.245 1.00491 0.204 1.00541 0.158 1.00578 0.107 1.00600 0.054 1.00388 O.222 1.00444 0.184 1.00489 O.I43 I.OO522 O.O97 1.00543 0.049 I.O 1.05 I.I 1-15 1.2 1.00821 o.oo 1.00810 0.073 1.00780 0.145 1.00731 0.213 1.00664 0.275 1.00742 o.oo 1.00733 0.066 1.00706 0.131 1.00661 0.192 1.00600 0.249 1.00671 o.oo 1.00663 0.060 1.00638 0.119 1.00598 0.174 1.00543 0.226 1.00607 o.oo 1.00600 0.054 1.00578 0.107 1.00541 0.158 1.00491 0.204 1.00549 o.oo 1.00543 0.049 I.OO522 O.O97 1.00489 O.I43 I.OO444 0.184 1.25 i-3 1-35 1.4 1-45 1.00580 0.331 1.00482 0.379 1.00372 0.417 1.00253 0-446 1.00128 0.463 1.00524 0.300 1.00436 0.343 1.00336 0.378 1.00229 0.403 1.00116 0.419 1.00474 0.271 1.00394 0.310 1.00304 0.342 1.00207 0.365 1.00105 0.379 1.00429 0.245 1.00356 0.281 1.00275 0.309 1.00187 0.330 1.00095 0.343 1.00388 O.222 1.00323 0.254 I.OO249 O.28O I.OOl69 O.299 1.00086 0.310 5 55 .6 -65 7 1 .00000 0.468 0.99872 0.463 0.99748 0.446 0.99630 0.417 0.99521 0.379 i .00000 0.424 0.99884 0.419 0.99772 0.403 0.99665 0.378 0.99566 0.343 1 .00000 0.383 0.99895 0.379 0.99793 0.365 0.99697 0.342 0.99607 0.310 1 .00000 0.347 0.99905 0.343 0.99813 0.330 0.99726 0.309 0.99645 0.281 i .ooooo 0.313 0.99914 0.310 0.99831 0.299 0.99752 0.280 0.99679 0.254 75 .8 .85 9 95 0.99424 0.331 0.99341 0.275 0.99275 0.213 0.99226 0.145 0.99196 0.073 0.99478 0.300 0.99404 0.249 0.99343 0.192 0.99299 0.131 0.99272 0.066 0.99528 0.271 0.99460 0.226 0.99406 0.174 0.99366 0.119 0.99341 0.060 0-99573 0.245 0.99511 0.204 0.99462 0.158 0.99426 0.107 0.99404 0.054 0.99613 0.222 0.99558 0.184 0.99513 0.143 0.99480 O.O97 0.99460 O.O49 2.0 0.99186 o.oo 0.99263 o.oo 0-99333 o-o 0.99396 o.oo 0.99454 0.00 Note. Negative quantities are in heavy type. Examples, tanh (2.9 + i 0.5) = i.oooo/o.347 = i.oooo /o.26'.49 ir . tanh (2.95 + * 1.75) = 0.99613 \o.222 = 0.99613 \o.i3'.i9'. TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r CONTINUED X = . 5.0 x = 3 05 * = . J-i * = 3 15 * = . 5-2 9 r 7 r y r 7 r 7 f 7 o o o o 0-9950S o.oo 0.99552 O.OO 0-99595 O.OO 0.99633 O.OO 0.99668 O.OO 0.05 0.99511 0.044 0.99558 O.O4O 0.99600 0.037 0.99638 0.033 0.99672 0.030 O.I 0.99530 0.088 0-99575 0.079 0.99615 O.O72 0.99651 0.065 0.99685 0.059 0.15 0-99559 0.129 0.99601 O.II7 0.99639 0.106 0.99673 0.096 0.99704 0.086 O.2 0.99600 0.167 0.99638 O.ISI 0.99672 0.137 0.99703 0.124 0.99732 0.112 0.25 0.99650 O.2OI 0.99683 0.182 0.99714 0.165 0.99741 0.149 0.99765 0.135 o-3 0.99709 0.229 0-99737 O.2O8 0.99762 o.i 88 0.99784 0.170 0.99805 0.154 o-3S 0-99775 0.253 0.99797 O.229 0.99816 0.207 0.99833 o.i 88 0.99849 0.170 0.4 0.99847 O.27O 0.99862 0.244 0.99875 O.22I 0.99887 O.2OO 0.99897 0.181 0-45 o.999 2 3 0.281 0.99930 0.254 0-99937 0.230 0-99943 O.2O8 0.99948 o.i 88 o-5 I.OOOOO 0.284 I.OOOOO 0.257 I.OOOOO 0.233 I.OOOOO O.2II I.OOOOO 0.191 0-55 1.00078 0.28l 1.00070 0.254 1.00064 0.230 1.00058 O.2O8 1.00052 o.i 88 0.6 1.00153 0.27O 1.00139 0.244 1.00125 0.221 1.00114 O.2OO 1.00103 0.181 0.65 1.00225 0.253 1.00204 O.229 1.00185 O.2O7 1.00167 o.i 88 1.00151 0.170 0.7 1.00292 O.229 I 00264 0.208 1.00239 o.i 88 1.00216 0.170 1.00196 0.154 0-75 1.00351 O.2OI 1.00318 0.182 1.00287 0.165 1.00260 0.149 1.00235 0.135 0.8 1.00402 0.167 1.00363 O.I5I 1.00329 0.137 1.00297 0.124 1.00269 O.I 1 2 0.85 1.00443 O.I29 1.00400 O.II7 1.00360 o.i 06 1.00328 0.096 1.00297 0.086 0.9 1.00473 0.088 1.00427 0.079 1.00387 0.072 1.00350 0.065 1.00316 0.059 o-9S 1.00491 O.O44 1.00444 O.O4O 1.00399 0.037 1.00363 0.033 1.00329 0.030 .0 1.00497 O.OO 1.00450 O.OO 1.00407 O.OO 1.00368 O.OO 1.00333 O.OO 05 1.00491 0.044 1 .00444 0.040 1.00399 0.037 1.00363 0.033 1.00329 0.030 .1 1.00473 0.088 1.00427 0.079 1.00387 0.072 1.00350 0.065 1.00316 0.059 15 1.00443 0.129 1.00400 0.117 1.00360 o.i 06 1.00328 0.096 1.00297 0.086 .2 1.00402 0.167 1.00363 0.151 1.00329 0.137 1.00297 0.124 1.00269 O.II2 25 1.00351 0.201 1.00318 0.182 1.00287 0.165 1.00260 0.149 1.00235 0.135 3 1.00292 O.229 1.00264 0.208 1.00239 0.188 1.00216 0.170 1.00196 0.154 35 1.00225 0.253 1.00204 0.229 1.00185 0.207 1.00167 0.188 1.00151 O.I7O 4 1.00153 O.27O 1.00139 0.244 1.00125 O.22I 1.00114 O.2OO 1.00103 O.lSl 1-45 1.00078 0.281 1.00070 0.254 1.00064 O.23O 1.00058 O.2O8 1.00052 o.i 88 i-5 I.OOOOO 0.284 I.OOOOO 0.257 I.OOOOO 0.233 I.OOOOO O.2II I.OOOOO 0.191 i-55 0.99923 0.28l 0.99930 0.254 0-99937 0.230 0.99943 O.2OS 0.99948 0.188 1.6 0.99847 O.27O 0.99862 0.244 0.99875 O.22I 0.99887 O.200 0.99897 0.181 1.65 0.99775 0.253 0.99797 0.229 0.99816 O.2O7 0.99833 0.188 0.99849 0.170 i-7 0.99709 O.220 0.99737 0.208 0.99762 0.188 0.99784 0.170 0.99805 o.i54 1-75 0.99650 O.2OI 0.99683 0.182 0.99714 0.165 0.99741 0.149 0.99765 0-I3S 1.8 0.99600 O.l67 0.99638 0.151 0.99672 0.137 0.99703 0.124 0.99732 O.II2 1.85 0.99559 O.I29 0.99601 0.117 0.99639 0.106 0.99673 0.096 0.99704 0.086 1.9 0.99530 0.088 0-99575 0.079 0.99615 0.072 0.99651 0.065 0.99685 0.059 1-95 0.99511 0.044 0.99558 0.040 0.99600 0.037 0.99638 0.033 0.99672 O.O3O 2.0 0.99505 O.OO 0.99552 O.OO 0-99595 O.OO 0.99633 O.OO 0.99668 O.OO Note. Negative quantities are in heavy type. Examples, tanh (3.2 + 10) = 0.99668 /o. tanh (3.2 + i 1.05) = 1.00329 \o.o3o = 1.00329 \o.i'.48". [134] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r % CONTINUED = 3-25 = 3-35 * = 3-4 * = 3-45 o 0.99700 o.oo 0.99728 o.oo 0.99754 o.oo 0.99777 o.oo 0.99799 o.oo 0.05 0.99703 0.027 0.99732 0.024 0.99757 0.022 0.99780 O.O2O 0.99801 0.018 O.I 0.99714 0.053 0.99741 0.048 0.99766 0.044 0.99788 0.039 0.99809 0.036 0.15 0.99732 0.078 0.99758 0.071 0.99781 0.064 0.99802 0.058 0.99821 0.052 O.2 0-99757 O.IOI 0.99780 0.092 0.99801 0.083 0.99820 0.075 0.99837 0.068 0.25 0.99788 0.122 0.99807 o.iio 0.99826 0.100 0.99843 0.090 0.99858 0.082 0-3 0.99823 0.139 0.99840 0.126 0.99855 0.114 0.99869 0.103 0.99882 0.094 0-35 0.99864 0.153 0.98877 0.139 0.99888 0.126 0.99899 0.114 0.99909 0.103 0.4 0.99907 0.164 0.99916 0.148 0.99924 0.134 0.99931 O.I 21 0.99938 o.iio 0-45 0-99953 0.170 0.99958 0.154 0.99962 0.139 0.99965 0.126 0.99968 0.114 0-5 i.ooooo 0.172 i.ooooo 0.156 i.ooooo 0.141 i.ooooo 0.128 i.ooooo 0.116 0-55 1.00047 0.170 1.00042 0.154 1.00039 0.139 1.00035 0.126 1.00032 0.114 0.6 1.00093 0.164 1.00084 0.148 1.00076 0.134 1.00069 O.I 21 1.00062 o.iio 0.65 1.00137 0.153 1.00124 - I 39 I.OOII2 0.126 i.ooioi 0.114 1.00092 0.103 0.7 1.00177 0.139 1.00160 0.126 1.00145 0.114 1.00131 0.103 1.00118 0.094 0-75 I.OO2I3 O.I22 1.00193 o.iio 1.00174 0.100 1.00158 0.090 1.00143 0.082 0.8 1.00244 o.ioi I.OO22O 0.092 1.00199 0.083 1.00180 0.075 1.00163 0.068 0.85 1.00268 0.078 1.00243 0.071 I.OO22O 0.064 1.00199 0.058 1.00180 0.052 0.9 I.O0286 0.053 1.00259 0.048 I.OO234 O.O44 I.OO2I2 0.039 1.00192 0.036 0-95 I.O0297 O.O27 1.00269 0.024 I.OO243 O.O22 I.OO22O O.O2O 1.00199 0.018 I.O 1.00301 o.oo I.OO273 'O 1.00246 o.oo 1.00223 o.oo 1.00202 O.OO 1.05 I.OO297 0.027 1.00269 0.024 I.OO243 O.O22 I.OO22O O.O2O 1.00199 0.018 .1 1.00286 0.053 1.00259 0.048 I.OO234 O.O44 I.OO2I2 0.039 1.00192 0.036 15 I.OO268 O.O78 1.00243 0.071 I.OO22O O.OO4 I.OOI99 O.O58 1.00180 0.052 .2 1.00244 o.ioi 1.00220 O.O92 I.OOI99 0.083 I.OOlSo O.O75 1.00163 .0.068 25 I.OO2I3 O.I22 1.00193 o.iio 1.00174 o.ioo I.OOI58 O.O9O 1.00143 0.082 3 I.OOI77 O.I39 1.00160 0.126 I.OOI45 O.II4 I.OOI3I O.IO3 1.00118 0.094 35 I.OOI37 0.153 1.00124 0.139 I.OOII2 O.I26 i.opioi 0.114 1.00092 0.103 1.4 I.OOO93 0.164 1.00084 0.148 1.00076 O.I34 I.OOO69 O.I2I 1.00062 o.iio i-45 I.OOO47 O.I7O 1.00042 0.154 1.00039 - I 39 I.OOO35 O.I26 1.00032 0.114 5 i.ooooo 0.172 i.ooooo 0.156 i.ooooo 0.141 i.ooooo 0.128 i.ooooo 0.116 55 0.99953 0.170 0.99958 0.154 0.99962 0.139 0.99965 O.I26 0.99968 0.114 .6 0.99907 0.164 0.99916 0.148 0.99924 0.134 0.99931 O.I2I 0.99938 o.iio .65 0.99864 0.153 0.99877 0.139 0.99888 0.126 0.99899 O.II4 0.99909 0.103 7 0.99823 0.139 0.99840 0.126 0.99855 0.114 0.99869 0.103 0.99882 0.094 75 0.99788 O.I22 0.99807 o.iio 0.99826 o.ioo 0.99843 0.090 0.99858 0.082 .8 0.99757 o.ioi 0.99780 0.092 0.99801 0.083 0.99820 O.O75 0.99837 0.068 .85 0.99732 O.O78 0.99758 0.071 0.99781 0.064 0.99802 O.O58 0.99821 0.052 9 0.99714 0.053 0.99741 0.048 0.99766 0.044 0.99788 O.O39 0.99809 0.036 95 0.99703 0.027 0.99732 0.024 0-99757 0-022 0.99780 0.020 0.99801 0.018 2.0 0.99700 o.oo 0.99728 o.oo 0.99754 o.oo 0.99777 o.oo 0.99799 o.oo Note. Negative quantities are in heavy type. Examples, tanh (3.4 + * 0.7) = 1.00131 /o.io3 = 1.00131 /o .o6'.ii*. TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r /r CONTINUED = 3-5 x = 3-55 * = 3-6 x = 3-6S = 3-7 o 0.99818 o.oo 0.99835 o.oo 0.99851 o.oo 0.99865 o.oo 0.99878 o.oo 0.05 0.99820 0.016 0.99837 0.015 0.99853 0.013 0.99867 O.OI2 0.99879 o.on O.I 0.99827 0.032 0.99843 0.029 0.99858 0.026 0.99872 0.024 0.99884 O.O22 0.15 0.99838 0.047 0.99853 0.043 0.99867 0.039 0.99880 0.035 0.99891 0.032 O.2 0.99853 0.06 1 0.99867 0.055 0.99879 0.050 0.99891 0.046 0.99901 0.041 0.25 0.99871 0.074 0.99883 0.067 0.98894 0.061 0.99905 0.055 0.99914 0.050 0-3 0.99893 0.084 0.99903 0.076 0.99912 0.069 0.99921 0.063 0.99928 0.057 0-35 0.99917 0.093 0.99925 0.084 0.99932 0.076 0.99939 0.069 0.99945 0.063 0.4 0.99944 0.099 0.99949 0.090 0.99954 0.081 0.99958 0.074 0.99962 0.067 0-45 0.99972 0.103 0.99974 0.093 0.99977 0.085 0.99979 0.076 0.99981 0.069 0-S i.ooooo 0.104 i.ooooo 0.095 i.ooooo 0.086 i.ooooo 0.077 i.ooooo 0.070 0-55 1.00028 0.103 1.00026 0.093 1.00023 0.085 I.OOO2I 0.076 1.00019 0.069 0.6 1.00056 0.099 1.00051 0.090 1.00046 0.081 1.00042 0.074 1.00038 0.067 0.65 1.00083 o '93 1.00075 0.084 .00068 0.076 1. 0006 1 0.069 1.00056 0.063 0.7 1.00107 0.084 1.00097 0.076 .00088 0.069 1.00079 0.063 1.00072 0.057 0-75 1.00129 0.074 1.00117 0.067 .00106 0.061 1.00096 0.055 1.00087 0.050 0.8 1.00148 0.061 1.00134 0.056 .OOI 2 1 O.O5O 1.00109 0.046 1.00099 0.041 0.85 1.00163 0.048 1.00147 0.043 .00133 0.039 I.OOI2O O.035 1.00109 0.032 0.9 1.00174 0.032 1.00157 0.029 .00142 0.026 1.00129 0.024 I.OOII7 O.O22 o-9S 1.00180 0.016 1.00163 0.015 1.00148 0.013 I.OOI34 O.OI2 I.OOI2I O.OII I.O 1.00183 o 'o 1.00165 'o 1.00149 o.oo 1.00135 o.oo I.OOI22 O.OO 1.05 1.00180 0.016 1.00163 0.015 1.00148 0.013 I.OOI34 O.OI2 I.OOI2I O.OII i.i 1.00174 0.032 1.00157 0.029 1.00142 0.026 1.00129 0.024 I.OOII7 O.O22 i-iS 1.00163 0.048 1.00147 0.043 1.00133 -30 I.OOI2O O.O35 I.OOI09 O.O32 1.2 1.00148 0.061 1.00134 0.056 I.OOI2I O.O5O 1.00109 0.046 I.OOO99 O.O4I 1-25 1.00129 0.074 1.00117 0.067 1.00106 0.061 1.00096 0.055 I.OOO87 O.O5O i-3 1.00107 0.084 1.00097 0.076 1.00088 0.069 1.00079 0.063 I.OOO72 O.O57 i-35 1.00083 0.093 1.00075 0.084 1.00068 0.076 1. 0006 1 0.069 1.00056 O.O63 1.4 1.00056 0.099 1.00051 0.090 1.00046 0.08 1 1.00042 0.074 1.00038 O.067 I- 45 1.00028 0.103 1.00026 0.093 1.00023 0.085 I.OOO2I 0.076 I.OOOI9 O.069 i-5 i.ooooo 0.104 i.ooooo 0.095 i.ooooo 0.086 i.ooooo 0.077 i.ooooo 0.070 i-55 0.99972 0.103 0.99974 0.093 0.99977 0.085 0.99979 0.076 0.99981 0.069 1.6 0.99944 0.099 0.99949 0.090 0.99954 0.081 0.99958 0.074 0.99962 0.067 1-65 0.99917 0.093 0.99925 0.084 0.99932 0.076 0.99939 0.069 0.99945 0.063 i-7 0.99893 0.084 0.99903 0.076 0.99912 0.069 0.99921 0.063 0.99928 0.057 i-7S 0.99871 0.074 0.99883 0.067 0.99894 0.061 0.99905 0.055 0.99914 0.050 1.8 -99 8 53 0.061 0.99867 0.055 0.99879 0.050 0.99891 0.046 0.99901 0.041 1.85 0.99838 0.047 0.99853 0.043 0.99867 0.039 0.99880 0.035 0.99891 0.032 1.9 0.99827 0.032 0.99843 0.029 0.99858 0.026 0.99872 0.024 0.99884 O.O22 i-9S 0.99820 0.016 0.99837 0.015 0.99853 0.013 0.99867 0.012 0.99879 o.oii 2.0 0.99818 o.oo 0.99835 o.oo 0.99851 o.oo 0.99865 o.oo 0.99878 o.oo Note. Negative quantities are in heavy type. Examples, tanh (3.6 -f io) = 0.99851 /o tanh (3.7 + i 1.7) = 0.99928 \o.o57 = 0.99928 \o.o3'.25". [136] TABLE XII. HYPERBOLIC TANGENTS, tanh (x + iq) = r /y. CONTINUED x = . 5-75 x = 3-8 x = ; [-85 x = 3-9 * = : 95 ? r 7 r 7 r 7 r 7 r 7 o o o o 0.99889 o.oo 0.99900 O.OO 0.99909 O.OO 0.99918 0.00 0.99926 O.OO 0.05 0.99891 O.OIO 0.99901 0.009 0.99911 0.008 0.99919 0.007 0.99927 0.007 O.I 0.99895 O.O2O 0.99905 0.018 0.99914 0.016 0.99922 0.015 0-99930 0.013 0.15 0.99902 0.029 0.99911 0.026 0.99919 0.024 0.99927 O.O2I 0-99934 0.019 0.2 0.99911 0.037 0.99919 0.034 0.99927 0.031 0-99934 O.O28 0.99940 0.025 0.25 0.99922 0.045 0.99929 0.041 0.99936 0.037 0.99942 0.033 0.99948 0.030 o-3 0-99935 0.051 0.99941 0.046 0-99947 0.042 0.99952 0.038 0.99956 0.034 0.35 0.99950 0.056 0-99955 0.051 0-99959 0.046 0.99963 O.O42 0.99966 0.038 0.4 0.99966 0.060 0.99969 0.055 0.99972 0.049 0-99975 0.045 0.99977 0.040 0.45 0.99983 0.063 0.99984 0.057 0.99986 0.051 0.99987 0.046 0.99988 0.042 o-5 I.OOOOO 0.063 I.OOOOO 0.057 I.OOOOO 0.052 I.OOOOO 0.047 I.OOOOO 0.043 0-55 1.00017 0.063 1.00016 0.057 1.00014 0.051 1.00013 0.046 I.OOOI2 0.042 0.6 1.00034 0.060 1.00031 0.055 1.00028 0.049 1.00025 0.045 1.00023 0.040 0.65 1.00050 0.056 1.00045 0.051 1.00041 0.046 1.00037 O.042 1.00034 0.038 0.7 1.00065 0.051 1.00059 0.046 1.00053 0.042 1.00048 0.038 1.00044 0.034 o.7S 1.00078 0.045 1.00071 0.041 1.00064 0.037 1.00058 0.033 1.00052 0.030 0.8 1.00089 0.037 1.00081 0.034 1.00073 0.031 1.00066 O.O28 1.00060 0.025 0.85 1.00099 0.029 1.00089 0.026 1.00081 0.024 1.00073 O.O2I 1.00066 0.019 0.9 1.00105 O.O2O 1.00095 0.018 1.00086 0.016 1.00078 O.OI5 1.00071 0.013 0-95 1.00109 O.OIO 1.00099 0.009 1.00089 0.008 1.00081 O.OO7 1.00073 0.007 I.O I.OOIII O.OO I.OOIOO O.OO 1.00090 O.OO .00082 0.00 1.00074 0.00 1.05 1.00109 O.OIO 1.00099 0.009 1.00089 0.008 .00081 0.007 1.00073 0.007 i.i 1.00105 O.O2O 1.00095 0.018 i. '00086 0.016 .00078 0.015 1.00071 0.013 i.'S 1.00099 O.029 1.00089 0.026 1.00081 0.024 .00073 O.O2I 1.00066 0.019 1.2 1.00089 0.037 1.00081 0.034 1.00073 0.031 .00066 O.O28 1.00060 0.025 1-25 1.00078 0.045 1.00071 0.041 1.00064 0.037 1.00058 0-033 .00052 0.030 1-3 1.00065 O.O5I 1.00059 0.046 1.00053 0.042 1.00048 0.038 .00044 0.034 1-35 1.00050 O.O56 1.00045 0.051 1.00041 0.046 1.00037 O.042 .00034 0.038 1.4 1.00034 O.06O 1.00031 0.055 1.00028 0.049 1.00025 0.045 .00023 0.040 i -45 1.00017 0.063 1.00016 0.057 1.00014 0.051 1.00013 0.046 .00012 0.042 i-5 I.OOOOO 0.063 I.OOOOO 0.057 I.OOOOO 0.052 I.OOOOO 0.047 I.OOOOO 0.043 i-SS 0.99983 O.O63 0.99984 0.057 0.99986 0.051 0.99987 0.046 0.99988 0.042 1.6 0.99966 O.06O 0.99969 0.055 0.99972 0.049 0-99975 0.045 0-99977 0.040 1.65 0.99950 0.056 0-99955 0.051 0-99959 0.046 0.99963 O.O42 0.99966 0.038 i-7 0-99935 0.051 0.99941 0.646 0-99947 0.042 0.99952 O.O38 0.99956 0.034 1-75 0.99922 0.045 0.99929 0.041 0.99936 0.037 0.99942 0.033 0.99948 0.030 1.8 0.99911 0.037 0.99919 0.034 0.99927 0.031 0-99934 0.028 0.99940 0.025 1.85 0.99902 O.O29 0.99911 0.026 0.99919 0.024 0.99927 O.O2I 0-99934 0.019 1.9 0.99895 O.O2O 0.99905 0.018 0.99914 0.016 0.99922 O.OI5 0.99930 0.013 i -95 0.99891 O.OIO 0.99901 0.009 0.99911 0.008 0.99919 O.OO7 0.99927 0.007 2.O 0.99889 O.OO 0.99900 0.00 0.99909 O.OO 0.99918 O.OO 0.99926 O.OO Note. Negative quantities are in heavy type. Examples, tanh (3.95 + 0.9) = 1.00071 /o.oij tanh (3.95 + i 1.9) TABLE XIII. FUNCTIONS OF 4 + ig. f (4 + ?) = + sinh cosh tanh 9 u v U V u v 3 27.28992 o.oo 27.30823 o.oo 0.99933 o.oo 3.05 27.20579 2.14258 27.22405 2.14114 0.99934 o.oooio 3.1 26.95392 4-27195 26.97202 4.26908 0.99936 O.OOO2I 3-IS 26.53588 6.37498 26.55370 6.37071 0.99940 0.00030 D.2 25-95425 8.43871 25.97166 8.43305 0.99946 0.00039 3.2S 25.21260 10.45041 25.22951 10.44340 0.99953 0.00047 3-3 24.31551 12.39768 24.33181 12.38935 0.99961 0.00054 5-35 23.26848 14.26851 23.28410 14.25895 0.99970 0.00060 3.4 22.07800 16.05138 22.09282 16.04061 0.99979 0.00064 5-45 20.75141 17-73528 20.76534 17.72339 0.99989 0.00066 3-5 19.29688 19.30983 19.30983 19.29688 i.ooooo 0.00067 3-55 I7-72339 20.76534 17.73528 20.75141 i.ooon 0.00066 3.6 16.04061 22.09282 16.05138 22.07800 I.OOO2I O.00064 3.65 14.25895 23.28410 14.26851 23.26848 i .0003 o o . 00060 3-7 12.38935 24.33181 12.39768 24.31551 1.00039 0.00054 3.75 10.44340 25.22951 10.45041 25.21260 1.00047 0.00047 3.8 8.43305 25.97166 8.43871 25.95425 1.00054 0.00039 3.85 6.37071 26.55370 6.37498 26.53588 1.00060 0.00030 3.9 4.26908 26.97202 4.27195 26.95392 I.OOO64 O.OOO2I 3-95 2.14114 27.22405 2.14258 27.20579 1.00066 o.ooon I.O o.oo 27.30823 o.oo 27.28992 1.00067 o.oo 05 2.14114 27.22405 2.14258 27.20579 1.00066 o.ooon .1 4.26908 26.97202 4.27195 26.95392 I.OOO64 O.0002I 15 6.37071 26.55370 6.37498 26.53588 1.00060 0.00030 .2 8.43305 25.97166 8.43871 25.95425 1.00054 0.00039 25 10.44340 25.22951 10.45041 25.21260 1.00047 0.00047 3 12.38935 24.33181 12.39768 24.31551 1.00039 0.00054 35 14.25805 23.28410 14.26851 23.26848 1.00030 0.00060 4 16.04061 22.09282 16.05138 22.07800 I.OOO2I O.OOO64 45 17.72339 20.76534 17.73528 20.75141 i.ooon 0.00066 5 19.29688 19.30983 19.30983 19.29688 i.ooooo 0.00067 55 20.75141 17.73528 20.76534 17-72339 0.99989 0.00066 .6 22.07800 16.05138 22.09282 16.04061 0.99979 0.00064 65 23.26848 14.26851 23.28410 14.25895 0.99970 0.00060 7 24.31551 12.39768 24.33181 12.38935 0.99961 0.00054 75 25.21260 10.45041 25.22951 10.44340 0-99953 0.00047 .8 25-95425 8.43871 25.97166 8.43305 0.99946 0.00039 .85 26.53588 6.37498 26.55370 6.37071 0.99940 0.00030 9 26.95392 4-27195 26.97202 4.26908 0.99936 O.O002I 95 27.20579 2.14258 27.22405 2.14114 0.99934 o.oooio 2.0 27.28992 0.00 27.30823 o.oo 0-99933 o.oo Note. Examples. Negative quantities are in heavy type. sinh (4 + i 0.7) = cosh (4 + *' i. 25) 12.38935 + * 24.33181. * 10.45041 + i 25.21260. [138] TABLE XIII. FUNCTIONS OF 4 + iq. / (4 + iq) = r sin' l cos h tan) 1 9 f 7 r 7 r T o o o o 27.28992 o.oo 27.30823 o.oo 0-99933 o.oo 0.05 27.29002 4-503 27.30810 4-497 0.99934 0.006 O.I 27.29036 9.006 27.30780 8.994 0.99936 O.OI2 0.15 27.29090 13-509 27.30723 I3-49I 0.99940 O.OlS O.2 27.29166 18.011 27.30650 17.989 0.99946 0.023 0.25 27.29260 22.514 27-3055 22.486 0-99953 O.027 -3 27.29370 27.016 27.30445 26.984 0.99961 0.031 0-35 27.29492 31-517 27.30324 31-483 0.99970 0.034 0.4 27.29624 36.018 27.30190 35-982 0.99979 0.037 0-4S 27.29764 40.519 27.30050 40.481 0.99990 0.038 o-5 27.29908 45.019 27.29908 44.981 I.OOOOO 0.038 0-55 27.30050 49-519 27.29764 49.481 I.OOOIO 0.038 0.6 27.30190 54.018 27.29624 53-982 I.0002I 0-037 0.65 27.30324 58.517 27.29492 58-483 1.00030 0.034 0.7 27.30445 63.016 27.29370 62.984 1.00039 0.031 o.7S 27-30550 67.514 27.29260 67.486 1.00047 O.O27 0.8 27.30650 72.011 27.29166 71.989 1.00054 0.023 0.85 27.30723 76.509 27.29090 76.491 1.00060 O.OlS 0.9 27.30780 81.006 27.29036 80.994 1.00064 O.OI2 o.QS 27.30810 85-503 27.29002 85-497 1.00066 O.OO6 1.0 27.30823 90 27.28992 90 1.00067 0.00 1.05 27.30810 94-497 27.29002 94-503 1.00066 0.006 i.i 27.30780 98.994 27.29036 99.006 1.00064 0.013 i-i5 27.30723 103.491 27.29090 103.509 1.00060 O.OlS 1.2 27.30650 107.989 27.29166 108.011 1.00054 O.O23 I.2S 27-30550 112.486 27.29260 112.514 1.00047 O.027 i-3 27-30445 116.984 27.29370 117.016 1.00039 0.031 I-3S 27.30324 121.483 27.29492 121.517 1.00030 0.034 1.4 27.30190 125.982 27.29624 126.018 I.O002I 0.037 1-45 27-30050 130.481 27.29764 130-519 I.OOOIO 0.038 5 27.29908 134.981 27.29908 I35- 01 9 I.OOOOO 0.038 55 27.29764 139.481 27.30050 139-519 0.99990 0.038 .6 27.29624 143.982 27.30190 144.018 0.99979 0.037 .65 27.29492 148.483 27.30324 148.517 0.99970 0.034 7 27.29370 152.984 27.30445 153.016 0.99961 O.O3I 75 27.29260 157.486 27-30550 I57-5I4 0-99953 O.O27 .8 27.29166 161.989 27.30650 162.011 0.99946 O.O23 1.85 27.29000 166.491 27.30723 166.509 0.99940 O.OlS 1.9 27.29036 170.994 27.30780 171.006 0.99936 O.OI2 1-95 27.29002 175-497 27.30810 I75-503 0-99934 O.006 2.0 27.28992 180 27.30823 180 0-99933 - Note. Negative quantities are in heavy type. Examples, sinh (4 + * i.o) = 27.30823 /go . tanh (4 + * 1.5) = i.oooo \o.038 = i.oooo \o.o2'.i7* T'ABLE XIV. SEMI-EXPONENTIALS, and logw e* . e* x - log - e* . e x x - log.o - e* . e* x - lo glo - 4.00 27.299 1.4361479 4.01 27.573 1.4404909 4.02 27.851 1.4448338 4.03 28.130 1.4491768 4.04 28.413 1.4S3SI97 4.50 45-009 1.6532952 4.51 45.461 1.6576381 4.52 45.918 1.6619811 4-53 46.379 1.6663240 4.54 46.845 1.6706669 5.00 74-207 1.8704424 5.01 74.952 1.8747854 5.02 75.706 1.8791283 5.03 76.467 1.8834712 5.04 77.235 1.8878142 4.05 28.699 1.4578627 4.06 28.987 1.4622056 4.07 29.278 1.4665485 4.08 29.573 1.4708915 4.09 29.870 1.4752344 4-55 47-3 1( 5 1.6750099 4.56 47-792 1.6793528 4.57 48.272 1.6836958 4.58 48.757 1.6880387 4.59 49.247 1.6923817 5.05 78.011 1.8921571 5.06 78.795 1.8965001 5-07 79-587 1.9008430 5.08 80.387 1.9051860 5.09 81.195 1.9095289 4.10 30.170 1.4795774 4.11 30.473 1.4839203 4.12 30.780 1.4882633 4.13 31.089 1.4926062 4.14 31.401 1.4969492 4.60 49.742 1.6967246 4.61 50.242 1.7010676 4.62 50.747 1.7054105 4.63 51.257 1.7097535 4.64 51.772 1.7140964 5.10 82.011 1.9138719 5.11 82.835 1.9182148 5.12 83.668 1.9225577 5.13 84.509 1.9269007 5.14 85.358 1.9312436 4.15 31.717 1.5012921 4.16 32.036 1.5056350 4.17 32.358 1.5099780 4.18 32.683 1.5143209 4.19 33.011 1.5186639 4.65 52.292 1.7184393 4.66 52.818 1.7227823 4-67 53-349 1.7271252 4.68 53.885 1.7314682 4.69 54-427 1.7358m 5.15 86.216 1.9355866 5.16 87.082 1.9399295 5.17 87.957 1.9442725 5.18 88.841 1.9486154 5.19 89.734 1.9529584 4-20 33.343 1-5230068 4.21 33.678 1.5273498 4.22 34.017 1.5316927 4-23 34-359 1-5360357 4.24 34.704 1.5403786 4.70 54-974 I-740IS4I 4.71 55-526 1.7444970 4.72 56.084 1.7488400 4.73 56.648 1.7531829 4.74 57.217 1.7575258 5.20 90.636 1.9573013 5.21 91.547 1.9616443 5.22 92.467 1.9659872 5-23 93.396 1.9703301 5.24 94.335 I-974673I 4-25 35-053 I.54472I5 4.26 35.405 1.5490645 4-27 35.76i 1.5534074 4.28 36.120 1.5577504 4.29 36.483 1.5620933 4-75 57-792 1.7618688 4-76 58.373 1.7662117 4.77 58.960 1.7705547 4.78 59.552 1.7748976 4.79 60.151 1.7792406 5.25 95.283 1.9790160 5.26 96.241 1.9833590 5.27 97.208 1.9877019 5.28 98.185 1.9920449 5.29 99.172 1.9963878 4.30 36.850 1.5664363 4.31 37.220 1.5707792 4-32 37-594 1.5751222 4-33 37-972 1.5794651 4-34 38.354 1.5838081 4.80 60.755 1-7835835 4.81 61.366 1.7879265 4.82 61.983 1.7922694 4.83 62.605 1.7966123 4.84 63.235 1.8009553 5.30 100.168 2.0007308 5.31 101.175 2.0050737 5.32' 102.192 2.0094166 5.33 103.219 2.0137596 5.34 104.256 2.0181025 4-35 38.739 1.588151 4.36 39.129 1.5924939 4-37 39-522 1.5968369 4-38 39-919 1.6011798 4-39 40-320 1.6055228 4.85 63.870 1.8052982 4.86 64.512 1.8096412 4.87 65.160 1.8139841 4.88 65.815 1.8183271 4.89 66.477 1-8226700 5-35 105.304 2.0224455 5.36 106.362 2.0267884 5.37 107.431 2.0311314 5.38 108.511 2.0354743 5.39 109.602 2.0398173 4.40 40.725 1.6098657 4.41 41.135 1.6142087 4.42 41-548 1.6185516 4.43 41.966 1.6228946 4-44 42.387 1.6272375 4.90 67.145 1.8270130 4.91 67.820 1.8313559 4.92 68.501 1.8356989 4.93 69.190 1.8400418 4.94 69.885 1.8443847 5.40 110.703 2.0441602 5.41 111.816 2.0485031 5.42 112.940 2.0528461 5.43 114-075 2.0571890 5.44 115.221 2.0615320 4.45 42.813 1.6315804 4.46 43.244 1.6359234 4.47 43-678 1.6402663 4.48 44.117 1.6446093 4.49 44-561 1.6489522 4.95 70.587 1.8487277 4.96 71.297 1.8530706 4.97 72.013 1.8574136 4.98 72.737 1.8617565 4.99 73.468 1.8660995 5.45 116.379 2.0658749 5.46 117-549 2.0702179 5.47 118.730 2.0745608 5.48 119.923 2.0789038 5.49 121.129 2.0832467 4.50 45-009 1.6532952 5.00 74.207 1.8704424 5.50 122.346 2.0875897 e 4 '* /c 4 ' 20 \ Example. = 33.343 logio \) = 1.5230068. [140] TABLE XIV. SEMI-EXPONENTIALS. - and lo glo ( - ). CONTINUED * 7 lo g"> 7 e x . e* x Iog 10 - 2 2 e* . e x x - lo glo - 5.50 122.346 2.0875897 5.51 123.576 2.0919326 5.52 124.818 2.0962755 5.53 126.072 2.1006185 5-54 127.339 2.1049614 6.00 201.714 2.3047369 6.01 203.742 2.3090785 6.02 205.789 2.3134222 6.03 207.858 2.3177667 6.04 209.947 2.3221098 6 -5 332-57I 2.5218844 6-51 335.9I3 2.5262268 6.52 339-289 2.5305699 6.53 342.699 2.5349128 6.54 346.143 2.5392556 5.55 128.619 2.1093044 5.56 129.911 2.1136473 5.57 131.217 2.1179903 5.58 132.536 2.1223332 5.59 133.868 2.1266762 6.05 212.057 2.3264527 6.06 214.188 2.3307951 6.07 216.340 2.3351368 6.08 218.514 2.3394793 6.09 220.711 2.3438220 6-55 349-622 2.5435988 6-56 353.135 2.5479408 6-57 356-685 2.5522849 6.58 360.270 2.5566281 6-59 363-890 2.5609701 5.60 135-213 2.1310191 5.61 136.572 2.1353620 5.62 137-945 2.1397050 5.63 139.331 2.1440479 5.64 140.731 2.1483909 6.10 222.929 2.3481666 6. it 225.169 2.3525086 6.12 227.432 2.3568516 6.13 229.718 2.3611951 6.14 232.027 2.3655386 6.60 367.547 2.5653129 6.61 371.241 2.5696560 6.62 374.973 2.5740000 6.63 378.741 2.5783424 6.64 382.547 2.5826849 5.65 142.146 2.1527338 5.66 143-574 2.1570768 5.67 i45- OI 7 2.1614197 5.68 146.475 2.1657627 5.69 147.947 2.1701056 6.15 234.359 2.3698817 6.16 236.714 2.3742240 6.17 239.093 2.3785669 6.18 241.496 2.3829100 6.19 243.923 2.3872527 6.65 386.391 2.5870270 6.66 390.275 2.5913708 6.67 394.197 2.5957134 6.68 398.160 2.6000576 6.69 402.161 2.6044000 5.70 149-434 2.1744485 5.71 150.936 2.1787915 5.72 152.452 2.1831344 5-73 I53-985 2.1874774 5-74 I55-532 2.1918203 6.20 246.375 2.3915966 6.21 248.851 2.3959394 6.22 251.352 2.4OO2824 6.23 253.877 2.4046234 6.24 256.429 2.4089672 6.70 406.202 2.6087420 6.71 410.285 2.6130856 6.72 414.409 2.6174292 6.73 418.574 2.6217721 6.74 422.780 2.6261144 5.75 157-095 2.1961633 5.76 158.674 2.2005062 5.77 160.269 2.2048492 5.78 161.880 2.2091921 5-79 163.507 2.2135351 6.25 259.006 2.4133099 6.26 261.609 2.4176526 6.27 264.239 2.4219970 6.28 266.894 2.4263388 6.2Q 269.576 2.4306813 6.75 427.030 2.6304584 6.76 431.321 2.6348006 6.77 435-656 2.6391436 6.78 440.034 2.6434862 6.79 444.457 2.6478298 5.80 165.150 2.2178780 5.81 166.810 2.2222209 5.82 168.486 2.2265639 5.83 170.179 2.2309068 5.84 171.890 2.2352498 6.30 272.285 2.4350237 6.31 275.022 2.4393675 6.32 277.786 2.4437104 6.33 280.578 2.4480536 6.34 283.398 2.4523967 6.80 448.923 2.6521719 6.81 453-435 2.6565151 6.82 457-993 .2.6608589 6.83 462.595 2.6652009 6.84 467-244 2.6995437 5.85 173-617 2.2395927 5.86 175-362 2.2439357 5.87 177.124 2.2482786 5.88 178.905 2.2526216 5.89 180.703 2.2569645 6.35 286.246 2.4567394 6.36 289.123 2.4610826 6.37 292.029 2.4654260 6.38 294.964 2.4697690 6.39 297.928 2.474III4 6.85 471.940 2.6738868 6.86 476.683 2.6782296 6.87 481.474 2.6825728 6.88 486.312 2.6869150 6.89 491.200 2.6912584 5.90 182.519 2.2613074 5.91 184.353 2.2656504 5.92 186.206 2.2699933 5.93 188.077 2.2743363 5.94 189.967 2.2786792 6.40 300.922 2.4784540 6.41 303.947 2.4827979 6.42 3O7.OO2 2.4871412 6.43 310.087 2.4914836 6.44 313.203 2.4958260 6.90 496.137 2.6956016 6.91 501.123 2.6999443 6.92 506.160 2.7042878 6.93 511.246 2.7086299 6.94 516.386 2.7129744 5.95 191.877 2.2830222 5.96 193.805 2.2873651 5.97 195.753 2.2917081 5.98 197.720 2.2960510 5.99 199.707 2.3003939 6.45 316.352 2.5001705 6.46 319.530 2.5045II6 6.47 322.742 2.5088555 6.48 325.985 2.5I3I977 6.49 329.262 2.5I754I6 6.95 521.575 2.7173168 6.96 526.816 2.7216589 6-97 532.112 2.7260030 6.98 537-459 2.7303454 6.99 542.860 2.7346878 6.00 201.714 2.3047369 6-50 332.571 2.5218844 7.00 548.317 2.7390317 gb.60 Example. - = 135-213 2 >- 1310191. [141] FABLE XIV. SEMI-EXPONENTIALS. ^ and Iog i0 (-\ CONTINUE e z e x x log 10 2 2 e x e* x log 10 2 2 e' e' x - lo glo - 7.00 548.317 2.7390317 7.01 553.827 2.7433741 7.02 559-393 2.7477170 7.03 565.015 2.7520600 7.04 570.694 2.7564033 7.50 904.021 2.9561785 7.51 913.107 2.9605217 7.52 922.284 2.9648647 7-53 931-553 2.9692076 7.54 940.915 2.9735504 8.00 1490.479 3.1733259 8.01 1505.457 3.1776683 8.02 1520.589 3.1820118 8-03 1535.870 3.1863546 8.04 1551.306 3.1906976 7.05 576.429 2.7607458 7.06 582.223 2.7650893 7.07 588.074 2.7694320 7.08 593-984 2.7737747 7.09 599.954 2.7781180 7-55 950.371 2.9778932 7.56 959.923 2.9822364 7-57 969-570 2.9865792 7.58 979-3I4 2.9909220 7.59 989.157 2.9952653 8.05 1566.895 3-1950399 8.06 1582.645 3.1993836 8.07 1598.552 3.2037268 8.08 1614.617 3.2080694 8.09 1630.841 3.2124116 7.10 605.984 2.7824612 7.11 612.074 2.7868039 7.12 618.225 2.7911467 7.13 624.439 2.7954901 7.14 630.714 2.7998325 7.60 999.098 2.9996080 7.61 1009.139 3.0039510 7.62 1019.281 3.0082939 7.63 1029.525 3.0126369 7.64 1039.872 3.0169799 8.10 1647.234 3.2167554 8.ii 1663.789 3.2210981 8.12 1680.510 3.2254412 8.13 1697.400 3.2297842 8.14 1714.458 3.2341270 7.15 637.053 2.8041755 7.16 643.456 2.8085189 7.17 649.922 2.8128612 7.18 656.454 2.8172043 7.19 663.052 2.8215476 7.65 1050.323 3.0213229 7.66 1060.879 3.0256659 7.67 1071.541 3.0300088 7.58 1082.310 3.0343517 7.69 1093.187 3.0386944 8.15 1731-690 3.2384703 8.16 1749.092 3.2428130 8.17 1766.672 3.2471560 8.18 1784.427 3.2514988 8.19 1802.364 3-2558425 7.20 669.715 2.8258901 7.21 676.446 2.8302331 7.22 683.245 2.8345765 7.23 690.111 2.8389189 7.24 697.047 2.8432620 7.70 1104.174 3.0430376 7.71 1115.271 3.0473806 7.72 1126.480 3.0517234 7-73 ii37-8oi 3.0560663 7.74 1149.236 3.0604092 8.20 1820.476 3.2601848 8.21 1838.774 3.2645284 8.22 1857.251 3.2688706 8.23 1875.914 3.2732129 8.24 1894.770 3-2775566 7.25 704.052 2.8476047 7.26 711.128 2.8519478 7.27 718.275 2.8562908 7.28 725-494 2.8606338 7.29 732.785 2.8649766 7.75 1160.786 3.0647523 7.76 1172.452 3.0690950 7.77 1184.236 3.0734383 7.78 1196.137 3.0777810 7.79 1208.159 3.0821242 8.25 1913.812 3.2188994 8.26 1933.047 3.2862424 8.27 1952.473 3.2905850 8.28 1972.098 3.2949284 8.29 1991.913 3.2992704 7.30 740.150 2.8693197 7.31 747.589 2.8^36629 7.32 755.102 2.8780056 7.33 762.691 2.8823487 7-34 770.356 2.8866915 7.80 1220.301 3.0864670 7.81 1232.565 3.0908098 7.82 1244.953 3.0951531 7.83 1257.465 3.0994961 7.84 1270.102 3.1038386 8.30 2011.936 3.3036142 8.31 2032.158 3-3079575 8.32 2052.580 3.3123000 8.33 2073.206 3.3166425 8.34 2094.045 3.3209860 7-35 778-098 2.8910343 7.36 785.918 2.8953772 7-37 793-8i7 2.8997205 7.38 801.795 2.9040633 7.39 809.853 2.9084062 7.85 1282.867 3.1081818 7.86 1295.760 3.1125246 7.87 1308.783 3.1168677 7.88 1321.936 3.1212105 7.89 1335.222 3-1255535 8.35 2115.092 3-3253293 8.36 2136.347 3.3296718 8.37 2157.819 3.3340150 8.38 2179.505 3-3383578 8.39 2201.409 3.3427008 7.40 817.992 2.9127491 7.41 826.213 2.9170920 7.42 834.517 2.9214351 7.43 842.904 2.9257782 7.44 851.375 2.9301209 7.90 1348.641 3.1298964 7.91 1362.195 3.1342394 7.92 1375-886 3.1385826 7.93 1389.713 3.1429254 7.94 1403.680 3.1472680 8.40 2223.533 3-3470436 8.41 2245.881 3-35I3868 8-42 2268.452 3.3557296 8.43 2291.250 3.3600725 8-44 2314.277 3.3644154 7.45 859.932 2.9344641 7.46 868.574 2.9388068 7-47 877.303 2.9431496 7.48 886. 120 2.9474925 7.49 895.026 2.9518356 7.95 1417-787 3-1516110 7.96 1432-036 3-1559539 7.97 1446.429 3.1602971 7.98 1460.966 3.1646402 7.99 1475.648 3.1689827 8-45 2337.536 3-3687583 8.46 2361.030 3.3731014 8.47 2384.752 3-3774433 8.48 2408.725 3.3817872 8.49 2432.926 3.3861290 7.50 904.021 2.9561785 8.00 1490-479 3-I733259 8.50 2457.383 3-3904730 e 7 - 10 Example. 2 / C 7.1(K = 605.984 log 10 (^ , ) = 2.7824612. [142] TABLE XIV. SEMI-EXPONENTIALS. - and Iog 10 ( ^ CONTINUED X e* i e * log 10 X e* log 10 - X e? g* log 10 - 2 2 2 2 2 2 8.50 2457.383 3.3904730 9-00 4051.543 3.6076204 9-50 6679.863 3.8247676 8.51 2482.082 3.3948162 9-OI 4092.263 3.6119636 9-51 6746.988 3.8291101 8.52 2507.027 3.3991590 9-02 4I33-388 3.6163062 9-52 6814.805 3-8334534 8-53 2532.221 3.4035016 9-3 4174.929 3.620649! 9-53 6883.295 3.8377964 8-54 2557.672 3.4078448 9.04 4216.889 3.6249922 9-54 6952475 3.8421394 8-55 2583-380 34I2I882 9-S 4259.264 3.6293345 9-55 7022.345 3.8464822 8.56 2609.341 3.4165308 9.06 4302.076 3.6336780 9-56 7092.923 3.8508252 8-57 2635.562 3.4208732 9.07 4345.302 3.6380200 9-57 7164.203 3.8551679 8.58 2662.052 3.4252166 9.08 4388.982 3.6423638 9-58 7236.210 3.8595112 8-59 2688.810 3.4295601 9.09 4433-098 3.6467073 9-59 7308.929 3.8638537 8.60 2715.830 3.4339026 9.10 4477.646 3.6510498 9.60 7382.390 3.8681970 8.61 2743.126 3.4382458 9.11 4522.647 3.6553927 9.61 7456.583 3.8725398 8.62 2770.693 3.4425884 9.12 4568.100 3-6597356 9.62 753I-526 3.8768830 8.63 2798.535 3.4469308 9-13 4614.016 3.6640791 9-63 7607.221 3.8812260 8.64 2826.665 3.4512744 9.14 4660.383 3.6684216 9.64 7683.672 3.8855688 8.65 2855.070 3.4556167 9-iS 4707.211 3.6727637 9.65 7760.882 3.8899111 8.66 2883.767 3.4599602 9.16 4754.528 3.6771074 9.66 7838.890 3.8942546 8.67 2912.745 3.4643025 9.17 4802.308 3.6814500 9.67 7917.680 3.8985980 8.68 2942.023 3.4686462 9.18 4850.577 3.6857934 9.68 7997.247 3.9029406 8.69 2971.592 3.4729891 9.19 4899.328 3.6901365 9.69 8077.622 3.9072835 8.70 3001.456 34773320 9.20 4948.563 3.6944792 9.70 8158.802 3.9116264 8.71 3031.621 3.4816749 9.21 4998.284 3.6988209 9.71 8240.792 3.9159690 8.72 3062.088 3.4860178 9.22 5048.532 3.7031652 9.72 8323.623 3.9203124 8-73 3092.852 3.4903592 9-23 5099.272 3.7075082 9-73 8407.262 3.9246546 8.74 3123.948 3.4947038 9.24 5150.519 3.7II85IO 9-74 8491.770 3.9289982 8-75 3*55-337 3.4990458 9- 2 5 5202.272 3.7161930 9-75 8577.112 3-93334II 8.76 3187.054 3.5033896 9.26 5254.569 3.7205370 9.76 8663.316 3.9376842 8-77 3219.085 3-5077325 9.27 5307-367 3.7248791 9-77 8750.384 3.9420270 8.78 3251.440 3.5120756 9.28 5360.716 3.7292228 9.78 8838.326 3.9463700 8.79 3284.114 3.5164182 9.29 54I4-587 3-7335654 9-79 8927.154 3.9507131 8.80 3317.122 3.5207614 9-30 5469.009 3.7379086 9.80 9016.875 3.9550560 8.81 3350.460 3.5251044 9-31 5523-975 3.7422517 9.81 9107.481 3.9593983 8.82 3384-I33 3-5294474 9-32 5579-491 3.7465946 9.82 9199.026 3.9637418 8.83 3418.141 3.5337900 9-33 5635-563 3-7509373 9-83 9291.480 3.9680850 8.84 3452.496 3-538I332 9-34 5692.203 3.7552804 9.84 9384.860 3.9724278 8.85 3487.197 3.5424766 9-35 5749405 3.7596229 9-85 9479.163 3.9767701 8.86 3522.243 3.5468192 5807.194 3.7639664 9.86 9574-444 3.9811136 8.87 3557-63I 3.5511609 9-37 5865.555 3.7683091 9.87 9670.678 3.9854569 8.88 3593-395 3-5555050 9-38 5924.507 3.7726522 9.88 9767.860 3.9897994 8.89 3629.512 3.5598482 9-39 5984.054 3.7769956 9.89 9866.020 3.9941420 8.90 3665-986 3.5641908 9.40 6044.19! 3.7813382 9.90 9965.186 3.9984854 8.91 3702.820 3.5685326 9.41 6104.922 3.7856801 9.91 10065.350 4.0028289 8.92 3740.045 3.5728768 9.42 6l66.29O 3.7900240 9.92 10166.494 4.007I7I2 8-93 3777-635 3.5772201 9-43 6228.269 3.7943674 9-93 10268.667 4.0II5I4I 8.94 38I5-597 3.5815626 9-44 6290.860 3.7987100 9-94 10371.873 4.0158572 8-95 3853-937 3.5859044 9-45 6354.080 3.8030526 9-95 10476.107 4.0201999 8.96 3892.678 3.5902486 9.46 64I7-943 3.8073958 9.96 10581.397 4.0245430 8-97 3931-795 3.5945909 9-47 6482.450 3.8II7392 9-97 10687.745 4.0288860 8.98 3.5989344 9.48 6547.591 3.8l6o8l6 9.98 10795.160 4.0332290 8.99 4011.228 3.6032773 9-49 6613.388 3.8204240 9-99 10903.652 4.0375721 9.00 4051.543 3.6076204 9-50 6679.863 3.8247676 IO.OO 11013.233 4.0419148 gS-M Example. i = 3665.986 log 10 ( ) = 3 .5641908. [143] TABLE XV REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + to Sinh0 Cosh0 Tanh0 Cothe Sech0 Cosech 6 o.oo o.oo I.OO o.oo 00 I.OO 00 0.00 O.OI O.OIOOOO 1.000050 O.OIOOO IOO. 0.9999 IOO. O.OI 0.02 O.O20OOI I.OOO2OO O.O2OOO 50. 0.9998 50. O.O2 0.03 0.030005 I.OOO45O 0.02999 33-34 0.9995 33-333 0.03 0.04 0.04001 1 1.000800 0.03998 25.013 0.9992 24.99 O.O4 O.OS 0.050021 I.OOI25O 0.04996 20.016 0.9987 19.992 0.05 0.06 0.060036 1.001801 0.05993 16.686 0.9982 16.657 O.O6 O.O7 0.070057 1.002451 0.06989 14.308 0-9975 14.274 O.O7 0.08 0.080085 1.003202 0.07983 12.527 0.9968 12.487 0.08 O.OQ 0.090122 1.004053 0.08976 11.141 0-9959 11.097 0.09 O.IO 0.100167 1.005004 0.09967 10.033 0.9950 9-983 O.IO O.II O.IIO222 1.006056 0.10956 9.128 0.9940 9-073 O.II O.I 2 0.120288 1.007209 0.11943 8.373 0.9928 8.314 O.I 2 0.13 0.130366 1.008462 0.12927 7-735 0.9916 7.669 0.13 0.14 0.140458 1.009816 0.13909 7.189 0.9902 7.120 O.I4 O.IS 0.150563 1.011271 0.14888 6.716 0.9888 6.642 0.15 0.16 0.160684 1.012827 0.15865 6.303 0.9873 6.223 0.16 0.17 O.I7O82O 1.014485 0.16838 5-939 0.9857 5-854 0.17 0.18 0.180974 1.016244 0.17808 5-6i5 0.9840 5-525 0.18 0.19 O.I9II45 1.018104 0.18775 5-325 0.9822 S-232 0.19 O.2O 0.201336 1.020067 0.19737 5.067 0.9803 4.967 O.2O O.2I 0.2II547 1.022131 0.20696 4.832 0.9784 4.726 O.2I O.22 O.22I779 1.024298 0.21652 4.618 0.9763 4-509 O.22 0.23 0.232033 1.026567 0.22603 4-425 0.9742 4-3io 0.23 O.24 O.2423II 1.028939 0.23549 4.246 0.9719 4.127 O.24 0.25 O.2526I2 1.031413 0.24492 4.083 0.9695 3-959 0.25 O.26 0.262939 1.033991 0.25430 3-932 0.9671 3-803 0.26 O.27 0.273292 1.036672 0.26363 3-793 0.9646 3-659 O.27 0.28 0.283673 1.039457 0.27290 3.664 0.9620 3-52S 0.28 0.29 0.294082 1.042346 0.28214 3-544 0.9591 3.400 O.29 0.30 0.304520 1-045339 0.29131 3-433 0.9566 3-284 0.30 0.31 0.314989 1.048436 0.30043 3-328 0-9537 3- r 75 0.31 0.32 0.325489 1.051638 0.30951 3-231 0.9511 3-072 0.32 0-33 0.336022 1.054946 0.31852 3.140 0.9479 2.976 0-33 0-34 0.346589 1-058359 0.32748 3-053 Q-9447 2.885 o-34 0-35 0.357190 1.061878 0.33637 2-973 0.9416 2.800 o-35 0.36 0.367827 1.065503 0.34522 2.897 0-9385 2,719 0.36 0-37 0.378500 1.069234 0.35399 2.825 0-9353 2.642 0-37 0.38 0.389212 1-073073 0.36271 2-757 0.9319 2.569 0.38 0-39 0.399962 1.077019 0.37136 2.693 0.9285 2.500 o-39 0.40 0.410752 1.081072 0.37995 2.632 0.9250 2-434 0.40 O.4I 0.421584 1.085234 0.38847 2-574 0.9215 2.372 0.41 O.42 0-432457 1.089504 0.39693 2-519 0.9178 2.312 0.42 0-43 0-443374 1.093883 0.40532 2.467 0.9141 2.256 0-43 0.44 0-454335 1.098372 0.41365 2.417 0.9103 2.2OI 0.44 0-45 0.465342 1.102970 0.42190 2-370 0.9066 2.149 0-45 0.46 0.476395 1.107679 0.43009 2-325 0.0925 2.099 0.46 0.47 0.487496 1.112498 0.43820 2.282 0.8988 2.051 0.47 0.48 0.498646 1.117429 0.44624 2.241 0.8949 2.O06 0.48 0.49 0.509845 1.122471 0.45421 2.2O2 0.8909 I.96l 0-49 Example. sinh 0.25 = 0.252612. TABLE XV REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + io. CONTINUED e Sinhfl Coshtf Tanh0 Cothfl Sechfl Cosech & e 0.50 0.521095 1.127626 0.46211 2.164 0.8868 1.919 0.50 0.51 0-532398 1.132893 0.46995 2.128 0.8827 1.878 0.51 0.52 0-543754 1.138274 0.47769 2.093 0.8785 1.839 0.52 o-S3 o-555i64 1.143769 0.48538 2.060 0.8743 1.801 o-53 0-54 0.566629 1.149378 0.49299 2.028 0.8700 1-765 0-54 0-55 0.578152 1.155101 0.50052 1.998 0.8658 1-730 0-55 0.56 0-589732 1.160941 0.50797 1.969 0.8614 1.696 0.56 0-57 0.601371 1.166896 0-51536 1.940 0.8570 1.663 0-57 0.58 0.613070 1.172968 0.52266 i-9i3 0.8525 1.631 0.58 o-59 0.624831 1.179158 0.52990 1.887 0.8480 1.601 0-59 0.60 0.636654 1.185465 0.53704 1.862 0.8435 I-57I 0.60 0.61 0.648540 1.191891 0.54413 1-838 0.8390 1.542 0.61 0.62 0.660492 1.198436 0.55112 1.814 0.8344 i.SH 0.62 0.63 0.672509 1.205101 0.55805 1.792 0.8298 1.487 0.63 0.64 0.684504 1.211887 0.56490 1.770 0.8251 1.461 0.64 0.65 0.696748 1.218793 0.57166 1.749 0.8205 1-435 0.65 6.66 0.708970 1.225822 0.57836 1.729 0.8158 1.410 0.66 0.67 0.721264 1.232973 0.58498 1.709 0.8110 1-387 0.67 0.68 0.733630 1.240247 0.59152 1.690 0.8065 1-363 0.68 0.69 0.746070 1.247646 0.59798 1.672 0.8015 1.340 0.69 0.70 0.758584 1.255169 0.60437 1-655 0.7967 1.318 0.70 0.71 0.771174 1.262818 0.61067 1.637 0.7919 1.297 0.71 0.72 0.783840 1.270593 0.61691 1.621 0.7870 1.276 0.72 o.73 0.796586 1.278495 0.62306 1.605 0.7821 1.255 0-73 0.74 0.809411 1.286525 0.62914 1.590 0-7773 1.235 0.74 o.7S 0.822317 1.294683 0.63516 1-5744 0.7724 i. 216 0-75 0.76 0.835305 1.302971 0.64108 1-5599 0.7675 1.1972 0.76 0.77 0.848377 1.311390 0.64693 1-5457 0.7625 1.1787 0.77 0.78 0.861533 1.319939 0.65271 1-5320 0.7576 1.1607 0.78 0.79 0.874776 1.328621 0.65842 1.5188 0.7527 1.1431 0.79 0.80 0.888106 1-337435 0.66403 I.5059 0.7477 1.1259 0.80 0.81 0.901525 1-346383 0.66959 1-4934 0.7427 1.1092 0.81 0.82 0.915034 1.355466 0.67507 1.4813 0-7377 1.0928 0.82 0.83 0.928635 1.364684 0.68047 1.4696 0.7327 1.0768 0.83 0.84 0.942328 I-374039 0.68580 1.4582 0.7278. 1.0612 0.84 0.85 0.956116 1-383531 0.69107 1.4470 0.7228 1.0459 0.85 0.86 0.969999 1.393161 0.69626 1-4362 0.7178 1.0309 0.86 0.87 0.983980 1.402931 0.70137 1.4258 0.7128 1.0163 0.87 0.88 0.998058 1.412841 0.70642 1.4156 0.7078 I.OO2O 0.88 0.89 1.012237 1.422893 0.71139 1-4057 0.7028 0.9881 0.89 0.90 1.026517 1433086 0.71629 1.3961 0.6978 0-9737 0.90 0.91 1.040899 1.443423 0.72114 1.3867 0.6928 0.9607 0.91 0.92 1.055386 I-453905 0.72591 1.3776 0.6878 0-9475 0.92 0-93 1.069978 1.464531 0.73060 1-3687 0.6828 0.9346 o-93 0.94 1.084677 1.475305 0.73522 1.3600 0.6778 0.9219 0.94 o-95 1.099484 1.486225 0-73979 i.35i8 0.6728 0.9095 0-95 0.96 1.114402 I-497295 0.74427 1.3436 0.6678 0.8973 0.96 0.97 1.129431 1.508514 0.74870 1.3356 0.6629 0.8854 0.97 0.98 I-I44573 1.519884 0.75306 1.3279 0.6579 0.8737 0.98 0.99 1.159829 1.531406 0.75736 1.3204 0.6529 0.8621 0-99 Example. cosh 0.55 = 1.155101. [i45] TABLE XV REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + io. CONTINUED e Sinh0 Coshfl Tanh0 Cothfl Sech0 Cosech 6 e 1. 00 1.175201 1.543081 0.76159 1-3130 0.6480 0.8509 I.OO I.OI 1.190691 1.554910 0.76576 1-3059 0.6431 0.8395 I.OI 1.02 1.206300 1.566895 0.76987 1.2989 0.6382 0.8290 1.02 1.03 1.222029 I-579036 0.77391 1.2921 0.6333 0.8183 1-03 1.04 1.237881 i-59 I 336 0.77789 1.2855 0.6284 0.8078 I.O4 1.05 1.253857 1.603794 0.78181 1.2791 0.6235 0-7975 1.05 1. 06 1.269958 1.616413 0.78566 1.2728 0.6 1 86 0.7874 1. 06 1.07 1.286185 1.629194 0.78946 1.2666 0.6138 0.7777 1.07 1. 08 1.302542 1.642138 0.79320 1.2607 0.6090 0.7677 1. 08 1.09 1.319029 1-655245 0.79688 1-2549 0.6042 0.7581 I.O9 I.IO I-33S647 1.668519 0.80050 1.2492 0-5993 0.7487 I.IO I. II 1.352400 1.681959 0.80406 1-2437 0-5945 0-7393 I. II I.I2 1.369287 1-695567 0.80757 1.2382 0.5898 0.7302 1. 12 I-I3 1.386312 1.709345 0.81102 1.2330 0.5850 0.7215 I-!3 I.I4 I-403475 1.723294 0.81441 1.2279 0.5803 0.7125 1.14 I-IS 1.420778 I.7374I5 0.81775 1.2229 0-5755 0.7038 1-15 1.16 1.438224 1.751710 0.82104 i. 2180 0.5708 0.6953 1.16 1.17 1-455813 1.766180 0.82427 1.2132 0.5662 0.6869 1.17 1.18 1.473548 1.780826 0.82745 1.2085 0.5616 0.6786 1.18 1.19 1.491430 I.79565I 0.83058 1.2040 0-5569 0.6705 1.19 1.20 1.509461 1.810656 0.83365 I-I995 0-5523 0.6625 i. 20 I.2I 1.527644 1.825841 0.83668 1.1952 0-5477 0.6546 1. 21 1.22 1-545979 1.841209 0.83965 1.1910 o.543i 0.6468 1.22 1.23 1.564468 1.856761 0.84258 1.1868 0.5385 0.6392 1.23 1.24 i.583"5 1.872499 0.84546 1.1828 0-5340 0.6317 1.24 I.2S 1.601919 1.888424 0.84828 1.1789 0.5296 0.6242 1-25 1.26 1.620884 1.904538 0.85106 1.1750 0.5251 0.6170 1.26 1.27 1.640010 1.920842 0.85380 1.1712 0.5206 0.6098 1.27 1.28 1.659301 1-937339 0.85648 1.1675 0.5162 0.6026 1.28 I.2Q 1.678758 1.954029 0.85913 1.1640 0.5118 0-5957 1.29 1.30 1.698382 1.970914 0.86172 1.1605 0.5074 0.5888 1.30 I-3I 1.718177 1.987997 0.86428 1.1570 0.5030 0.5820 I-3I 1.32 1.738143 2.005278 0.86678 i-i537 0.4987 0-5753 1.32 1-33 1.758283 2.022760 0.86925 1.1504 0.4944 0.5687 i-33 1-34 I-778599 2.040445 0.87167 1.1472 0.4901 0.5623 i-34 I-3S 1.799093 2-058333 0.87405 1.1441 0.4858 0-5559 i-35 1.36 1.819766 2.076427 0.87639 1.1410 0.4816 0-5495 1-36 1-37 1.840622 2.094729 0.87869 1.1380 0-4773 0-5433 i-37 1.38 1.861662 2.113240 0.88095 L^Si 0.4732 0.5372 1-38 1-39 1.882887 2.131963 0.88317 1-1323 0.4690 0.53H 1-39 1.40 1.904302 2.150898 0.88535 1.1295 0.4649 0.5252 1.40 1.41 1.925906 2.170049 0.88749 1.1268 0.4608 0.5192 1.41 1.42 L947703 2.189417 0.88960 1.1241 0.4568 o.5i34 1.42 1-43 1.969695 2.209004 0.89167 1.1215 0.4527 0.5077 i-43 1.44 1.991884 2.228812 0.89370 1.1189 0.4486 0.5020 1.44 i. 45 2.014272 2.248842 0.89569 1.1165 0.4446 0.4964 1-45 1.46 2.036862 2.269098 0.89765 1.1140 0.4407 0.4909 1.46 1.47 2-059655 2.289580 0.89958 1.1116 0.4367 0.4855 1-47 1.48 2.082654 2.310292 0.90147 1.1093 0.4329 0.4802 1.48 1.49 2.105861 2-331234 0.90332 1.1070 0.4290 0.4749 1.49 Example. tanh 1.25 = 0.84828. [146] TABLE XV REAL HYPERBOLIC FUNCTIONS. / (x + io) = u + to. CONTINUED e Sinh0 Cosh0 Tanh0 Coth Sech 6 Cosechfl 6 1.50 2.129279 2.352410 0.90515 1.1048 0.4251 0.4697 1.50 1-51 2.152910 2.373820 0.90694 1.1026 0.4212 0.4645 i-Si 1-52 2.176757 2.395469 0.90870 1.1005 0.4174 0.4594 1.52 1-53 2.200821 2.417356 0.91042 1.0984 0.4137 0-4543 1-53 1-54 2.225105 2.439486 0.91212 1.0963 0.4099 0.4494 1-54 i-SS 2.249611 2.461859 0.91379 1.0944 0.4062 0.4444 1-55 1.56 2-274343 2.484479 0.91542 1.0924 0.4025 0.4398 1.56 1-57 2.299302 2.507347 0.91703 1.0905 0.3988 0.4350 1-57 1.58 2.324490 2-530465 0.91860 1.0886 0-3952 0.4302 1.58 1-59 2.349912 2.553837 0.92015 1.0868 0.3916 0.4255 1-59 i. 60 2-37SS68 2.577464 0.92167 1.0850 0.3879 0.4209 1.60 1.61 2.401462 2.601349 0.92316 1.0832 0.3844 0.4164 1.61 1.62 2.427596 2.625495 0.92462 1.0815 0.3809 0.4119 1.62 1.63 2-453973 2.649902 0.92606 1.0798 0-3774 0.4075 1.63 1.64 2.480595 2.674575 0.92747 1.0782 0-3739 0.4031 1.64 1-65 2-507465 2-699515 0.92886 1.0766 0.3704 0.3988 1.65 1.66 2.534586 2.724725 0.93022 1.0750 0.3670 0-3945 1.66 1.67 2.561960 2.750207 0.93155 1.0735 0.3636 0.3903 1.67 1.68 2-589591 2-775965 0.93286 , 1.0719 0.3602 0.3862 1.68 1.69 2.617481 2.802000 0.93415 1.0704 0-3569 0.3820 1.69 1.70 2.645632 2.828315 0-93541 1.0691 0.3536 0.3780 1.70 1.71 2.674048 2.854914 0.93665 1.0676 0-3503 0.3740 1.71 1.72 2.702731 2.881797 0.93786 1.0662 0.3470 0.3700 1.72 1-73 2.731685 2.908969 0.93906 1.0649 0.3438 0.3661 1-73 1.74 2.760912 2.936432 0.94023 1.0636 0.3405 0.3622 1.74 i-75 2.790414 2.964188 0.94138 1.0623 0-3373 0.3584 i-75 1.76 2.820196 2.992241 0.94250 1.0610 0.3342 0.3546 1.76 1.77 2.850260 3.020593 0.94361 1-0597 0.3310 0.3508 1.77 1.78 2.880609 3.049247 0.94470 1-0585 0.3279 o.347i 1.78 1.79 2.911246 3.078206 0.94576 1-0573 0.3248 0-3435 1.79 1.80 2.942174 3-107473 0.94681 1.0562 0.3218 0-3399 1.80 1.81 2-973397 3-137051 0.94783 1-0550 0.3187 0.3363 1.81 1.82 3.004916 3.166942 0.94884 1.0539 0.3158 0.3328 1.82 1-83 3-036737 3-r97i5o 0.94983 1.0528 0.3128 0.3293 1.83 1.84 3.068860 3.227678 0.95080 1.0517 0.3098 0.3258 1.84 1.85 3.101291 3-258528 0.95175 1.0507 0.3069 0.3224 1.85 1.86 3.134032 3.289705 0.95268 1.0497 0.3040 0.3191 1.86 1.87 3.167086 3.321210 0-95359 1.0487 0.3011 0.3157 1.87 1.88 3.200457 3-353047 0.95449 1.0477 0.2982 0-3125 1.88 1.89 3.234148 3.385220 0-95537 1.0467 0.2954 0.3092 1.89 1.90 3.268163 3-4I7732 0.95624 1.0458 0.2926 0.3059 1.90 1.91 3.302504 3-450585 0.95709 1.0448 0.2897 0.3028 1.91 1.92 3-337I76 3-483783 0.95792 1.0439 0.2870 0.2997 1.92 i-93 3.372181 3-5I7329 0.95873 1.0430 0.2843 0.2965 1-93 1.94 3.407524 3-55I227 0-95953 1.0422 0.2816 0.2935 1.94 I- 95 3.443207 3-58548i 0.96032 1.0413 0.2789 0.2904 i-95 1.96 3.479234 3.620093 0.96109 1.0405 0.2762 0.2874 1.96 1.97 3.515610 3-655067 0.96185 1.0397 0.2736 0.2844 1.97 1.98 3-552337 3.690406 0.96259 1.0389 0.2710 0.2815 1.98 1.99 3-5894I9 3-726115 0.96331 1.0380 0.2684 0.2786 1.99 Example. coth 1.70 = 1.0691. [147] TABLE XV REAL HYPERBOLIC FUNCTIONS, f (x + io) = u + io. CONTINUED e Sinh0 Cosh0 Tanhfl Cothfl Sechfl Cosech 2.OO 3.626860 3.762196 0.96403 J-0373 0.2658 0.2757 2.00 2.OI 3.66466 3-79865 0.96473 1-0365 0.2632 0.2729 2.OI 2.O2 3.70283 3-83549 0.96541 1.0358 0.2607 0.2701 2.O2 2.03 3-74I38 3.87271 0.96608 1-0351 0.2582 0.2673 2.03 2.04 3.78029 3.91032 0.96675 1.0344 0.2557 0.2645 2.04 2.05 3-81958 3-94832 0.96740 1-0337 0-2533 0.2618 2.05 2.O6 3.85926 3.98671 0.96803 1.0330 0.2508 0.2596 2.06 2.07 4.02550 0.96865 1.0323 0.2484 0.2565 2.07 2.08 3-93977 4.06470 0.96926 1.0317 0.2460 0.2538 2.08 2.O9 3.98061 4.10430 0.96986 1.0310 0.2436 0.2512 2.09 2.IO 4.02 16 4.14431 0.97045 1.0305 0.2413 0.2486 2.IO 2. II 4.06350 4.18474 0.97103 1.0298 0.2389 0.2461 2. II 2.12 4-10555 4-22558 0.97159 1.0293 0.2366 0.2436 2.12 2.13 4.14801 4.26685 0.97215 1.0286 0.2344 0.2411 2.13 2.14 4.19089 4-30855 0.97269 1.0280 0.2321 0.2386 2.14 2.15 4.23419 4-35067 0.97323 1.0275 0.2298 0.2362 2-15 2.l6 4.27791 4.39323 0-97375 1.0269 0.2276 0.2338 2.l6 2.17 4.32205 4-43623 0.97426 1.0264 0.2254 0.2314 2.17 2.18 4.36663 447967 0.97477 1.0259 0.2232 0.2290 2.18 2.19 4.41165 4.5 2 356 0.97526 1.0254 O.22II 0.2267 2.19 2.20 4-457" 4.56791 0-97574 1.0249 0.2189 0.2244 2. 2O 2.21 4.50301 4.61271 0.97622 1.0243 0.2168 0.2221 2.21 2.22 4-54936 4-65797 0.97668 1.0239 0.2147 0.2198 2.22 2.23 4.59617 4.70370 0.97714 1.0234 O.2I26 O.2I76 2.23 2.24 4.64344 4.74989 0.97758 1.0229 0.2105 0.2154 2.24 2.25 4.69117 4.79657 0.97803 1.0225 0.2085 O.2I32 2.25 2.26 4-73937 4.84372 0.97847 I.022O 0.2064 O.2IIO 2.26 2.27 4.78804 4.89136 0.97888 I. O2 1 6 0.2044 0.2089 2.27 2.28 4.83720 4.93948 0.97929 I.O2II O.2024 O.2067 2.28 2.29 4.88683 4.98810 0.97970 I.O2O7 0.2OO5 O.2047 2.29 2.30 4.93696 5.03722 0.98010 1.0203 0.1985 O.2O26 2.30 2.31 4.98758 5.08684 0.98049 I.OI99 0.1966 0.2005 2.31 2.32 5.03870 5^3697 0.98087 I.OI95 0.1947 0.1985 2.32 2-33 5.09032 5.18762 0.98124 I.Oigi 0.1928 0.1965 2-33 2-34 5^4245 5-23879 0.98161 I.OI87 0.1909 0.1945 2-34 2-35 5.19510 5.29047 0.98198 I.Ol84 0.1890 0.1925 2-35 2.36 5-24827 5-34269 0.98233 I.OlSo 0.1872 0.1905 2.36 2-37 5.30196 5-39544 0.98268 I.OI77 0.1854 0.1886 2-37 2.38 5.35618 5-44873 0.98302 I.OI73 0.1835 0.1867 2.38 2-39 5-41093 5-50256 0.98335 1.0169 O.l8l7 0.1848 2-39 2.40 5.46623 5-55695 0.98368 1.0166 0.1800 0.1829 2.40 2.41 5.52207 5.611-89 0.98399 1.0163 0.1782 0.1811 2.41 2.42 5-57847 5-66739 0.98431 1.0159 0.1765 0.1793 2.42 2-43 5-63542 5-72346 0.98462 1.0156 0.1747 0.1775 2-43 2-44 5.69294 5.78010 0.98492 1-0153 0.1730 0.1757 2-44 2-45 5-7SI03 5.83732 0.98522 1.0150 O.I7I3 0.1739 2-45 2.46 5.80969 5.89512 0.98551 1.0147 0.1696 0.1721 2.46 2-47 5.86893 5-95352 0.98579 1.0144 o.i 680 0.1704 2-47 2.48 5.92876 6.01250 0.98607 1.0141 0.1663 0.1687 2.48 2.49 5-98918 6.07209 0.98635 1.0138 0.1647 0.1670 2.49 Example. sech 2.00 = 0.2658. [148] TABLE XV REAL HYPERBOLIC FUNCTIONS. / (x + to) = u + io. CONTINTJED Sinhfl Coshfl Tanh0 Cothfl Sech0 Cosech 8 6 *-5 6.05020 6.13229 0.98661 1.0136 0.1631 0.1653 2-5 2.6 6.69473 6.76901 0.98903 I.OIII 0.1477 0.1494 2.6 2.7 7.40626 7-47347 0.99101 1.0091 0.1338 0.1350 2.7 2.8 8.19192 8.25273 0.99263 1.0074 O.I2I2 O.I22I 2.8 2.9 9-05956 9.11458 0.99396 1. 0061 0.1097 0.1104 2.9 3-o 10.01787 10.06766 0-99505 1.0050 0.0937 0.09982 3-o 3-i 11.07645 11.12150 0-99595 1.0041 0.0899 0.0003 3-i 3-2 12.24588 12.28665 0.99668 1.0033 0.0814 0.08 1 6 3-2 3-3 13.53788 I3-57476 0.99728 1.0027 0.0736 0.0739 3-3 3-4 14.96536 14.99874 0.99778 1.0022 0.0667 0.0668 3-4 3-5 16.54263 16.57282 0.99818 1.0018 0.0604 0.0604 3-S 3-6 18.28546 18.31278 0.99851 1.0015 0.0546 0.0547 3.6 3-7 20.21129 20.23601 0.99878 I.OOI2 0.0494 0.0495 3-7 3-8 22.33941 22.36178 0.99900 I.OOIO 0.0447 0.0448 3-8 3-9 24.69110 24.7"3S 0.99918 1.0008 0.0405 0.0405 3-9 4.0 27.28992 27.30823 0-99933 1.0007 0.0366 0.0366 4.0 4.1 30.16186 30.17843 0-99945 1. 0006 0.0331 0.0332 4-1 4-2 33-33567 33-35066 0-99955 1.0005 0.0300 0.0300 4.2 4-3 36.84311 36.85668 0.99963 1.0004 0.0271 0.0271 4-3 4.4 40.71930 40.73157 0.99970 1.0003 0.0245 0.0245 4-4 4-5 45.00301 45.01412 0-99975 1.0003 O.O222 O.O222 4-5 4.6 49-737I3 49.74718 0.99980 I.OOO2 O.O2OI O.O2OI 4.6 4-7 54.96904 54.97813 0.99983 I.OO02 0.0182 0.0182 4-7 4-8 60.75109 60.75932 0.99986 I.OOOI 0.0165 0.0165 4.8 4-9 67.14117 67.14861 0.99989 1. 0001 0.0149 0.0149 4-9 S- 74.20321 74.20995 0.99991 I.OOOI 0.0135 0.0135 5-o 5- 1 82.0079 82.0140 0-99993 1.00007 0.01219 0.01219 5-i 5-2 90.6334 90.6389 0.99993 1.00007 0.01103 0.01103 S-* 5-3 100.1659 100.1709 0-99994 1.00006 0.00998 0.00998 5-3 5-4 110.7009 110.7055 0.99995 1.00005 0.00903 0.00903 5-4 5-5 122.3439 122.3480 0.99996 1.00004 0.00818 0.00818 5-5 5-6 135.2114 135.2150 0-99997 1.00003 0.00740 0.00740 5-6 5-7 149.4320 J49-4354 0.99998 I.OOO02 0.00669 0.00669 5-7 5-8 165.1483 165.1513 0.99998 I.OOOO2 0.00606 0.00606 5-8 5-9 182.5174 182.5201 0.99998 1.00002 0.00548 0.00548 5-9 6.0 201.7132 201.7156 0-99999 I.OOOOI 0.00496 0.00496 6.0 6.1 222.9278 222.9300 I.OOO I.OOO 0.00449 0.00449 6.1 6.2 246.3735 246.3755 I.OOO I.OOO 0.00406 0.00406 6.2 6-3 272.2850 272.2869 I.OOO I.OOO 0.00367 0.00367 6-3 6.4 300.9217 300.9233 I.OOO I.OOO 0.00332 0.00332 6.4 6-5 332.5701 332.5716 I.OOO I.OOO 0.00301 0.00301 6-5 6.6 367.5469 367-5483 I.OOO I.OOO 0.00272 0.00272 6.6 6-7 406.2023 406.2035 I.OOO I.OOO 0.00246 0.00246 6.7 6.8 448.9231 448.9242 I.OOO I.OOO 0.00223 0.00223 6.8 6.9 496.1369 496.1379 I.OOO I.OOO O.OO2O2 0.00202 6.9 7.0 548.3161 548.3170 I.OOO I.OOO 0.00182 0.00182 7-o 7-i 605.9831 605.9839 I.OOO I.OOO 0.00165 0.00165 7-i 7-2 669.7150 669.7158 I.OOO I.OOO 0.00149 0.00149 7.2 7-3 740.1496 740.1503 I.OOO I.OOO 0.00135 0.00135 7-3 7-4 817.9919 817.9925 I.OOO I.OOO 0.00122 O.OOI22 7-4 7-5 904.0209 904.0215 I.OOO I.OOO O.OOIII O.OOI 1 1 7-S Example. cosech 2.50 = 0.1653. [149] TABLE XVI. SUBDIVISIONS OF A DEGREE AUXILIARY TABLE O.OI 00.36 O.4I 24.36 0.81 48.36 OI 0.0167 4i 0.6833 21 0.0058 O.O2 OI.I2 0.42 25.12 0.82 49.12 02 0.0333 42 O.7OOO 22 0.006 1 0.03 01.48 0-43 25.48 0.83 49.48 03 0.0500 43 0.7167 23 0.0064 O.O4 02.24 0.44 26.24 0.84 50.24 04 0.0667 44 0.7333 24 0.0067 O.OS 03.00 o-4S 27.00 0.85 51.00 05 0.0833 45 0.7500 25 0.0069 O.O6 03-36 0.46 27-36 0.86 51.36 06 O.IOOO 46 0.7667 26 0.0072 O.O7 O4.I2 0.47 28.12 0.87 52.12 07 0.1167 47 0.7833 27 0.0075 0.08 04.48 0.48 28.48 0.88 52.48 08 0.1333 48 O.SOOO 28 0.0078 O.O9 05.24 0.49 29.24 0.89 53-24 09 0.1500 49 0.8167 29 0.0081 O.IO O6.OO 0.50 30.00 0.90 54-oo IO 0.1667 So 0.8333 30 0.0083 O.I I 06.36 0.51 30-36 0.91 54.36 II 0.1833 Si 0.8500 31 0.0086 O.I2 07.12 0.52 31.12 0.92 5S- 12 12 0.2OOO 52 0.8667 32 0.0089 0.13 07.48 0-53 31.48 o-93 55-48 13 0.2167 53 0.8833 33 0.0092 O.I4 08.24 0-54 32.24 0.94 56-24 14 0.2333 54 0.9OOO 34 0.0094 O.IS 09.00 o-SS 33-00 0-95 57.00 15 0.2500 55 0.9167 35 0.0097 0.16 09.36 0.56 33-36 0.96 57-36 16 0.2667 56 0-9333 36 O.OIOO 0.17 IO.I2 0-57 34-12 0.97 58.12 17 0.2833 57 0.9500 37 0.0103 0.18 10.48 0.58 34-48 0.98 58.48 18 0.3000 58 0.9667 38 0.0106 0.19 11.24 o-S9 35-24 0.99 59-24 19 0.3167 59 0-9833 39 0.0108 0.20 12.00 0.60 36.00 I.OO 60.00 20 0-3333 60 I.OOOO 40 O.OI 1 1 a O.2I 12.36 0.61 36.36 21 O.3SOO OI 0.0003 41 0.0114 O.22 I-J.I2 0.62 O X/ "J V ' 37.12 22 oo^^ 0.3667 02 0.0006 42 0.0117 O.23 *O* * IV48 0.63 o / ** 27.48 23 v "O'-' v -' / 0.3833 03 0.0008 43 0.0119 ^"O O.24 J. J.4J.V 14.24 *" "O 0.64 O 1 *^** 38.24 *j 24 J JO 0.4000 o 4 O.OOII ~O 44 O.OI 2 2 o.2<; I^.OO 0.6? O" *^ 30.00 2? 0.4167 oc 0.0014 41; O.OI25 ** j 0.26 * J**" 1C. 76 ""o 0.66 o y 30.36 ~j 26 O.4333 J 06 0.0017 ~j 46 O.OI28 0.27 O O 16.12 0.67 ov o 40.12 . 27 V .*TOOO 0.4500 07 0.0019 47 O.OI3I 0.28 16.48 0.68 40.48 28 O.4.66? 08 O.OO22 48 O.O1 11 O.2Q . vs.tfi_r 17.24 0.60 *T ^ 4.1.24. 2O v.lfwu y O.4833 oo O.OO25 f** 4O WA Oi> 0.0136 y O.1O / *"T iS.OO ' y O.7O ^* *^ 42.OO *y 30 ^t^OO 0.5000 vy IO O.OO28 ty CO O.OI 3O *~*o ** f >* T^ o / // o w - WA oy 0.31 18.36 0.71 42.36 0.001 00.03.6 3i 0.5167 II 0.0031 51 0.0142 0.32 19.12 0.72 43.12 O.O02 00.07.2 32 0-5333 12 0.0033 52 0.0144 o-33 19.48 o-73 43-48 0.003 06.10.8 33 0.5500 13 0.0036 53 0.0147 0-34 20.24 0.74 44.24 0.004 00.14.4 34 0.5667 14 0.0039 54 0.0150 0-35 21.00 0-75 45-00 0.005 00.18 35 0-5833 15 O.OO42 55 0.0153 0.36 21.36 0.76 45-36 0.006 OO.2I.6 36 0.6000 16 O.OO44 56 0.0156 0-37 22.12 0.77 46.12 0.007 00.25.2 37 0.6167 17 O.0047 57 0.0158 0.38 22.48 0.78 46.48 0.008 00.28.8 38 0.6333 18 O.OO5O 58 0.0161 o-39 23.24 0.79 47.24 0.009 00.32.4 39 0.6500 19 0.0053 59 0.0164 0.40 24.00 0.80 48.00 O.OIO 00.36 40 0.6667 20 0.0055 60 0.0167 Examples. o.4i = o.24'.j6" o.oo5 = o.oo'.i8* o.4i'.oo* = o.6833. o.oo'.46* = o.oi28. [150] EXPLANATORY TEXT EXPLANATORY TEXT INTRODUCTION THE Tables in this book are designed primarily for presenting hyperbolic functions of a complex variable either in the rectangular coordinate form of that variable (x -f- iy) or the polar coordinate form (p/8). They are also designed secondarily for presenting circular functions of a complex variable. A few formulas are added as aids to the conversion of such functions. The most extensive range offered is in Tables VII to XIV inclusive, between which, the functions sinh (x + iy), cosh (x + iy), tanh (x -f iy), expressed in the result either in rectangular coordinates u + iv or in polar coordinate quantities r /y, may be obtained between the limits of o and 10 of x, and between the limits of o and for y. It is shown, moreover, to be an easy matter to extend the range of x beyond the offered range of 10, should such an extension be required. The practical need for tabulated values of hyperbolic functions of (x + iy) beyond the range of x = 10 appears to be so small that any such extension is left to the reader. As the author's applications for financial assistance in the computation of the Tables were unsuccessful, the steps in x and y (0.05 and 0.07854 respectively) are larger than were originally intended; i.e., for reducing the work of the user to the lowest practic- able limits. Consequently, interpolation must ordinarily be resorted to, when three or more significant digits are needed in the results. Such interpolations require an appre- ciable amount of time to effect in two dimensions; i.e., for both x and y. In order to render such interpolation unnecessary for ordinary engineering purposes, where three, or at most four, significant digits may be needed, a separate atlas of 23 large-scale charts, 45 cm. X 45 cm. over ruled areas, has been prepared, and is published as an adjunct to these Tables. The necessary interpolation can very swiftly be made on the charts, by inspection. COMPLEX QUANTITIES The following brief outline of complex quantities is offered in view of their funda- mental importance in connection with the Tables, for the assistance of those who have studied elementary mathematics, but who may not have become familiar with com- plex numbers. For a more comprehensive discussion of complex quantities, the reader must be referred to special treatises on the subject. Ordinary numerical quantities, or the numbers dealt with in ordinary arithmetic, may be considered to range between zero and either positive or negative infinity, by indefinitely small gradations. Such numbers may be represented geometrically by distances, in either direction, from a zero point on an infinite straight line. Thus in Fig. i, we may consider that the straight line XOX extends from minus infinity on the left, to plus infinity on the right, O being the zero point. The point x\. would then represent + i, and so on. That is, the number + i may be regarded as represented on the line XOX either by the position of the point xi with respect to the zero point 0; [153] EXPLANATORY TEXT or, as the vector Ox\; i e., the straight line drawn from the origin O to the point x\ and forming a part of the reference line XOX. Under these assumptions, the ordinary numbers of arithmetic may be represented geometrically as vectors; but such vectors are confined to a single straight-line direction from O towards .X" for positive numbers, and from O towards X for negative numbers. Complex quantities, or complex numbers, cannot be completely represented by reference to a single direction, or to vectors along one and the same straight line. They may, however, be represented geometrically by the position, in an infinite plane, of a movable point with respect to a fixed point as origin. Thus, in Fig. i, the plane XOY is the plane of reference, and the fixed point is the origin. Then any point PI in the plane represents a complex number, and any complex number may be represented by a point on the plane. -Y 2 X FIG. 2. Plane Vector 2.236 e"- 106 or FIG. i. Complex quantity i + a. designated by 2.236 763. 26'. A complex number may be specified either in rectangular coordinates, or in polar coordinates, as may be preferred. Thus, the same vector OP\ is represented in Fig. i to rectangular coordinates, and in Fig. 2 to polar coordinates. In Fig. i, the X axis XOX passing through the origin O is the fundamental reference axis, and the Y axis YOY, perpendicular thereto in the reference plane, immediately follows. Then the point Pi, measuring + i along OX, and -f- 2 along OF, may be defined by the expres- sion (i + 22), where the symbol i signifies measurement along 'the subordinate axis. It is shown in mathematical treatises that * = V i. The vector OP\ of Fig. i may therefore be expressed as (i + V~ 1-2) and a vector from to any point in the plane may be represented by x + V~i y x + *3S where x and y may have any positive or negative numerical values, including zero. In pursuance of time-honored terminology, the axis XOX is sometimes called the "real" axis, and YOY the "imaginary," axis; so that the ^-component of a complex number becomes the " real component," and the ^-component the " imaginary component." The symbol i still stands for the imaginary component. In mathematics as applied to electrical engineering, the symbol i commonly designates electric current- strength, and so, in order to prevent the possibility of confusion, the symbol j is frequently substituted as the sign of the imaginary. Under such a convention, the plane-vector, or complex quantity, OP\, would be represented as i -\- j 2. As, however, in this book we necessarily consider complex quantities from a broader viewpoint than that offered by electrical engineering, we shall use the symbol i to denote the imaginary component, perpendicularly rotated with respect to the fundamental X axis. [154] EXPLANATORY TEXT Complex quantities may also be expressed in polar coordinates, as in Fig. 2, where the fundamental reference axis OX is drawn in the positive direction in the reference plane, from the origin 0, and the circular angle 5i is measured in the positive or counter- clockwise direction from OX to OP\. The vector OPi is then specified in polar co- ordinates by its length pi and by its angle 5i. The length p\ is called the modulus of the vector, and the angle 81 is called the argument. This argument may be expressed in circular radians, in degrees-minutes-seconds, quadrants, or any other recognized unit of circular angle. Thus, in Fig. 2, the vector OPi may be represented to polar co- ordinates symbolically by pi/8\ or, using numbers, by 2. 236/63. 26', where 2.236 is the modulus to the same scale of linear measure as in Fig. i, and 63. 26' is the argument. If one and the same complex quantity be expressed both in rectangular and polar coordinates, as follows: x + iy = p/8 (i) it is evident that x = p cos 8, y = p sin 5, ;y/a; = tan 8, and p = V x 2 + y 2 , relations which enable the coordinates to be changed, at will, from one form to the other. ' Thus in Figs i and 2, xi = i, yi = 2, p\ VJ = 2.236, and 1 = tan -1 (2) = 63.26 / . -X FIG. 3. Complex quantity 2 i. FIG. 4. Plane-Vector 2.236 e* 3 - 608 . or 2.236^206^34'. Similarly, Figs. 3 and 4 represent the complex quantity or plane vector OPi to rec- tangular and polar coordinates respectively. Here x% = 2, y z = i, pz = Vs = 2.236 and 82 = 206. 34'. ADDITION OF COMPLEX QUANTITIES One vector quantity is added to another, by drawing it in the reference plane from the extremity of the latter as origin, and then drawing a vector from the origin to its -X -Y FIG. 5. Addition of two complex quantities (x + 2) + (- 2 - * i) = - i + i. FIG. 6. Complex Addition, Polar coordinates. Pl/Si +pJ/5i! = p3/53 2. 236/63 ".26' + 2.236/2o6.34' = x .414/135* Op + pP = OP free end. The last named vector is the required sum. Thus, in Fig. 5, the complex quantity OP 2 2 ii of Fig 3 is added to the complex quantity OP\ =1+12 EXPLANATORY TEXT of Fig i, giving the resultant vector OP = i + ii. Fig. 6 shows the corresponding operation with polar coordinate vectors. Here OP 2 = 2. 236/206. 34' of Fig. 4 is added to OPi = 2. 236/63. 26' of Fig. 2, to produce OP = 1.414/135 = p 3 /8z of Fig. 6. On the drawing-board, the graphic process of adding vectors is as easily effected when they are expressed in polar as in rectangular coordinates. But the arithmetical addition is much more easily made with rectangular coordinates. The rule is: find the vector sum by taking first the sum of the reals, and then the sum of the imaginaries; or + x z + . . . -f *) (xi + iyj + (x z + iyz) + - . + (x n + fy B ) = + i (y* + y* + . . . + ?) = 2x In the case of Figs. 5 and 6: (i + ) + (- 2 - ) = (+ i - 2) + i(2 - i) = - i + ii = SUBTRACTION OF COMPLEX QUANTITIES (2) /i3S. Reversing the sign of a rectangular complex quantity means reversing the sign of both its real and imaginary components. Reversing the sign of a polar complex quan- tity means changing its argument by 180. To subtract one complex quantity A from another B, reverse the sign of A, and then add it thus reversed to B, by the rules of addition. Y o FIG. 7. Complex Subtraction (i + 12) - (- 2 -*i) .. 3 +*3 =OP. FIG. 8. Complex Subtraction, Polar Coordinates pi /Si p-i/b-i = p3/5a 2.236 763 ".26' - 2.236/206^34' = 4.243/45" Op t + piP = OP. In Figs. 7 and 8, the vector P z of Figs. 3 and 4 is subtracted from the vector P\ of Figs, i and 2. In Fig. 7, we have OPi - OP 2 = OP. (i + *2) - (- 2 - *l) = (i + *2) + (2 + *l). = 3 + *3- In Fig. 8, pi/3i p2/5 2 = psAj 2.236/63. 26^ 2. 236/206. 34' = 2. 236/63. 26' + 2. 236/26. 34'. = 4.243/45. Here again the process of complex subtraction, which is only a slight modification of complex addition, is very easily made on the drawing board by purely geometric proc- esses, whether the quantities are rectangular or polar. If, however, the process is to be conducted algebraically, it is much more easily conducted with rectangular coordinates. [156] EXPLANATORY TEXT MULTIPLICATION OF COMPLEX QUANTITIES Two rectangular complex quantities may be multiplied algebraically by the ordinary rules of algebra, remembering that i 2 = i. Thus (*i + fyi) (*2 + iyz) = (xi x z - yi y 2 ) + i(xi y 2 + x z yi). (3) In Fig. 9, the vector OP\ of Figs, i and 2 is multiplied by the vector OP 2 of Figs. 3 and 4. The product is the broken line OPz. For (i + M) X (- 2 - ii) = (- 2 + 2) - i(i + 4) = - *5. FIG. 9. Product and Quotient of Complex Quantities Rectangular Coordinates (i +.2) X(-2 -ii) = -is =OP 3 (-2 +11) -M+i + 12) = -o.8+to.6 = OP 4 FIG. 10. Product of two Complex Quantities, Polar Coordinates 2.236 763. 26' X 2.236/2o6.34' = 5 /270 = OPs If the two quantities to be multiplied are polar; then Pi /Si X PZ/SZ = Pi Pz/8i + (4) Or the rule is form the product of the moduli and add the arguments. Thus in Fig. 10, OPi = Vj/63.26' and OP 2 = Vs/2o6^3j/ /. OP 3 = 5/27o.oo / . RECIPROCAL OF A COMPLEX QUANTITY The reciprocal of a rectangular complex quantity can be reduced to the standard algebraic form, by multiplying both numerator and denominator by the same complex quantity with reversed imaginary. Thus: i i X (x - iy) _ x - iy _ ( x \ _ .f y \ ,. 1 yt.2 / V /L-2 _J /t2 / \J/ * 2 + x + iy (x + iy) (x - iy) For example if x + iy = i + 12, I _ I /I J2\ _ !_ 1+^2 I + 12 \ I 12 ) I 12 I 12 = = 0.2 + 4 5 5 The reciprocal of a polar complex quantity is obtained by taking the reciprocal of its modulus, and reversing its argument. That is = -I- 8 = p p Ci57] (6) EXPLANATORY TEXT For example: V5/63. 2 6' VS QUOTIENT OF COMPLEX QUANTITIES To find the quotient of a complex quantity A divided by another B, form the re- ciprocal of B and then multiply this reciprocal by A . Thus to find (x\ + iyi)/(x 2 + fy 2 ) / x For example: \ __ 2 ii __ 2 ii /i i: \ i + *2 i -f- 12 \ i i'. _ (- 2 - 2) + t( o , , = o.o + i o.o. Thus, in Fig. 9, The quotient of two polar complex quantities is formed by taking the quotient of their moduli and the difference of their arguments. That is P2/02 P2 /E ' E /ox -=7= = /fc ~ Si. (8) Pi/5i Pi Thus in Fig. 1 1 we have the quotient of OP 2 of Figs. 3 and 4 divided by OPi of Figs, i and 2, or ^ = i /I 4 3.o8 / . * A' FIG. n. Quotient of two Complex Quantities, Polar Coordinates. 8.836 /2o6.34 > -5- 2.236 /63.26' = I.o/i43.o8' = O?4 Pz ^83 It is thus evident that in order to find either the sums or the differences of complex quantities, it is desirable to have them expressed in rectangular coordinates; while, on the other hand, in order to find products, reciprocals, or quotients, it is preferable to have them expressed in polar coordinates. [158] EXPLANATORY TEXT POWERS AND ROOTS OF COMPLEX QUANTITIES It will be evident from the foregoing that (p/8) n = p n /n8: and \fpjl = y~p/8/n (9) operations that are readily executed on polar complex quantities. CIRCULAR AND HYPERBOLIC FUNCTIONS GEOMETRICALLY COMPARED Since the Tables in this book are adapted for the evaluation of both circular and hyperbolic functions of a complex variable; that is, either of sin (a; -f- iy), cos (it + iy) and tan (x + iy); or of sinh (x + iy), cosh (x + iy) and tanh (x + iy), it may be advisable to consider some propositions in the comparative geometry of the circular and hyperbolic functions, both real and complex. REAL CIRCULAR AND HYPERBOLIC FUNCTIONS The geometry of the real circular functions sin x, cos x and tan x relates, as is well known, to the motion of a radius vector over a circle. The geometry of the real hyper- bolic functions sinh x, cosh x and tanh x relates to the motion of a radius vector over a rectangular hyperbola. In Fig. 12, A b c d E g is a circle x* + y 2 = i, assumed to have unit radius, and center 0. As the radius vector OA rotates in the positive or coun- terclockwise direction about the center 0, it describes a circular sector such as AOE, and a circular angle /3, the tangent Ef being always perpendicular to the radius vector OE. The magnitude of the circular angle /3 may be defined in either of two ways, namely: (1) By the ratio of the circular arc length 5 described during the motion, by the vector's terminal E, to the constant length p of the radius vector. (2) By the area of the circular sector AOE swept out by the radius vector during the motion. According to definition (i), if the radius vector generates any infinitesimal angle dfi circular radians, by moving its terminal over an infinitesimally small circular arc ds then . y /\ v --^NHT O.5 - "7 ^ - 4 ^ft HI 'PE RB OL C AN GLE Ljr Q ^ / 1 1 ^w ' __^ [162] EXPLANATORY TEXT BISECTION or CIRCULAR AND HYPERBOLIC ANGLES If we take any circular angle BOC Fig. 17, we may of course bisect this angle in either of two ways: FIG. 17. Bisection of a circular sector in the well-known manner by a radius vector through the intersection of ter- minal tangents, or through the midpoint of the chord be- tween terminal points. (i) By drawing tangents bb, cc, to the curve at the points B, C, respectively, and drawing the straight line Od from the center O through the point of intersection d. FIG. 18. Bisection of a hyperbolic sector by a radius vector through the intersection of terminal tangents, or through the midpoint of the chord between terminal points. [163] EXPLANATORY TEXT (2) By drawing the chord BC, and marking the radius OD through the midpoint 8 of this chord. Similarly, if we take any hyperbolic angle BDC Fig. 18, between the points B and C of a rectangular hyperbola, we may bisect this angle in either of two ways: (1) By drawing tangents bb, cc, to the curve at the points B, C, respectively, and drawing the straight line Od from the center O through the point of intersection d. (2) By drawing the chord, BC, and marking the radius OD through the midpoint 5 of this chord.* COMPARATIVE GEOMETRY OF COMPLEX CIRCULAR AND HYPERBOLIC FUNCTIONS We have seen that the real circular functions sin x, cos x, may be derived from a circle diagram, and that the real hyperbolic functions sinh x, cosh x, may be similarly derived from a rectangular hyperbola diagram. We shall see that both the complex circular functions sin (x + iy), cos (x + iy), and the complex hyperbolic functions sinh (x + iy), cosh (x -\- iy), may be derived from a combination circle and hyperbola diagram. COMPLEX CIRCULAR FUNCTIONS CONSTRUCTION FOR sin (x iy), AND FOR sin" 1 (u in) In Fig. 19, take OA = i along the negative side of the Y axis. From OA as initial line, mark off the circular angle x = AOB. From OB as initial line, mark off the hyper- bolic angle y and its sector BOD. Let C be the foot of the perpendicular from D on OB produced. Drop perpendiculars from C and D on the axis of reals OX, at c and d respectively. About c as center, rotate cd positively through 90 to cZ. Then will * This proposition is proved in GreenhilFs " Differential and Integral Calculus," Macmillan & Co., 1896, page 67, Fig. 16, for the particular case when the angle AOB, in our Fig. 18, is zero. The demonstration of proposition (i) for the general case of Fig. 18 is not difficult; but that found by the author is rather lengthy. The demonstration of the general proposition (2) is, however, brief and direct, as follows: Let 0i be the hyperbolic angle of the sector AOB. Let 02 be the hyperbolic angle of the sector AOC. Then it is required to show that S = hA_ = hA = Of OA i But from an inspection of the Figure, Be = sinh 0i, Cg = sinh 02, Oe = cosh 0i, Og = cosh 02, eB + gC sinh 0i + sinh 02 so that 2 2 Oe -4- Og cosh 0i + cosh 02 and Of = 2 2 _ jft _ sinh 0i -\- sinh 02 Of~ cosh 0i + cosh 02 io tions," 1909, p. XIV, Formula (49). (n I rt \ which is a known equivalent expression for tanh , see Becker and Van Orstrand's "Hyperbolic Func- [164] EXPLANATORY TEXT the complex vector OZ = Oc + icd be the required circular sine of the complex angle x + iy radians. In the case represented, sin (i + i i) = 1.299 + ^0.635 = 1.446 /26.o5. As y varies, Z moves along the hyperbola bZ : X 2 Y* sin 2 x cos z x and as x varies, Z moves along the ellipse : X 2 Y 2 = i (14) cosh 2 y ' sinh 2 y Both the hyperbola and the ellipse have as common foci FF', the points X = i, F = o. 7 FIG. 19. Constructions for sin (* + iy) and sin- 1 (u =*= I'D). From the same figure, we have also, if Oc = u and cZ = iv, sin" 1 (u iv) = sin" 1 Ob i cosh" 1 OE /(IT uY + v 2 - Vd - ) 2 + v 2 \ 2 / = sn i (16) ^ FZ ~ F ' Z A KV FZ + F ' Z since Ob = and OE = CONSTRUCTIONS FOR cos (x + iy) AND FOR cos" 1 (u + iv) In Fig. 20, take OA as unit distance along the real or X axis, in the positive direction. From OA as initial line, describe the circular angle x, or the circular sector AOB of area x/2. On OB as initial line, describe the hyperbolic angle y, or the hyperbolic sector area BOD of area y/2. Let C be the foot of the perpendicular from D on OB produced. Drop perpendiculars from C and D on the X axis at c and d respectively. With c as center, rotate the line cd in the positive direction through 90 into the position cZ; so that ~cZ = i.~cd. Then the complex quantity OZ = Oc + *'.cd will be the required circular cosine of the complex angle (x + iy) radians. [165] EXPLANATORY TEXT In the case represented, cos (i + i i) = 0.834 * 0.989 = i.293\49.866. As y varies, Z moves along the hyperbola bZ defined by X 2 Y 2 = i cos^ x sirr x and as x varies, Z moves along the ellipse ZE, defined by (18) cosh 2 y sinh 2 y Both the hyperbola and the ellipse have as common foci FF', the points X = i , Y = o. FIG. 20. Constructions for cos (x == iy) and cos" 1 ( in). From Fig. 20 we obtain: cos" 1 OZ = cos" 1 (u iv) = cos" 1 Ob =F cosh" 1 OE E 2 1 (19) snce = FZ ~ F ' Z and OE = COMPLEX HYPERBOLIC FUNCTIONS CONSTRUCTIONS FOR sinh (x iy) AND sinh" 1 (u ') In Fig. 21, take CM as unit length along the real or X axis in the positive direction. From OA as initial line, describe the circular angle y, or the circular .ector AOB of area y/z. From OB as initial line describe the hyperbolic angle x, or the hyperbolic sector BOD of area x/2. Let C be the foot of the perpendicular from D on OB pro- [166] EXPLANATORY TEXT duced. Drop perpendiculars from C and D on the F axis at c and d respectively. With c as center, rotate the line cd negatively, or clockwise, through 90 to cZ. The complex quantity OZ = Oc i.cd will be the required hyperbolic sine of the complex angle (x + iy) radians. In the case represented, sinh (i + * i) = 0.635 + * I - 2 9 8 5 = 1.446/63. 95. As x varies, Z moves along the hyperbola Zbz: F 2 X 2 = i sin" y cos" y and as y varies, Z moves along the ellipse XExy F 2 Y 2 7 i - V (20) (21) cosh 2 x sinh 2 # The hyperbola and ellipse are confocal at the points F and/ defined by X = o, F = i. FIG. 21. Constructions for sinh (* * iy) and sinh" 1 (u =t in). From Fig. 21 we also obtain sinh" 1 ( w) = sinh~ l (cZ Oc) = cosh- 1 OE * sin" 1 Ob. = cosh" 1 db i sin" 1 (22) since [167] EXPLANATORY TEXT CONSTRUCTIONS FOR cosh (x -f- iy) AND cosh" 1 (u + iv) In Fig. 22, take OA as unit distance along the real or X axis in the positive direction. From OA, as initial line, describe the circular angle y, or the circular sector AOB of area y/z. From OB, as initial line, describe the hyperbolic angle x, or the hyperbolic sector BOD of area x/2. Let C be the foot of the perpendicular from D on OB produced. Drop perpendiculars from C and D on the X axis at c and d respectively. About c, as FIG. 22. Constructions for cosh (x == jy) and cosh ' (u MI). center, rotate the line cd negatively, or clockwise, through 90 to cZ ; so that cZ = i.cd. Then the complex quantity OZ = Oc i.cd will be the required cosine of the complex angle (x + iy) radians. In the case represented, cosh (i + i i) = 0.834 + 20.989 = 1.293 749. 866. As x varies, Z moves along the hyperbola Zbz X 2 cos 2 y sin 2 }/ As y varies, Z moves along the ellipse EZez F 2 = i. i = i. (23) (24) cosh 2 x sinh 2 x The ellipse and hyperbola are confocal at the points A, a, denned by X = i, Y = O. Li68] EXPLANATORY TEXT From the same figure. If Oc u and cZ = iv cosh- 1 (u iv) = cosh" 1 (Oc i.cZ) = cosh- 1 OE i cos" 1 C _i_ ~_\ * cos" 1 = cosh" 1 V(l + ") 2 + ^ + V(l - U) 2 (25) CONSTRUCTIONS FOR tan (x fy) AND tan -1 (u *) In Fig. 23, lay off along the X axis a point A distant tan x from 0, and also a point B such that O.5 = cot x. Draw a circle through A and 5 having its center on OX at C. The distance OC measures cot 2x and the radius of the circle is cosec 2X. Any circle thus drawn will intersect the Y axis at two points e and / which are at unit dis- FIG. 23. Constructions for tan (* iy) and tan~' (u tances from 0. Then lay off the Y axis two points a and b, distant respectively tanh y and coth y from O. With center c on the Y axis, draw a circle through a and b. The distance Oc will be coth 2y, and the radius of the circle will be cosech 2j. Let Z be the point of intersection of the two circles. Then OZ is the required tangent of (x + iy). If x is kept constant but y is varied, the point Z moves over the circle AZB. If on the other hand y is kept constant, but x is varied, Z will move around the circle aZb and will make one complete revolution for each increase of TT units in x. [169] EXPLANATORY TEXT In the case represented, tan (i + i i) = 0.2718 + i 1.084 = 1.118/75. 916. From Fig. 23 it is evident that the angle AeO is equal to x, and angle eAO is thus the complement of x. Hence half the angle between r\_ and r 2 is the complement of x. Moreover ;y = log e V^/ tan -1 (u iv) = - tan- Therefore, if OZ = u + 1 ( ^ + tan- 1 \ v i (26) CONSTRUCTIONS FOR tanh (x iy) AND tanh- 1 (w n>) In Fig. 24 mark off on the axis of reals xOX two points T and -X" such that the former is distant by tanh x and the latter by coth x from the origin O. Find the point C mid- way between T and X. Incidentally, this point will be distant coth 2X from O. With FIG. 24. Constructions for tanh (x =*= 'y) and tanh 1 (w =*= i) center C and radius CT = CX = cosech 2x, draw the circle TXZ. Mark off on the axis of imaginaries yOY, two points t and y such that the former is distant by tan y and the latter by cot y from the origin O. Find the point c midway between them. Incidentally, this point will be distant cot 2y from O. With center c and radius ct = cy = cosec zy, draw the circle By At. This circle will cut the axis of reals at two points A and B distant each one unit from O. It will also intersect the circle TXZ perpen- dicularly at Z. Connect OZ. This vector OZ is the required hyperbolic tangent of the complex angle (x + iy) radians. [170] EXPLANATORY TEXT In the case represented, tanh (i -f- i i) = 1.084 + 20.2718 = 1.118/14. 084. As x varies, Z moves along the circle AtB. As y varies, Z m^ves along the circle TZX, performing one complete revolution for each TT units of increase in y, From the same Figure, if OZ u iv = op ipZ, we have tanh" 1 (u iv) = x iy. In this case x = log e V r i/ r 2 or * = $log.(ri/r a ). (27) and y = - - where a is the circular angle at Z between the radii vectores r, and r 2 . 2 Also Hence DEGREE OF PRECISION OF TABLES INTRODUCTION If a numerical quantity, freed from decimals, is correctly expressed to within say i part in 1000; i.e., i part in io 3 , then this degree of precision may conveniently be described as precision of the third order. In general, therefore, if a numerical quantity be correctly expressed to within i part in io n , where n is any real positive number, its precision is of the nth order. The weekly statement of the financial assets of a bank might be expressed as $186,257,361.26 which, assuming that it is to be taken as being numerically correct to a single cent, represents 18,625,736,126 cents, an apparent precision of i in lo 10 ' 27 , or of the 10.2 7th order. Physical and astronomical precisions are less ostensibly pretentious, however, and rarely exceed the 6th order. Engineering computations are commonly satisfied with a precision of the third order; although, on rare occasions, the order required may be the highest that physics can attain. The degree of precision corresponding to retaining a specified number of significant digits correct within unity, in Tables, can only be stated approximately ; since it varies with the values of the digits. Thus, if we have tables containing entries each of three significant digits, correct to the last digit, the lowest entry may be 100 and the highest 999. The precision would therefore be i in 100 in the former case, and practically i in 1000 in the latter. That is the order of precision would vary between the second and the third. The average precision might be stated as of the 2. 5th order. Such tables of n significant digits lay claim to an average precision of the (n i/2)th order. Many tables are, however, employed in which the last digit is stated to be correct to the nearest digit; that is within half of unity. On that understanding, the precision of say a three-digit table would vary between 1/2 in 100 to 1/2 in 999 or between the 2. 3rd and the 3. 3rd order, with a mean of the 2.8th order. Consequently, we may say that such tables, giving n significant digits, lay claim to an average precision of the (n o.2)th order. [171] EXPLANATORY TEXT DEGREE OF PRECISION PRESENTED IN THE FOLLOWING TABLES The tables of complex hyperbolic functions here presented have been prvffciied with a view to giving five decimal places regularly. This means five significant digits when the values of the results lie between o.i and unity, six significant digits when they lie between i and 10, four when they lie between o.i and o.oi and so on. Tables I to VI inclusive were computed with the aid of five-figure logarithms of real hyperbolic functions, so that their degree of precision is necessarily limited to, and must on the average fall below that of such logarithm tables, which, as we have seen, is of the 4. 8th FIG. 25. sinh (p/5) expressed in polar coordinates. order. Exclusive of such mistakes as may exist, they do not claim a degree of pre- cision beyond the 4. 5th order. Tables VII to XIII inclusive were, however, computed for the most part from Lig- owski's gudermannian angles which are tabulated by him for each thousandth of a hyperbolic radian, to the nearest hundredth of a second of circular arc. The logarithms of the corresponding real hyperbolic functions were then found in the eight-place tables of Bauschinger and Peters, which offer such logarithms for each and every second of circular arc. The results were computed in the formulas to at least six significant [172] EXPLANATORY TEXT digits and the sixth was then frequently discarded to meet the needs 9f the five-decimal table. Consequently in this group of tables, excluding such errors as may exist, the precision is on the average of the 4.8th order, and rises to the 5. 8th order, when the value of the result lies between i and 10. The average precision of the second group of tables is thus about half an order greater than that of the first group. PRECISION OF THE CHARTS IN THE ATLAS The charts of the accompanying Atlas have been prepared with a view to offering three digits in the deduced quantity, if reasonable care be taken in their use. This represents an average degree of precision of the 2. 5th order; or about equal to that fur- FIG. 26. sinh in polar coordinates. nished by an ordinary 25 cm. slide rule. When a higher degree of precision than this is needed, arithmetical interpolation in the Tables must ordinarily be resorted to; but even then it is desirable to obtain a preliminary approximate value from the Atlas, in order to furnish a check against gross error. [173] EXPLANATORY TEXT GRAPHIC REPRESENTATIONS Figs. 25 and 26 present the results obtained from Table I to true polar coordinates. Each intersection of the curves corresponds to an entry in the table. Fig. 25 relates to pages 2, 3, and Fig. 26 to the rest of the table. The curves of constant p intersect those of constant 5 perpendicularly. That is, each intersection occurs theoretically at right angles. If, however, an attempt is made to prepare plates corresponding to Figs. 25 and 26 on a large scale, for a reasonable degree of precision, in rapid interpola- tion by graphical inspection, difficulties present themselves. Firstly, it has been found impracticable to procure polar coordinate ruled sheets large enough. Secondly, regular polar coordinate charts of the type presented in Figs. 25 and 26' necessarily offer very little graphical interpolation precision at small radial distances from the origin of co- 1 2 3 ^" s* i \ \ ^._ 1 Z i > 4 FIG. 27. Polar Coordinate Diagram Regular Presentation on Circular Sheet. FIG. 28. Polar Coordinate Dia- gram Squared Presentation on Rectangular Sheet. ordinates, where the radial lines, sharply converging, crowd the diagram. On the other hand, they offer relatively great apparent interpolation precision at large dis- tances from the origin, as the radii diverge. In preparing interpolation charts, therefore, the author has devised the scheme of using squared paper sheets for presenting such polar coordinate quantities as appear in Figs. 25. and 26. Fig. 27 represents the regular polar coordinate r/_y_ diagram, in which the lines of constant r are circles concentric at 0, and the lines of constant 7 are radii diverging from O. Fig. 28 represents the corresponding squared polar coordinate diagram, in which the lines of constant r are the vertical straight lines i-i, 2-2, 3-3, and 4-4. The lines of constant 7 are the parallel horizontal lines, while the origin point O in Fig. 27 becomes expanded into the original straight line o-o in Fig. 28. The straight, broken line DU is transformed into the curved line du, and, in general, orthogonally intersecting curves of one diagram do not transform into orthogonally intersecting curves on the other. [i74] EXPLANATORY TEXT INTERPOLATION CHARTS Plates IA, IB, and Ic of the Atlas correspond to Table I and present, to squared polar coordinates, the results in that table. Each intersection of curves in the plates corre- sponds to one entry in the table. Plate IA includes the entries on pages 2 and 3 of the table; while Plates IB and Ic include the entries in the remainder of the table. The curves of constant p and constant 5 intersect one another at various angles, but the method of interpolation requires little explanation. The entering quantity will fall within some particular curvilinear parallelogram. The respective opposite sides may be subdivided into tenths in any of the three following ways: (i) by direct inspectional estimate, (2) by graphical subdivision on a sheet of tracing paper laid over the chart, (3) by means of a radiating decimal scale of lines, prepared in advance, on tracing paper or thin celluloid. It is not, in general, worth the effort of attempting a closer subdivi- sion than tenths of the sides of any parallelogram. The point of intersection of lines parallel to the sides, through the correct decimal points, is then to be marked on the covering tracing paper, or held with a blunt pointer, such as a knitting needle, on the chart itself, and the rectangular coordinates of this point read off from the parallel ruling or background of the plate. That is, the charts are always used with the entering variable on the curvilinear coordinates, and with the result found on the rectilinear frame- work in the background; except when inverse functions are sought, and the procedure consequently reversed. TABLE I sinh (p/8) = ry POLAR HYPERBOLIC SINES OF A POLAR VARIABLE Table I, pages 2 to 7, gives the hyperbolic sine of vectors up to 3.0 in modulus, by steps of o.i, for each degree of argument from 45 to 90. The results are also expressed in polar coordinates, as plane vectors, corresponding to the relation: sinh (p/5) = r/7 (30) or p/d_ = sinh- 1 (7/7). (31) The graphs of the results to true polar coordinates appear in Figs. 25 and 26, where the curves of constant p always intersect orthogonally the curves of constant d ; so that at any point of intersection the angles of intersection are right angles. In Plates IA, IB, and Ic of the Atlas, the same graphs are given to squared polar coordinates, the disadvantages of the distortion being more than outweighed by the advantages in facility of graphic interpolation. In these charts the curves of constant p do not inter- sect the curves of constant 8 orthogonally. [175] EXPLANATORY TEXT INTERPOLATION. FIRST CASE. IN MODULUS ONLY If Table I is entered with a vector quantity of more than one decimal in modulus and of some exact degree of argument, such as 2.76/70; then the result will lie nearly on the line between the results for 2.7/70 and 2.8/70; namely, between 1.2 and i. 2136/143. 005. A first approximation may be obtained by proportional parts between them, thus : Required sinh 2.76/70 by Table, sinh 2.80/70 = 1.2136/143. 005. by Table, sinh 2.70/70 = 1.2031/136. 489. Difference 0.10/70 = 0.0105 /6.5i6. Proportion for 0.06/70 = 0.0063 /3.Qio. sinh 2.70/70 = i. 2031/136. 489. Result sinh 2.76/70 = 1.2094/140. 399. The true value is 1.2086 /140.366. INTERPOLATION BY THE USE OF TAYLOR'S THEOREM When more precise interpolation is required than that by simple intermediate pro- portion, we may use Taylor's theorem in the following form; since -cosh*, =sinh0, etc. d0 sinh (0 + A0) = sinh + A0 cosh + sinh + - cosh + . . (32) 2! Let = p/5 and A0 = Ap/5. Then sinh \(p + Ap)/ j = sinh (p/6) +Ap/5. cosh (p/5) +~ M sinh (p/3) + . . . (33) The number of correction terms to be retained depends on the interval, and on the degree of precision desired. It is seldom that more than two correction terms have to be retained. Thus in the case already considered: sinh (2. 76/70) = sinh (2. 7/70) -f- 0.06/70. cosh (2. 7/70) +0.0018/140. sinh(2. 7/70). By Table II, page 13, cosh (2.7/70) = 1.3422/153. 322. Consequently dealing first with the first correction term only: sinh (2.76/70) = i.203i/i36.489 + 0.06/70 X 1.3422/153. 322 = i. 2031/136. 489 + 0.08053/223. 322 = i. 2031/136. 489 (i + 0.066937/86. 833) = i. 2031/136. 489 (i + 0.003698 + i 0.06684) = i 2031/136. 489 (i. 00592 /3.8io) = i. 2102/140. 299. [176] EXPLANATORY TEXT Taking next the second correction term into account. sinh (2.76/70) = i. 2102/140. 299 -}- 0.0018/140 X i. 203 1/136. 489 = i. 2102/140. 299 + 0.002166/276. 489 = i. 2102/140. 299 (i -+ 0.001789/136. 190) = i. 2102/140. 299 (i 0.00129 + i 0.00124) = i. 2102/140. 299 (0.99871 -f-io. 00124) = i. 2102/140. 299 (0.99871 /o. 067) = 1.2086/140.366. The correct result is 1.2086/140.366. (34) SECOND AND GENERAL CASE. INTERPOLATION BOTH IN MODULUS AND ARGUMENT Let the entered quantity be sinh (1.025/80.75). We have from Table I the four nearest results as follows: sinh 1.0/80 = 0.85125/83^.489. sinh 1.1/80 = 0.90416/84^.286. sinh 1.0/81 = 0.84940/84. 156. sinh 1.1/81 = 0.90172/84. 877. Difference for i = 0.00185 + o.667. Proportion for 0.75 = 0.001388 + 0.500. sinh i. 0/80. 75 = 0.84986/83. 989. sinh i.i/8o sinh i.i /8o.75 = 0.90233/84. 729. Difference for o.i = + 0.05247 /o.74o. Proportion for 0.025 = + 0.01312 /o.i85. sinh i.Q25/8o.75 = 0.86298/84. 174. The true value = 0.863 72/84. 166. Diff. for i = 0.00244 + o-59i- 0.00183 + 0.443. = 0.90233 (35) DUAL INTERPOLATION BY THE USE or TAYLOR'S THEOREM Let the nearest tabular function be sinh = sinh (p/5_) and the required function sinh (0 + A0) = sinh | (p + Ap)/d + A8\. Then (p + Ap) /S + A3 = p/S + (Ap + * P A5). (36) where the increment Ap + *pA5 is taken with reference to the vector axis p/5. Refer- ring this increment to the initial axis of reference, + A0 = (p + Ap)/5+AS = p/5 + V(Ap 2 + ( P A6) 2 /& + tan' 1 - (37) /& So that A0 = V (Ap) 2 + (pAS) 2 /8 + tan" 1 (pA3/Ap). (38) When, however, A5 is not very small, the last formula may contain an appreciable error, and the following method of deducing A0, using rectangular complex quantities, is to be preferred. [i77] EXPLANATORY TEXT Let = p/_8 = x + iy. and + A0 = (p -f Ap) /8 + A3 = x + A* + * (y + Ay). Then A0 = A* + i. (39) (40) = V(Aa) 2 + (Ay) 2 /tan- 1 (Ay/As). (41) We then have by Taylor's theorem, as before, sinh (0 + A0) = sinh + A0. cosh + - - . sinh + - - . cosh + . . . (42) 2 3.2 a series in which two correcting terms only need ordinarily be retained. Thus, in the example last considered, = 1.0/80 and + A0 = 1.025/80. 75. If we form A0 by the use of (37), we have Ap = 0.025, A5 = o.75 = 0.01309 radian, pA5 = 0.01309. A0 = V(o-25) 2 + (o.oi309) 2 /8o + tan" 1 (0.01309/0.025) = 0.02822/80 + 27. 637 = 0.02822 /107.637. If we form A0 by the use of the rigid formula (41) + A0 = 1.025/80. 75 = 0.164761 + i 1.0116715. = 1.0/80 = 0.173648 + ^0.9848078. A0 = 0.008887 + 30.0268637 = 0.028295/108 .306. Entering now the correction formula (42), we find in the tables: sinh 1.0/80 = 0.85125/83. 489, cosh 1.0/80 = 0.57991/14. 521. so that sinh 1.025/80. 75 = sinh 1.0/80 + 0.028295 /io8.3o6 X cosh 1.0/80 6 = 0.85125/83. 489 -fr- 0.028295 /io8.3o6 X 0.57991/14. 521 + 0.0004003/2 1 6. 6 1 2 X 0.85125/83. 489 + o.ooo ooi /324.9i8 X 0.57991/14. 521. It is evident that in conformity with the precision of the tables, only the first two correction terms need be included. Taking the first into account, we have: sinh i. 025/80. 75 = 0.85125/83^.489 + 0.028295/108. 306 X 0.57991/1. = 0.85125/83^.489 + o.oi64o8/i22.827 = 0.85125/83/^489 (i + o.oi9276/39.338) = 0.85125/83^.489 (i + 0.014909 + 30.012219) = 0.85125/83. 489 (1.014909 + 30.012219) = 0.85125/83^.489 X i. 01498/0. 690 = 0.86400/84. 1 79. [178] EXPLANATORY TEXT Taking up the second correction term: sinh 1.025/80. 75 = 0.86400 784. 179 + 0.0004003 /2i6.6i2 X 0.851 25 783. 489 = 0.86400/84. 179 + 0.0003405 /3oo. IQI = 0.86400/84. 1 79 (i + 0.000395/215. 922) = 0.86400/84. 1 79 (i 0.00032 20.000232) = 0.86400/84. 1 79 (0.99968 to. 000232) = 0.86400/84. 1 79 X 0.99968X0. 013 = 0.863727 /84.166. The true value is 0.86372 /84.166. CONCLUSIONS In general, dual interpolation by simple proportion, as in (35), will give a result of the third order of precision. In order to secure precision of the fourth order, inter- polation by the use of Taylor's theorem as in (42) may be required. EXTENSION or TABLE BY USE or FORMULA FOR 28 Although Table I is only carried as far as 3.0 in modulus (p 3) ; yet it may be used with a little additional calculation in conjunction with Table II, for obtaining the hyp. sines of plane vector quantities of moduli up to 6.0, by means of the formula: sinh 29 = 2 sinh cosh 6 (43) Example: Required sinh 5.0/77, a quantity outside of Table I. Here 6 = 2.5/77 is within the limits of the Table; so that sinh 5-0/77 = 2 X sinh 2.5/77 X cosh 2.5/77 (44) = 2 X 0.87843 /i 20. 891 X o.96459/i56.524 = 2 X 0.87843 X 0.96459/277^.415 = 1.75686 X 0.96459/277^.415 = 1. 69465 /277.41S. This method ordinarily calls for interpolation both in sinh 6 and cosh 6. For this reason, it may be preferable to obtain the required result by the use of either Table VII or Table X, the limits of which are less restricted. TABLE II cosh (p/J>) = r[i_ POLAR HYPERBOLIC COSINES OF A POLAR VARIABLE Table II gives the value of cosh p/5 between the limits of p = o and p = 3.0 by steps of o.i, and the limits 5 = 45 and 5 = 90, by steps of i. The graphs of these quantities, to squared polar coordinates, appear in Plates HA and HB of the Atlas. [179] EXPLANATORY TEXT INTERPOLATION BY SIMPLE PROPORTION In general, as in the case of Table I, a very fair degree of precision in interpolation can be obtained by taking first simple proportional parts in argument, and then simple proportional parts in modulus. Example: Required cosh (0.931 05 1 We have from Table II: - cosh 0.9/57 = 0.88922/23. 140. cosh 0.9/58 = 0.87602/23. 003. Difference for i = 0.01320 /o.i37. Diff. for 0.518 = - 0.00685 /o.o7i. cosho. 9/57. 518 = 0.88237 /23.o69. cosh 1.0/57 = cosh 1.0/58 = Difference for i = 0.01626 /o.o94. Diff. for 0.518 = 0.00844 /o.Q49. cosh i.o/57.5i8 = 0.87132 /28.868. cosh 0.9 /57-5i8 = 0.88237 /23.o69. Difference for o.i = 0.01105 /5-799- Diff. for 0.3105 = 0.00343 /i.8oo. cosh 0.93105/57. 518 = 0.87894 /24.869. The correct value is 0.87837 /24.803. INTERPOLATION OF TAYLOR'S THEOREM When a higher degree of precision is required than can be expected from simple proportional parts, we may use Taylor's Theorem in the following form: cosh (0 + A0) = cosh + A0 sinh + cosh + *=f- sinh + . . . (45) " ; 3- Example: Required cosh 0.93105/57. 518 having given in Table II cosh 0.9/57 = 0.88922/23. 140 and in Table I sinh 0.9/57 = 0.85414/64^.^8. 0.93105/57.518 = 0.500 + * 0.785398. 0.900 /57 = 0.49018 + 20.754804. A0 = 0.00982 + 20.030594 = 0.03214/72. 196. cosh 0.93105/57. 518 = cosh 0.9/57 + 0.03214/72. 196 X 0.85414/64. 218 + o. 00052 /i44.392 X 0.88922/23. 140. [180] EXPLANATORY TEXT It is evident that for the Tables here considered only two correction terms need be included. Taking up the first correction term, cosh o.93io5/57.5i8 = 0.88922 / 2^.140 + 0.03214/22^10)6 X o.854i4./64.2i8 = 0.88922/23. 140 + 0.02745/136. 414 = 0.88922/23. 140 (i + 0.03087 /i 13. 274) = 0.88922/23. 140 (i 0.01220 + * 0.02835) = 0.88922/23. 140 (0.98780 + i 0.02835) = 0.88922/23. 140 X 0.98780 (i + i 0.02870) = 0.88922/23. 140 X 0.98780 X i. 00041/1. 645 = 0.87873 /24.?85. Taking up the second correction term: cosh 0.93105/57. 518 = 0.87873/24. 785 + 0.00052/144. 392 X 0.88922 /23.i40 = 0.87873/24. 785 + o.ooo46/i67.532 = 0.87873/24. 785 (i + o.ooo524/i42.747 = 0.87873/24. 785 (i 0.000416 -f- i 0.000317) = 0.87873 /24.785 (0.999584 + * 0.0003 1 7) = 0.87873 X 0.999584 /24^7_8 (i + i 0.00032) = 0.87837 /2 4 .785 X 1/0.018 = 0.87837 /24.8Q3. The correct value is 0.87837 /24.8Q3. GRAPHICAL INTERPOLATION For rapid but less precise work, interpolation may be made by proportional parts on Plate HA or Plate HB, without arithmetical computation. TABLE III tanh (p/3) = r/y POLAR HYPERBOLIC TANGENTS OF A POLAR VARIABLE Table III gives in polar coordinates the value of tanh p/3 between the limits p = o and p = 3.0 by steps of o.i, and the limits 8 = 45 and 8 = 90, by steps of i. The graphs of these quantities, to squared polar coordinates, appear in Plates HlA and Ills of the Atlas. INTERPOLATION BY SIMPLE PROPORTION In general, as in the cases of Tables I and II, a very fair degree of precision can be obtained by taking first simple proportional parts in argument and then simple propor- tional parts in modulus. Example: Required tanh (0.93105/57. 518). [181] EXPLANATORY TEXT We have from Table III: tanh 0.9/57 = 0.96056 /4i.o78. tanh 0.9/58 = 0.97069/42. in. Difference for i = 0.01013 /i.o33. Diff. for 0.518 = 0.00525 /o.535. tanh 0.9/57. 518 = 0.96581/41. 613. tanh 1.0/57 = 1.06648 /37.Q3S. tanh 1.0/58 = 1.08054 /38.oo4. Difference for i = 0.01406 /o.g6g. Diff. for 0.518 = 0,00728 /o.5Q2. tanh i.o/57.5i8 = 1.07376 /37-537- tanh 0.9/57.518 = 0.96581 /4i.6i3. Difference for o.i = 0.10795 / 4-Q76. " for 0.3105 = O.03352/ i.266. Inferred value of tanh 0.93105 /57.5i8 = 0.99933 /40.347. Correct value of tanh 0.93105/57^.518 = 1.0000 /40.395. INTERPOLATION BY TAYLOR'S THEOREM For a higher degree of interpolation precision than by simple proportion, we may use Taylor's theorem in the following form: (A0) 2 tanh (0 A0) = tanh 6 + A0 sech 2 - (A0) 3 2! 2 sech 2 tanh0 2 sech 2 (sech 2 - 2 tanh 2 0) (46) Example: Required tanh 0.93105/57. 518. having given in Table I sinh 0.9/57 = 0.85414/64.218. II cosh 0.9/57 = 0.88922/23. 140. Ill tanh 0.9/57 = Q.96o56/4i.o78. Here A0 = 0.03214/72. 196, as given by (41). Hence by Taylor's theorem as far as the second correction term inclusive, 0.03214 2.i6 tanh 0.93105/511518 = tanh 0.9/52! + (0.8892 2) (o.o32i4) 2 /i 44 .392 - (0.88922) 2 / 4 6. 2 8o Taking up the first correction term: tanh o.93iQ5/57.5i8 = 0.96056 / 4 i.Q78 + = 0.96056/41. 078 + 0.04065/25. 916 = 0.96056 /4i.o78,(i + 0.04084 20.01107) = 0.96056/41. 078 (1.04084 i 0.01107) = 0.96056 /4I.O78 (1.04090X0. 609) = 0.99985 /4o.469. [182] EXPLANATORY TEXT Taking up next the second correction term: tanh o.93io5/57.5i8 = 0.99985/40^469 0.00103 /i44-3Q2 X 0.96056/41^078 (o.88922) 2 /46.28o = 0.99985/40^.469 + 0.00103 X 0.96056X40. 810 0.79071 = 0.99985/40. 469 + 0.00126X40. 810 = 0.99985/40. 469 (i + 0.00126X81.279) = 0.99985/40. 469 (i + 0.00019 ~~ * 0.00125) = 0.99985/40. 469 (1.00019 ~ * 0.00125) = 0.99985/40. 469 X i.oooi9\o.o72 = i.oooo /4o.- Correct value = 1.0000 /40.395. When more than two correction terms have to be retained, it is often easier to deter- mine sinh (8 + A0) and cosh (6 + A0) by Taylor's theorem, as already described, and then to take their ratio for tanh (6 + A0). TABLE IV sinh POLAR RATIO - FOR POLAR VALUES OF v Table IV has been prepared by dividing the values of sinh found successively in Table I by their respective values of 0. The object of the table is to facilitate the computation of the equivalent T or II of any uniform alternating-current line of known electrical constants.* That is, the table pertains more particularly to the applications of hyperbolic functions than to the fundamental properties of those functions. The sinh (p/5) table gives the vector value of - - for the range p = o to p = 3 by steps of o.i, p/5 and for 5 = 45 to 5 = 90 by steps of i. The graphs of the values contained in the tables are plotted to squared polar coordinates in Charts IVA and IVs of the Atlas, for rapid graphic interpolation. INTERPOLATION BY SIMPLE PROPORTION A fair degree of precision in interpolation can ordinarily be obtained by first taking simple proportional parts in argument and then simple proportional parts in modulus. sinh (1.025/80. 75) Example: Required - *"The Application of Hyperbolic Functions to Electrical Engineering Problems," by A. E. Kennelly, University of London Press, 1914, Chap. III. [183] EXPLANATORY TEXT We have from Table IV the following values of - : u For 1.0/80 = 0.85125 /3.48g. For 1.1/80 = 0.82196 /4.286. 1.0/81 = 0.84940 /3.i56. 1.1/81 = 0.81975 /3.8y7. Difference for i = 0.00185 / -333- Difference for i = 0.00221 / o. 0.75 = O.OOI39/ o.25o. -75 = ~~ o. 00166 / o. For i.i/8o.75 = 0.82030 /^.g 0.84986 /3.239. For i.o/8o.75 = 0.84986 /3.2 Difference for o.i = 0.02956 /o.74o. for 0.025 = 0.00739 /o.i85. For i. 025/80. 75 = 0.84247 /3.424. Correct value = 0.84265 /3.416. When a higher degree of precision is required than can be expected from proportional parts, the proper value of sinh (6 + A0) should be obtained by Taylor's theorem as already explained in connection with Table I, and this value divided by (6 + Ad) ; because the expansion of , . directly, by Taylor's theorem, does not lend itself conveniently for computation. EXTENSION FOR THE RANGE OF THE TABLE BY THE USE OF FORMULA FOR 26 Although Table IV is only carried as far as 3.0 in modulus (p = 3); yet it may be used with a little additional calculation, in conjunction with Table II, for obtaining , for vector values of 6 with moduli up to 6.0, by means of the formula: a sinh 26 = 2 sinh 6. cosh 6 (47) sinh 26 sinh 6 whence - = - . cosh 6. (48) 20 6 Consequently, to find - for the double of any quantity within the range of Table u IV, find the value of - - for the quantity, by interpolation directly in Table IV, and V multiply the result by the hyperbolic cosine of the quantity as obtained from Table II. Corresponding steps may be taken with Charts II and IV. sinh (5.0/77) Example: Required -; 5 , this being outside of the limits of Table IV; * iii, but not outside twice the value therein obtainable. Here for = 2.5/77 is by Table IV 0.35137 e and cosh 6 " " " " " II 0.96459 l [184] Hence sinh (5.0/77) 5-0/77 EXPLANATORY TEXT = 0-35137 /43-89i X Q.96459/i56.524. = 0.33893 /200 .415. sin 6 This procedure calls for interpolation both in - - and in cosh 6. For this reason e it may be preferable to obtain the required result by the use of either Table VII or Table X, the limits of which are less restricted. TABLE V tanh 6 POLAR RATIO - FOR POLAR VALUES OF u Table V, like Table IV, has been prepared for electrical engineering applications of hyperbolic functions, rather than for developing these functions alone. It gives the tanh (p/6) vector value of p/5 for the range p o to p = 3.0 in modulus, by steps of o.i, and for the range 5 = 45 to 8 = 90 in argument, by steps of i. It was computed directly from Table III by dividing the resulting values successively by their respec- tive values of 6. The graphs of the values in Table V are presented to squared polar coordinates in Chart V, for rapid graphic interpolation. INTERPOLATION BY SIMPLE PROPORTION Except where a high degree of precision in interpolation is required, it is prefer- able to interpolate first by simple proportion in argument, and then by simple pro- portion in modulus; although this order of operations may be inverted. Example: Required -r for 6 = 0.93105^ 9 We have from Table V: For 6 0.9/57 = 1.06729 \i5.922. 6 = 0.9/58 = 1.07854 \i5.88 9 . For 6 = 1.0/57 = i- 06648 \i^< Difference for i = o.oii25\ 0.033. for 0.518 = o.oo583\ 0.017. For = 0.9/57. 518 = J -73 12 -95- 6 = 1.0/58 = i.o8o54\i9.996. Difference for i = 0.01406 \o.o3i. " for o.5i8 = 0.00728 \o.oi6. For = i.o/57.5i8 = 1.07376X19. 981. 6 = o.9/57-5i8 = 1.07312 \i5-905. Difference for o.i = 0.00064 \4-o76. " for 0.03105 = 0.00020 \i.266. For0 = o.93iQ5/57-5i8 = 1.07332X17^171. Correct value, [185] 1. 07406 \17.123. EXPLANATORY TEXT When a higher degree of precision is needed than simple proportion can give, it is preferable to find the proper interpolated value for tanh Q from preceding tables and then to divide by 0; since the function - -. r does not lend itself to expansion (0 + A0) by Taylor's theorem in a simple form. Tables IV and V jointly, with their respective graphs in the Atlas, enable the equiva- lent T or II of any uniform alternating-current line in the steady state, at a single fre- quency, to be completely determined, provided does not exceed six radians in modulus (5 lying between 45 and 90) ; because although in both tables, is not carried beyond three radians; yet can be found by extension up to six radians, and in the formu- las for deducing the equivalent T or II, - - has only to be carried to half the modulus , sinh of -0-' The following example may illustrate the use of Tables IV and V either with or without the aid of the graphic interpolation Charts IV and V of the Atlas. An alter- nating-current line of uniform electrical constants is 250 km. long and has, at a certain frequency, a total conductor impedance of 565.711 784. 777 ohms, associated with a total distributed insulation admittance of 4.3707 X io~ 3 /9o mhos. Its hyperbolic angle is therefore Vs. 65711 X 4.3707 X io~ 1 /i74 .777 = 1.5724 787. 388 hyperbolic radians. Interpolating either from the tables or the Charts IV and V, we obtain sinh0 .tanh () tanh 0.7862/87^88 = 0.638 / . _ . 786a/8 If we multiply the conductor impedance by - , we have u 565. 711 784. 777 X 0.638/2. 6 = 360.69/87. 377 ohms, and if we multiply half the insulation admittance by ff) . g , we have Iff/ 2.1854 X io- 3 /9o X i.27\i.5 = 2.78 X io- 3 /88.g mhos = 359-77\88.5 ohms. If now we apply an artificial condenser leak of 2.78 X io-y88.5 mhos to each end of a localised impedance coil of 3 60. 6 9/87^.3^7 ohms, we obtain the "equivalent II" of the line at the frequency considered, and such a combination of localised impedance and ad- mittances would behave exactly like the line, at its terminals, or outside them, so as to be capable of replacing the line in any electrical system, at that frequency. [186] EXPLANATORY TEXT TABLE VI POLAR FUNCTIONS OF POLAR SEMI-IMAGINARY QUANTITIES A semi-imaginary quantity is a complex numerical quantity which, when expressed in rectangular coordinates, has equal real and imaginary components; or, when ex- pressed in polar coordinates, has an argument of 45. That is x/ 45 = a rt i a. The interest of the table pertains primarily to the application of hyperbolic functions to uniform alternating-current lines of negligibly small linear inductance and leakance, a case approximated to by cabled lines at low frequencies. The table was first pub- lished by the author in the transactions of the International Electrical Congress of St. Louis (1904). The arguments of the results are given in degrees and minutes, and not in degrees and decimals like the rest of the tables. The table gives the hyperbolic sine, cosine, tangent, cosecant, secant, and cotangent of the vector x/4$ for the range x = o to x '= 20.5, by steps in x of o.i up to x = 6, and of 0.05 beyond that point. At x = 6, the values of the hyp. sine and cosine so nearly coincide, that they are taken as equal in the table, thus bringing sech x and cosech x into equality as well as tanh x = coth x = i. Graphs of the functions are given in Chart VI as far as x = 4, approximately. INTERPOLATION BY SIMPLE PROPORTION In general, interpolation may be quickly effected by simple proportional parts of modulus since the argument is constant at 45. This procedure is sufficiently evident to require no exemplification. INTERPOLATION OF TAYLOR'S THEOREM When precise interpolation is necessary, we have the following expansions for f (0 + A0) sinh (* + A*) /4S! = sinh (2/45) + (Ax) / 4S . cosh (*/ 4 5) + sinh (*/45) + -=r/ilS. cosh (*/ 4 5) + ..... (49) O " cosh | (* + A*)/45} = cosh (x/4S + (A*)/45_. sinh(*/45) + ^fl^L< (Ar^ 3 cosh (*/45) + ^fr/ias!- sinh ( *^ 0) + (5o) , tanh (* + A*)/4,S! = tanh 2 sech 2 (/4S). tanh (s/ 4 Q - Zl^. . 2. sech 2 (*/45). J sech 2 (*/ 45 ) - 2 tanh 2 (s/ 4S ) *+.... (51) Example: Required cosh (3.1/45), having given in Table VI sinh (3.0/45) = 4.1986 /i 2o.48 / . cosh (3. 0/45) = 4-U43 /i22.i6 / . [187] EXPLANATORY TEXT Here cosh (3.1/45!) = 4.i443/i22.i6' + 0.1/45 X 4-1986 /i2o. 4 8' 0.01/90 o.ooi , + - - X 4.i443/i22.i6' H /i35 X 4.1986 / 120. 48' = 4.1443 /I22.l6' + 4.1986/120. 48 (0.1/45 + 0.00017/135) + 4. 1443 /I 22. 1 6' (0.005/90) = 4. 1443 /i 2 2. 1 6' (i + 20.005) + 4.i986/i65.48' (o.i + 20.00017) = 4.i443/i22.i6' ( i. 0000/0. 1 7') + 4. 1986/165. 48' (o.i /o.oi') = 4.1443 /i 22^33' + o.4i99/i65.49 / :2.33 / (i + o. 10132/43. 16') = 4.1443 /i 22.33 / (i + 0.07378 + * 0.06944) = 4.1443 /I22.33 / (1.07378 + * 0.06944) = 4.I443/I22.33' X 1.0760 /3.42 ; which is in substantial agreement with the tabulated value of cosh 3.1. Beyond x = 6, the value of either sinh (3/45) or cosh (#/45!) was computed from the formula: sinh (s/45) = cosh (s/45) = J /T/T radians. (52) v ^ where e = 2.71828 . . . X 6 141.14 Thus, with x = 7, T= = 4.9498, - = - = 70.57 at the argument of 4.9498 cir- \j 2 ^2 cular radians = 283. 36'; so that: sinh (7/45) = cosh (7/4.5!) = 7Q.57/28 3 .36 / which coincides with the tabulated value in Table VI. TABLE VII sinh (x + iy) = u + vo RECTANGULAR HYPERBOLIC SINES OF A RECTANGULAR VARIABLE Tables I to VI contain certain restrictions in range which limit their general appli- cation. They are primarily designed to cover particular applications of hyperbolic functions to electrical engineering. Tables VII to XIV, however, are free from such restrictions, and are intended to furnish the circular as well as the hyperbolic sine, cosine, and tangent of a complex angle, and to furnish this result either in the rectangular or polar form. That is, they furnish: sinh (x + iy) \ . , . . . . . , , ; . ; > in the f orm u + vo ; also in the form r /j or sin (x + iy) j cosh (x -\- iy)\ . ^ f , ,1 e / . ( } in the form u + vo\ also in the form r /y or cos (x + iy) J tanh (x + iy) \ . . , , , . ( > in the form u + tv: also in the form r /y or tan (x + ly) } [188] EXPLANATORY TEXT between the limits, for the hyperbolic functions, x = o and x = 10, by steps of 0.05 and between the limits y = o and y = oc , by steps of 0.07854 = TT/^O. PERIODIC PROPERTIES OF THE RECTANGULAR COMPLEX HYPERBOLIC SINES AND COSINES It is well known that sinh \x + i (y -f 2mr)\ = sinh \x + iy\ (53) and cosh \x + i (y + 2mr) \ = cosh \x + iy\ (54) where n is any integer. This means that, keeping x constant, the values of the hyp. sine and hyp. cosine repeat themselves as iy passes through increments of i.2ir\ or they are periodic functions of iy, having the period 2ir i. The matter may be visualised more clearly from geometrical reasoning. Consider- ing the exponential form of the hyperbolic cosine, c * + 'V _L e -<* + ') cosh (x + iy) =- . (55) e* . e~ 1 This may written in the form: . e iy -\ e~ iv . If x be kept constant, we require to study the changes produced in this form of the hyp. cosine by varying y. q' $ %sZ**>* -a d ^~ -A P FIG. 29. Geometrical constructions for cosh (x + iy) and sinh (* + iy). In Fig. 29, OA is an initial line and OP a radius vector of length or modulus */ 2 > multiplied by e '; that is rotated positively about O, from OA through a circular angle of y radians. Similarly, op is a vector of length or modulus ~*/ 2 rotated negatively through a circular angle of y radians from the initial line Oa. The equation (55) states that the hyperbolic cosine is the plane vector sum of OP and op, or O'p' in the Figure. If now we steadily increase the value of y, leaving x constant, we cause OP to rotate steadily counterclockwise, and also op to rotate steadily clockwise, through Ay cir- cular radians. When Ay = 2ir, both OP and op will have made one complete revo- lution and will have returned to their initial positions indicated. Consequently, the value of cosh \x + i (y + 271-) \ repeats that of cosh \x + iy \. x + iy _ f -(x + iy) Since sinh (x + iy) = - (56) the same reasoning applies; but the vector op is added in the negative or reversed di- rection; so that O'q' is the hyperbolic sine of x + iy. The above mentioned periodic property of the hyp. sine and cosine has been utilized for shortening the tables of those functions by reducing the circular angle y of Fig. 29 [189] EXPLANATORY TEXT from radians to quadrants. That is, any complex angle x + iy represented by a point P, and radius vector OP, in the complex plane XY, Fig. 30, is first transferred to a new complex plane XQ, Fig. 31, at the point p = x, q, by keeping x the same in both planes, but making the points -, ,,,... etc., on the Y axis of the XY diagram, 2222 become the points i, 2, 3, 4 ... etc., on the Q axis of the XQ diagram. Thus if x + iy = 2.5 + * 6. 2832, x + iq = 2.5 + i 4.00 where 4.00 is underscored to indicate quadrant measure, instead of the ordinary radian measure. "Y FIG. .30. FIG 3 r. Transference of a Complex Quantity from the XY to the XQ Plane. In the case indicated by Fig. 30, x + iy = 3 + i 9 and x + iq 3 + * 5.74 in Fig. 31. Consequently, after a complex angle has been transferred from the complex plane XY to the complex plane XQ, the values of either sinh (x + iq) or cosh (x + iq) exactly repeat themselves for each 4 units of increase in q; or with reference to Fig. 31 for each 4 quadrants of increase in the circular angle instead of 6.2832 . . . radians. The operation of transferring the complex angle from the XY to the XQ plane may there- fore be described as quadranting y; i.e., changing the expression of y from circular radian units to circular quadrant units. All of the Tables VII to XIII inclusive require to be entered in terms of x + iq; so that the complex entering value has to be quadranted by dividing its imaginary or y-component by the numeric v/2 = 1.57079 This preliminary step occupies a certain extra time and effort; but it actually economises the total time and effort involved. If the tables were computed for x -f- iy, they would have to be repeated in bulk for each TT radians, or 2 quadrants, increase in y. In electrical engineering appli- [ 190] EXPLANATORY TEXT cations, y frequently rises to 100 radians, and might easily be much greater than 100. In order to go up to 100 radians, the bulk of the Tables VII to XIII would have to be increased about thirty fold. Altogether, aside from the greatly increased bulk and expense of such tables, the extra time and effort consumed in turning over the numerous pages would be comparable with that saved by eliminating the preliminary step of quadranting the imaginary component or dividing it by ir/2. RULES FOR THE USE OF TABLE VII Express the " angle " whose hyperbolic sine is required in the form of an ordinary rectangular complex quantity x + iy. Quadrant the imaginary component y through the process of dividing it by ir/2; i.e., transfer the quantity from the XY to XQ plane; so that the new expression of the complex quantity is x + iq; where q = y/i. 57079. . . . If q is greater than 4.6, divide by 4 and retainonly the remainder. If the remainder exceeds 2, subtract 2 therefrom, and apply a negative sign to the result found in the table. A change of 2 quadrants simply reverses the sign of the result. If the remainder on the other hand does not exceed 2, enter Table VII with it, and take out the result with unchanged sign. Example: Required the hyperbolic sine of 0.65 + i 25.75. Here x = 0.65 and y = 25.75. That is y is 25.75 circular radians. Reduce this to quadrants through dividing by 1.57079. . . . 25-75 * log 25.75 = 1.4107772. 1.57079 ... log TT/2 = 0.1961199. log 16.393 = 1.2146573. The quadranted yalue x + iq = 0.65 + i 16.393 NOTE. It is found convenient to underscore quadrantal quantities to distinguish them from radianal quantities. Rejecting quadrant multiples of 4, i.e., 16 in this case, we enter Table VII with x -|- iq = 0.65 + i 0.393. The nearest entry to this is x = 0.65, q = 0.4, for which the hyperbolic sine is 0.56368 + * 0.71639, an ordinary rectangular complex quantity on the U V plane. Interpolation should be made in this result to meet the change from q = 0.40 to q = 0.393, as w iH De explained later. Second Example: Required sinh (x + iy) = sinh (1.15 + i 10.10). . / . 10.10 \ Quadranting the imaginary, sinh (x + iq) = sinh ( 1.15 + * = 1 = sinh (1.15 + i 6.430). Rejecting 4'$ from the imaginary = sinh (1.15 + i 2.430). Deducting 2 from the residual and chang- ing the sign = sinh (1.15 -f *'o-43)- * This operation would ordinarily be effected with the slide-rule, when a high degree of precision is not aimed at. [191] EXPLANATORY TEXT We now enter Table VII with x = 1.15 and q = 0.43. The nearest entry is x = 1.15 q = 0.45, the result for which is 1.08037 + * 1.12836. But we must apply a negative sign to the whole of this result because of the 2 rejected in the quadrantal residuum. Hence, sinh (1.15 + i 10. 10) = (1.08037 + * 1-12836) = 1.08037 i 1.12836 = u + i v except for the interpolation from q = 0.45 to q = 0.43. The operation of interpolation will be discussed later on. Third Example: Required sinh (x + iy) = sinh (3.60 + i 18.1). Quadranting the imaginary, sinh (x + iq) = sinh ( 3.60 + i '^ } * \ i-57 sinh (0.15 -f 10.25) = sinh (o.i + i 0.2) j i -f- - p + . > + cosh (o.i + * 0.2) j A'0 + ^^ + . . . | \ 3 / = (0.09526 + i 0.31056) (0.99816 + i 0.00393) = 0.09386 + i 0.31036 + (0.95582 + i 0.03095) (0.04987 + i 0.07856) = 0.04524 + i 0.07664 = 0.13910 + i 0.38700. The correct tabulated value is = 0.13910 + i 0.38700. EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY Table VII expresses the relation sinh (x + iq) = u + iv. (63) (a) If x be taken with negative sign, we have sinh ( x + iq) = u + iv (64) so that changing the sign of the real component entering the table changes the sign of the real component in the result; but leaves the sign of the imaginary component unchanged. (b) If q be taken with negative sign, we have sinh (x iq) = u iv (65) so that changing the sign of the imaginary component in the entering quantity changes the sign of the imaginary component in the result, leaving the sign of the real com- ponent unchanged. (c) If both x and q be taken with negative sign, we have sinh ( x iq) = sinh \ (x + iq) \ = u iv = (u + (66) so that changing the sign of the total entering quantity changes the sign of the total result. The facts may be summed up by saying that changes in the sign of the entering quantity produce corresponding changes of sign in the result. [195] EXPLANATORY TEXT CIRCULAR SINES OF COMPLEX " ANGLES " Since, as is well known: sin 6 = i sinh (iff) (67) we have sin (x + iy) = i sinh (ix y) (68) = * sinh (y ix). (69) Consequently, in order to find the circular sine of the complex quantity (x + iy), enter Table VII for sinh (y + ix), which on being quadranted, becomes sinh \ y + ix/(ir/2) \ and let the result be (u + iv). Then sinh (y ix) = u iv and sin (x + iy) = v + iu. In other words, invert the entering components, and then invert the components of the result. Example: Required sin (i + i 2) from Table VII. Here 6 = (i + i 2). Enter the Table with sinh (id) = sinh ( id) = sinh (2 i i). Quadranting the imaginary, we enter the table with (x iq) = (2 i 0.6366). The nearest entry is (2 i 0.65), for which the hyp. sine is given as 1.89503 i 3.20780. Consequently, sin (i + i 2) = 3.2078 -f- i 1.89503, except in so far as interpolation is needed to reduce sinh (2 i 0.6366) from sinh (2 3' 0.65). In this way any circular sine of a complex quantity can always be obtained from the table of hyperbolic sines, between the limits of o and 4 in y, and of o and o in x. GRAPHIC INTERPOLATION BY MEANS OF CHARTS VIlA, Vila, VIIc Charts VII-VIII A, B, and c, serve for the evaluation of either sinh (x + iq) or cosh (x + iq), according to the axis of reference selected. Thus, taking Chart VII- VIIIs, if this is held with the. line SS as the axis of reference or initial line; then by comparison with the entries in Table VII, it will be found that sinh (x + iq) can be read from it directly over the range q = o to q = 4, beyond which the values repeat themselves indefinitely. On the other hand, if the chart be turned through 90, so as to bring the line CC as the axis of reference, it will be found by comparison with the entries of Table VIII, that cosh (x + iq) can be read from it directly over the range q = o to q = 4. Chart VII-VIIlA gives sinh (x + iq) and cosh (x + iq) for values of x up to about 0.9. Chart VII-VIIIs gives the corresponding results for values of x up to about x = 2. Finally, Chart VII-VIIIc provides for values of x up to x = 4. In all of these charts, interpolation can be made for both x and q to o.oi, by direct inspection. The graphs on these charts are undistorted, since they give complex functions as results, in rectangular coordinates. The curves therefore always intersect orthogonally, and they represent a confocal system of ellipses and hyperbolas, the common foci being at two points at unit distances from the center, along one of the reference axes. The curvi- linear rectangles into which the charts are divided have pairs of sides the ratio of whose lengths tends to the value ir/2. If the preliminary process of quadranting the imaginary of the entering quantity were not adopted; that is, if the graphs were entered in terms of (x + iy), instead of [196] EXPLANATORY TEXT (* + iq) ; then it would be necessary* to have a new chart for each range of 2ir units in 3>; or some 16 sets of Charts A, B, and C, in order to reach y = 100. That is, 48 charts would have to be computed, prepared, drawn, lithographed, bound, sold and operated instead of the 3 charts actually presented. Moreover, if y were needed greater than 100, the set of 48 would fail; whereas, working with quadrant imaginaries, the three charts serve up to indefinitely great values of q and y, GRAPHIC CHART VII-VIIlA This chart corresponds to Tables VII and VIII at least as far as x = 0.9, or for pages 42 to 45, and 58 to 61 of this book. To find hyperbolic sines from the chart, place it facing the observer with the axis OO vertical. This is the major axis of all the ellipses shown. Starting from this central axis towards the right hand, the successive ellipses marked o.i, 0.2, 0.3, etc., represent values of x; while the successively rising hyperbolas o.i, 0.2, represent values of q. These values of q will be found to extend over two quadrants. Enter the chart on the curvilinear coordinates for x and q. At the proper intersection read off the u and v coordinates of the rectilinear ruling, u being the abscissas and v the ordinates. Conversely, to find sinh" 1 (u + iv) within the limits u = o and = i,=oto v = 2.0, enter the chart with the same aspect on the rectilinearly ruled coordinates and read off at the proper intersection the curvilinear values taking x on the ellipses and q on the hyperbolas. To find hyperbolic cosines from the chart, rotate it clockwise 90; so as to have the axis OO horizontal. Then enter on the curvilinear coordinates with x on the ellipses and q on the hyperbolas. The first and fourth quadrants only will be presented to the observer; but from the symmetry of the diagram, it will be easy to reverse the chart, so as to present the second and third quadrants. Read off the result on the rectilinear background using u for abscissas and v for ordinates. Conversely, to find cosh" 1 (u + iv) from the chart with the axis OO horizontal, enter on the rectilinear background and read off at the proper intersection from the curvilinear coordinates in x and q, taking the ellipses as parts of the it-system and the hyperbolas as part of the ^-system. GRAPHIC CHART VII-VIIIs This chart gives the graph of the functions sinh (x + iq) and cosh (x + iq) from x = 0.8 at least as far as x = 2.05 along the ellipses and from q = o to q = o by virtue of successive rotations. In this and the following charts, the numerical values of q are all underscored, an indication which may serve readily to distinguish the imaginaries q, from the reals, x. * A single set of charts entered in terms of (x + iy) could be used up to y 6.2832 in one revolution, and could be used for all larger values by throwing out multiples of 2*-. This operation of dividing y by 2T would, however, take as much time as the operation of quadranting, and would also lead to a dissymmetrical chart in the hyperbolas. EXPLANATORY TEXT In all of the Charts VII to IX inclusive, the curvilinear rectangles all tend to have sides in the ratio TT : 2; that is the long side approximates to being 1.57 times the short side. In IXA exceptions are found; because extra curvilinear coordinates are supplied. To find sinh (x + iq) from Chart VII-VIIlB, hold the minor axis 55 horizontal. Enter on the curvilinear coordinates with x on ellipses and g on hyperbolas. At the proper intersection read off on the rectilinear background in u and v. Proceed inversely to find the inverse function sinh" 1 (u + iv). To find cosh (x + iq) from the same chart, hold the major axis CC horizontal. Enter on the curvilinear coordinates with x on ellipses and q on hyperbolas. Read off on the rectilinear background. All four quadrants appear in this and the following chart, so that it is not necessary to limit the value of q to less than 2 quadrants. GRAPHIC CHART VII-VIIIc This Chart gives the graph of sinh (x + iq) and cosh (x + iq) from x = 2.0, at least as far as x = 3.90. The procedure is precisely the same as that for VII-VIIlB already described. TABLE VIII cosh (x + iq) = + iv RECTANGULAR HYPERBOLIC COSINES or A RECTANGULAR VARIABLE Table VIII may be regarded as an inversion of Table VII; because: cosh 6 = i sinh (6 + iir/z) (70) or in quadrant imaginaries, cosh 6 = i sinh (6 + i i). (71) That is the hyp. cosine of any complex quantity (x + iq) is i times the hyp. sine of that quantity with an additional quadrant in the imaginary. Thus cosh (0.5 + io,6) = isinh (0.5 + i 1.6) = * ( 0.42158 + 20.66280) = + 0.66280 + * 0.42158. All of the entries in Table VII thus reproduce themselves by inversion in corresponding parts of Table VIII, a fact which serves as a numerical check upon both. In order to find the value of cosh (x + iy), quadrant the imaginary quantity y, by dividing it with IT/ 2, as in entering Table VII. The complex quantity (x + iy) will now be expressed as (x -\- iq); or will in effect have been transferred from the XY to the XQ plane. Next throw out multiples of 4 from q, so as to leave a remainder less than 4. If this remainder exceeds 2, deduct 2 from it, but change the sign of the total result thereupon deduced. If the remainder, however, is not greater than 2, then, the result is taken directly from the table. Example: To find cosh (1+25 ) = cosh (x + iy). Quadranting, we have cosh (i + 13.183) = cosh (x + iq). Deducting 2 from q, cosh (i + i 1.183). [198] EXPLANATORY TEXT With this we enter Table VIII. The nearest entry is x + iq = i +^1^2, the result for which is 0.47684 + i 1.11768. This has to be corrected by interpolation from q = 1.2 to q = 1.183. Reverse the sign of the result to 0.47684 i 1.11768 for the deduction of 2 quadrants. Example 2: Required cosh (0.25 + 2*30 ) = cosh (x + iy). Quadranting, this becomes cosh (0.25 + i 19.099) = cosh (x + iq). Rejecting imaginary quadruples = cosh (0.25 + 2*3.099) = cosh (x + iq). Deducting 2 quadrants = cosh (0.25 + i 1.099) = cosh (x + iq). The nearest entry is 0.25 + ii.i for which the result is ( 0.16135 + * 0.24950). Applying the negative sign on account of the two deducted quadrants, the final result is, neglecting interpolation, cosh (0.25 + i 30) = 0.16135 ~~ * o- 2 495o = u iv. INTERPOLATION BY SIMPLE PROPORTION A first approximation can be obtained by interpolating according to simple pro- portion. Example: Required cosh (0.55 + 2*0.55) = cosh (x + iq) having given cosh (0.6 + 20.5) = 0.83825 + i 0.45018. cosh (0.6 + i 0.6) = 0.69680 + 10.51506. cosh (0.5 + 2*0.5) = 0.79735 + i 0.36847. cosh (0.5 + 20.6) = 0.66280 + 2*0.42158. Diff. for x o.i = 0.04090 + 20.08171. Diff. for x o.i = 0.03400 -f- 20.09348. Diff. for x 0.05 = 0.02045 ~l~ 20.04086. Diff. for x 0.05 = 0.01700 + 2*0.04674. cosh (0.55 + i 0.6) = 0.67980 + 20.46832. cosh (o.55 + 2o.jj) = 0.81780 + 20.40933. cosh (0.55 + 2*0.5) = 0.81780 + 10.40933. Diff. for q cxi = 0.13800 + 2*0.05899. Diff. for (70.05 = 0.06900 + 2*0.02950. cosh (0.55 + 2*0.55) = 0.74880 + 20.43883. Correct value = 0.75018 + * 0.43963. INTERPOLATION BY TAYLOR'S THEOREM For a higher degree of precision than simple proportion affords, reference may be had to Taylor's theorem in the following form : cosh (6 + A0) = cosh 6 + A0 sinh Q + - cosh 6 + sinh + . . . 2! 3 ! cosh { (x -f- iy) + (Ax + 2' Ay) \ = cosh (x + iy) + (Ax + iAy) sinh (x + iy) + (^iM! cosh ( , + ^ + (a-M*y)' sinh ( , + iy )+... (7ia) EXPLANATORY TEXT Quadranting imaginaries on both sides, or transferring to the XQ plane, cosh \ (x + iq) + (A* + iAq) | = cosh (x + iq) + (Ax + iAq ir/2) sinh (x + iq) (Ax + iAq 7T/ 2 ) 2 + - * - u t \ cosh (* + ^q) , (Ax + 3*A<7 ir/2) 3 . , , . . + - f*- - sinh (x + 39) + . O " = cosh (x + iq) -f A'0 sinh (x + iq) cosh (x + iq) + . . . where A'0 = (Ax + * Aj) = (Ax + * A^ x/a). (7ib) Example: Required cosh (0.5 + 3*0.55) = cosh (x + iq) having given cosh (0.5 + ^0.5) = 0.79735 + i 0.36847 in Table VIII and sinh (0.5 + ^0.5) = 0.36847 + 0.79735 in Table VII. Here Ax = o, Aq = 10.05, A'0 = 30.05 X 1.5708 = 30.07854. cosh (0.5 + io^) = cosh (0.5 + io. i + + + . . + sinh (0.5 + 3'o^ A'0 A'0 = 3*0.07854. (A'0) 2 (A'0) 2 = 0.00617. ~~T~ = ~ 0-00309 2! (A'0) 3 (A'0) 3 = 3*0.00048. p = 3*0.00008. (A'0) 4 (A'0) 4 = + 0.00004. i = o.ooooo. 4 ! cosh (0.5 + 3*0.55) = (o.79735 + ^ 0.36847) (i - 0.00309) + (0.36847 + ^' 0.79735) (30.07854 3*0.00008) = (-79735 + * 0.36847) 0.99691 + (0.36847 + i 0.79735) 3*0.07846 = 0.79489 -f- 3*0.36733 + 3*0.02891 0.06256 = 0.73233 + 3*0.39624. The tabulated value = 0.73233 + 0.39624. In view of the similarity of the interpolation operations by Taylor's theorem to those already discussed in relation to Table VII, further examples are probably not needed. EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY Since if cosh ( x + iy) = u + iv (72) cosh ( x + iy} = u iv (73) cosh ( x iy) = u iv (74) cosh ( x iy) = u + iv (75) changing the 'sign of either the real or imaginary entering component only changes the sign of the imaginary component in the result; while changing the sign of the enter- ing quantity as a whole, has no effect on the sign of the result. [ 200] EXPLANATORY TEXT CIRCULAR COSINES OF COMPLEX " ANGLES " It is well known that if 6 be any angle, real or complex, cos 6 = cosh (i6). (76) Consequently, cos (x + iy) = cosh ( y -f ix) (77) or, quadranting the imaginary component, cos (x + iy) = cosh ( y -f- i 2x/ir) = u + iv. (78) To find the circular cosine of any complex quantity x + iy, we enter Table VIII with ( y -f * x/ 1. 5 708). The result is the desired cosine. Example: Required cos (0.4 + i 1.2). Thus we require cosh ( 1.2 + i 0.2546). We now enter Table VIII with x = 1.2 and q = 0.2546 the nearest entry being x = 1.2 and q = 0.25, for which the result is 1.67283 * 0.57765. Hence cos (0.4 + * 1.2) = 1.67283 i 0.57765 neglecting interpolation from q = 0.25 to q = 0.2546. GRAPHIC CHART INTERPOLATIONS The use of the Graphic Charts VII-VIIlA, B, c, for hyperbolic cosines has already been described in connection with sines, on pages 197-198. TABLE IX tanh (x + iq) = u + iv RECTANGULAR HYPERBOLIC TANGENTS OF A RECTANGULAR VARIABLE Entering Process Let tanh (x + iy} be the required function. Quadrant the imaginary component, as described under Tables VII and VIII; that is, divide y by IT/ 2; so that y/(*/2) q. The required function is now expressed in the form tanh (x + iq). Throw out mul- tiples of 2 from q and retain only the remainder as q. Enter Table IX with (x + iq), and find the result directly as u iv. It is a well-known property of tanh (x + iy), that it is periodic in iy, and that the period is i.ir circular radians; or, in quadrants, i.2. That is tanh I x + * (y -f mr) \ = tanh (x + iy). (79) where n is any integer; or, in quadrant measure of the imaginary, tanh \ x + i (q + 2n) j = tanh (x + iq). (80) Example: Required tanh (0.25 + *3o) = tanh (x + iy). Quadranting, tanh (0.25 + i 19.099) = tanh (x + iq). Rejecting multiples of 2, tanh (0.25 + i 1.099) = tan ^ ( x + *'?) We now enter Table IX with x = 0.25 and q = 1.099, the nearest entry to which x = 0.25, q = i.i. The result is 2.95122 i 1.75011. [201] EXPLANATORY TEXT INTERPOLATION Interpolation may be approximately effected by simple proportion, first in x and then in q, as indicated in connection with Tables VII and VIII; or, when a higher degree of precision is required, recourse may be had to Taylor's theorem in the following form: tanh (0 + A0) = tanh 6 + A0 sech 2 - ?f- 2 sech 2 tanh + - 2 sech 2 (2 tanh 2 - sech 2 0) + . . . (81) o * or tanh i (* + iy) + (A* + iAy) \ = tanh (x + iy) + (A* + iAy)' . _ _ cosh 2 (as + and quadranting, tanh \ (x + iq) + (A* + i&q) \ = tanh (x + iq) + - . cosh 2 (x + i so that as far as the second correction term: tanh \ (x + iq) + (A* + *Ag) } = tanh (* + iq) + cosh 2 tanh (, + {)+ ... (83) \IQ (A'O)* tanh (ag + iq) cosh 2 (* + iq) where A'0 = (A* + iAy) = \Ax + iAq (ir/z) } (85) Example: Required tanh (0.5 + 3' 0.5 5) = tanh (a: + ^) having given cosh (0.5 + ^'0.5) = 0.79735 + * 0.3684 7 by Table VIII = 0.87837 /2 4 .8o3 by Table XI and tanh (0.5 + ^'0.5) = 0.76159 -f- i 0.64805 by Table IX = i. 0/40. 395 by Table XII. Here A'0 = (o + ^'0.05 X 1.5708) = (o + ^'0.07854). *' 0.07854 , 0-00617 X i /4Q. tanh (0.5 + .0^5) = 0.76x59 + .0.64805 + 2 ^ + = 0.76159 + i 0.64805 + 0.77153 0.77153 = 0.76159 + i 0.64805 + 0.10180/40. 394 + o.oo8oo\9.2ii = 0.76159 + i 0.64805 + 0.07753 + * 0.06597 ~J~ 0.00790 i 0.00128 = 0.84702 + i 0.71274. The correct value is 0.84752 + * 0.71229. As the third correction term is inconvenient for computation, it is often preferable to obtain a precise interpolation of tanh (x + iy) in working out the correct inter- polations of sinh (x + iy) and cosh (x + iy) by the methods already illustrated, and then to take their ratio. [202] EXPLANATORY TEXT EFFECTS OF CHANGES OF SIGN IN THE ENTERING QUANTITY If tanh ( x + iy) u + iv (86) then tanh ( x iy) = u iv (87) tanh ( x + iy) = u + iv (88) and tanh ( x iy) = (u + iv). (89) Consequently, changes in the sign of the entering quantity produce corresponding changes of sign in the result. FIG. 32. Graphs of x + iy and tanh (* + iy) in the XY and UV planes respectively. CIRCULAR TANGENTS OF COMPLEX " ANGLES " Since tan 6 = ita.nh (id), (go) It follows that tan (x + iy) = stanh ( y + ix) (91) or, quadranting the imaginary: tan (x + iy) = i tanh ( y + i 2x/ir) = i tanh ( y + iq) = i ( u + iv) = v + iu. (92) Consequently, to find tan (x iy) from Table IX, enter it with y as x and with x/(ir/2) as q. Invert the components of the result and the required function is obtained. Thus [203] EXPLANATORY TEXT required tan (i+ 2* 2). We enter with x = 2 and q = 0.6366. The nearest entry is x 2.0 and q = 0.65 for which u + iv = 1.01623 + i 0.03318. Therefore, inverting, tan (i + i 2) = 0.03318 + i 1.01623 neglecting the interpolation from q = 0.65 to q = 0.6366. GRAPHIC INTERPOLATION BY MEANS or CHARTS IXA, IXs, AND IXc These charts contain all of the entries in Table IX, and also a certain number of additional results. They present circles intersecting circles orthotomically; i.e., by rectangular intersection. It is clear that for values of x less than o.io, the curves run off Chart IXA. In fact the first curve shown of x = o.oi extends as far as u = 100. By taking x small enough, the corresponding values of u and v may become indefinitely great. The entire UV plane is covered to infinity once between x = o and x = o, q = o and q = ^ It is covered once more for each 2 quadrants increase in q. When entering for tanh ( x iq); or for the inverse operation tanh" 1 ( u ), it must be remembered that the confocal conic-section diagrams VII and VIII are complete for negative as well as for positive values of x and q; but that only half of the UV plane is presented in Charts IX. The full graph is indicated in Fig. 32, by the aid of which the functions corresponding to negative real values are readily apprehended. TABLE X sinh (x + iq) = r /y POLAR HYPERBOLIC SINES or A RECTANGULAR VARIABLE This table corresponds completely to Table VII, already considered, except that it offers results in polar instead of rectangular coordinates. To find sinh (x + iy) expressed in polar coordinates, quadrant the imaginary, and express the entering variable as (x + iq). Reject multiples of 4 in q, and if the re- mainder exceeds 2, reject 2 but change the sign of the total result. INTERPOLATION BY SIMPLE PROPORTION Required sinh (o. 1 5 + i 0.25) having given sinh (0.2 + 1*0.2) = o.36882/58.723. sinh (0.2 + 20.3) = 0.49663/68. 825. sinh (o.i + 20.2) = 0.32485 /72.947. sinh (o.i + ^'0.3) = 0.46491 778. 932. Diff. for o.i x = +0.043977 14. 224. Diff. for o.i x = +0.031727 10. 107. Diff. for 0.05 x = 0.02I99/ 7. 112. Diff. for 0.05 x = 0.015867 5.Q54. sinh (0.15 + foj) = o.48o77/73.878. sinh (0.15 + 1*0.2) = 0.34684 /65.83S. sinh (0.15 + 1*0.2) = 0.34684/65. 835. Diff. for i o.i = 0.133937 8-43- Diff. for 2*0.015 = o. 06697 / 4. 022. sinh (0.15 + 2*0.25) = 0.41381/69. 857. Correct Value 0.41 124/70. 229. [ 204] EXPLANATORY TEXT INTERPOLATION BY TAYLOR'S THEOREM For a higher degree of precision than is obtainable by simple proportion, it is con- venient to use rectangular coordinates and apply formula (62). Thus, required sinh (0.15 + 20.25). Here referring to Table VII and to the work on page 195, we find for the result u + iv = 0.13911 + i 0.38701. Here ^0.38701 = 1.5877110 yo .^'.^ = jo.22g ^0.13911 = 1.1433584 log 0.13911 = 1.1433584 log sec. 7o.229 = 0.4707400 0.4443526 log 0.41124= 1.6140984 log tan 70. 13' = 0.4440674 Result 0.41124/70. 229. 2853 Correct Value 0.41 124 /70.229. 0-72 3968 INTERPOLATION BY CHARTS X-XlA AND X-XIs These charts present the polar coordinate results on rectangular coordinate sheets, so that they are not true graphs, but are merely to be regarded as interpolation dia- grams. To find sinh (x + fy), proceed as in the use of the tables and quadrant the imaginary so as to obtain the entering quantity in the form (x + iq}> Enter with the curvilinear coordinates, taking the more nearly vertical wavy lines for x and the more nearly hori- zontal lines for q, starting from the line SS as the zero of q. Read off the result on the rectangular background to the left-hand scale of ordinates. When we leave X-XlA and enter X-XlB,it is noticeable that the curves of constant x approach vertical straight lines and the curves of constant q approach horizontal straight lines. At and beyond x = 3.0, we may approximate to the modulus r at any required q, by taking the value of r at q = 0.5 and simple proportional parts between this and r at q = o or r at q = i.o. The change in modulus r between q = 0.5, and either of the above limits is very nearly e~ z / 2 - Thus at x = 3.5 and q = 0.5, r by the tables is 16.55774. At q = o, r = 16.54263, a change of 0.01511, and at q = i.o r = 16.57282, a change of + 0.01508. The value of c~ 3 - 6 /2 will be found to be 0.01510, and over the entire range of q from o to i.o, the change in r follows in nearly simple proportion. Beyond x = 3.2, the limit of Chart X-XIs, the values of r can be obtained by the above rule applied to Table X and with the aid of Chart VII-VIIIc. The values of the amplitude y beyond x = 3 closely approximate to q quadrants. That t x is sinh (x -f- iq) approximates to ^, with q in quadrant measure. TABLE XI cosh (x -\- iq) * /7 POLAR HYPERBOLIC COSINES OF A RECTANGULAR VARIABLE Table XI corresponds completely with Table VIII, except that it gives results expressed in polar instead of rectangular coordinates. It is entered with (x -f iq} just as in Table VIII. [205] Interpolation may be effected by simple proportion, as in the case of Table X, or when a higher degree of precision is required, it may be carried on by Taylor's theorem. In the latter case, it is more convenient to refer to the corresponding entries in Table VIII, interpolating according to formula (716). The rectangular coordinates duly interpolated are then transformed into polar coordinates, as in the last example on page 225. INTERPOLATION BY CHARTS X-XlA AND X-XIs When Charts X-XI are used to find cosh (x + iy) , the imaginary is first quadranted by dividing with ?r/2, so as to obtain the entering variable in the form cosh (x + iq). Starting then from q = o at the horizontal line CC, near the middle of the chart, the underscored figures correspond to q for a little more than the first quadrant. The manifest repetition of the curves enables the lower half of the sheet, however, to be used for the second quadrant. The result is read off on the rectangular background to the right-hand scale of argument. Beyond x = 3.2, the limit of Chart X-XlB reference may be had to Chart VII- VIIlB; or the approximate formula may be used: e x cosh (x + iq} = /. (93) the argument q of the result being interpreted in quadrant measure and converted into degrees. TABLE XII tanh (x + iq) = r /y POLAR HYPERBOLIC TANGENTS OF A RECTANGULAR VARIABLE Table XII corresponds completely with Table IX, except that it gives results expressed in polar instead of rectangular coordinates. If we desire to find tanh (x + iy), we must first divide y by IT/ 2 so as to obtain the entering quantity in the form (x + iq). Multiples of 2 are then rejected in q leaving a remainder less than 2. With this remainder the table is entered. Interpolation may be made by simple proportion to a moderate degree of precision. GRAPHIC INTERPOLATION BY MEANS OF CHARTS XIlA, B, c, D These charts cover between them the full range of Table XII. To find tanh (x + iq) from them with q less than 2, find the proper chart, and enter orr'the curvilinear co- ordinates keeping the underscored number for q. Read off the result on the rectilinear background. For tan (x + iy) and also for the effects of changes of sign, see directions in the dis- cussion on Table IX. To find tanh"" 1 (r/r), enter immediately on the rectangular background of r and 7 in the proper chart, and read off at the correct intersection the corresponding values on the curvilinear coordinates. The result will appear in terms of (x + iq). The imaginary q must be dequadranted, or multiplied by 7r/2, in order to be expressed in terms of (x + iy). [ 206] EXPLANATORY TEXT TABLE XIII / (4 + iq) = u + iv or r /y RECTANGULAR AND POLAR FUNCTIONS OF THE RECTANGULAR VARIABLE (4 + iq) In this table the hyperbolic sine, cosine and tangent of (4 + iq) are collected from q o to q = 2.0. The results are expressed both in rectangular coordinates (u + iv), and in polar coordinates r /y. It will be seen that the moduli of the tangents vary between 0.99933 an d 1.00067, or differ from unity by two thirds of one per mil, at most. The arguments also differ from o by less than 0.04, or about 2'. 17" of arc. Beyond x = 4, it is evident that the hyp. sine and cosine differ by so small a per- centage, that no tabulation of these differences would ordinarily be required. TABLE XIV e*/2 and log 10 (e x /2) SEMI-EXPONENTIALS This table enables the hyp. sine or cosine of any rectangular variable (x + iq) to be found for values of x greater than 4 and less than 10. It is shown in the preceding table that when x reaches 4, the ratio of the sine to the cosine never differs from unity by more than two-thirds of i per mil. This deviation from unity rapidly diminishes as x is further increased. Consequently, the sine and cosine may each be computed from the formula. sinh (x + iq) = cosh (x -f- iq) = /q. (94) Example: Required the value of sinh (8.51 -f- i 25.75). The first step is to quad- rant the imaginary by dividing with x/2, as on page 191. This gives the required func- tion in the form sinh (8.51 + i 16.393). Rejecting multiples of 4.0 in q, we may then write it sinh (8.51 + 3*0.393). Turning to the top of page 143, we find e x /2 = 2482.082 for x = 8.51; so that the result is 2482.082 70.393 quadra-nt. Expressing the argument in .degrees by multiplying with 90 and we have 0.393 X 90 = 35.37. Thus sinh (8.51 + i 16.393) = cosh (8.51 + i 16.393) = 2482.082 /35.37. INTERPOLATION IN x . Since ^ = ?^ = <- I+Al+ (M! + +... (95) it follows that when Ax is a small quantity, it suffices to multiply the tabular value of e r /2 by (i -f Ax) in order to arrive at the interpolated result unless (A*) 2 / 2! the second correction term, is of sufficient magnitude to need consideration. [207] EXPLANATORY TEXT Example: To find sinh (8.51 + iq), having given that sinh (8.50 + iq) = 2457.383 /. Here Ax = o.oi. 2457-383 X 1 = 1 2457.383 2457-3 8 3 X Ax = o.oi 24.574 (A*) 2 2457-383 X - = 0.00005 -123 2482.080 Result 2482. 080 /q. Tabulated value 2482.082 [^ TABLE XV /(* + * o) REAL HYPERBOLIC FUNCTIONS This is a short table of real, as distinguished from complex hyperbolic functions for convenience of reference. It was prepared and published by the author in 1903 in re- lation to continuous-current electric circuit applications, taking the sines, cosines, and tangents from Ligowski's tables, and adding the corresponding computed reciprocals for the cosecants, secants, and cotangents. Much more extensive tables of real hyper- bolic functions are, however, available. See Bibliography, page 211. TABLE XVI SUBDIVISIONS or A DEGREE This is a short table for convenience in changing the expression of a circular angle from decimals of a degree to minutes and seconds, or inversely. By its aid, three- decimal subdivisions of a degree may be converted into minutes and seconds of arc, by direct inspection; or minutes and seconds may be read off as decimals of a degree ta three-digit accuracy. METHODS EMPLOYED IN COMPUTATION Tables I to V, inclusive, were computed as one group, and Tables VII to XIII, inclusive, as a separate group. Tables I to V were computed, at first, by using the formulas: sinh (x + iy) = V sinh 2 x + sin 2 y /tan" 1 (tan y / tanh x) = ri /yi. (96) cosh (x + iy) = Vcosh 2 # sin 2 ;y /tan" 1 (tan y tanh x) = r 2 /7 2 . (97) tanh (x + iy) = (ri/r 2 ) /TI ~ 72- (98) At a later stage of the work, the following formulas, kindly suggested by Professor Bouton, were substituted: sinh (x + iy) = Vcosh 2X sin z /tan" 1 (tan y / tanh x) = ri fji. (99) cosh (x + iy) = V cosh sx cos z /tan" 1 (tan y tanh x) = r% /j z . (100) tanh (x + iy) = tan z /y\ 72- (101) [208] EXPLANATORY TEXT Where the auxiliary circular angle z is defined by : cos 2y - = COS 2Z. (102) cosh 2X The arithmetical work was conducted with the aid of five-place logarithms, and was checked by tabulating successive first and second differences in the tabulated results. Tables VII to XII were computed by means of the following formulas: sinh (x + iy) = sinh x cos y -f- i cosh x sin y. (103) cosh (x -f- iy) = cosh x cos y + * sinh x sin y. (104) , f . . sinh 2X sin 2y tanh (x + ty) = r- - -f * ^ (105) cosh 2# -f cos 2y cosh 2* + cos 2y A standard schedule was prepared and seven-place logarithms used in the compu- tation. The value of tanh (x + iy) was arrived at in two ways, first by dividing (103) by (104), and second by the independent formula (105). If these two methods did not give identical results for tanh (x -f- iy) to five decimal places, when expressed both in rectangular and polar coordinates, the steps of the computation were gone over afresh.* Complete agreement being secured, leads to the inference that the values of sinh, cosh, and tanh (x + iy) are correct, at least as far as their logarithms. Finally, all of the tables have been reduced to graphic form in the Atlas, each entry of the tables being marked off on its proper chart with a sharp needle, and the ruling pen drawn through the successive punctures. In this process a certain number of errors were discovered and rectified. The tables were then set up in type from the MSS. used in making the charts, and were proofread three times. By this procedure it is hoped that the outstanding errors are neither large nor numerous. BIBLIOGRAPHY AND APPLICATIONS OF HYPERBOLIC FUNCTIONS Hyperbolic functions of a real variable are employed extensively in mathematics generally. In particular, they are used in the solution of cubic equations. In navigation, real hyperbolic functions enter in connection with Mercator sailing. In cartography, real hyperbolic functions are used in preparing maps on certain pro- jections, especially on Mercator's projection, which appears to have been the first application of hyperbolic functions. In statics, real hyperbolic functions naturally present themselves in relation to the properties of the catenary and of the funicular polygon; also in the discussion of the forms and stresses of elastic bodies. In dynamics, the same functions present themselves in the theory of vibrations, and in the motion of bodies through a resisting medium. * The author desires to express his acknowledgement of the care and painstaking effort of his assistants engaged in computation, namely, Miss Ethel Smith, A.B. Radcliffe, 1911. Miss A. F. Daniell, A.B. Radcliffe, 1911. Miss Mary M. Devlin, A.B. Radcliffe, 1912. Miss Hope M. Hearn, A.B. Radcliffe, 1912. C 209] EXPLANATORY TEXT A good summary of the historical development of real hyperbolic functions is given in Becker and van Orstrand's "Hyperbolic Functions," Smithsonian Mathematical Tables, 1909, together with a fine compendium of formulas involving these functions.* In electrical engineering, the earliest published application of real hyperbolic func- tions is perhaps in T. H. Blakesley's "Alternating Currents of Electricity," London, 1889, which also appends a short table of these (real) functions. The real functions were also introduced by Sir J. J. Thomson, in "The Electrician," Vol. XXVIII, page 599, 1891. "On the Heat Produced by Eddy Currents in an Iron Plate Exposed to an Alternating Magnetic Field." The fundamental differential equation of the alternating potential-current, steady- state distribution along a uniform conductor, involving hyperbolic functions, nominally real, seems to have been first published by O. Heaviside in 1893, "Electromagnetic Theory," Vol. I, page 450. The first published application of complex hyperbolic functions to the last-named problem was by the author, "On the Fall of Pressure in Long-Distance Alternating- Current Conductors," Electrical World, N. Y., Vol. XXIII, page 17, January, 1894 and "The Electrician," London (abstract), Vol. XXXII, page 239, January 5, 1894. Complex hyperbolic functions also present themselves in the discussion of Hertzian- wave reflections, and in other branches of electrical engineering. They naturally enter the subject of confocal ellipses and hyperbolas, such as Captain Weir's Azimuth diagram of these confocals, for indicating the azimuth of a celestial object in terms of the hour- angle, latitude and declination. (Godfray's "Astronomy," 222.) The mathematical discussion of hyperbolic functions is found in GreenhilTs "Differ- ential and Integral Calculus," Macmillan and Co., 1896; Ligowski's "Tafeln der Hyper- belfunctionen und der Kreisfunctionen," Berlin, Ernst & Korn, 1890; McMahon's "Hyperbolic Functions," Wiley and Sons, N. Y., 1896; Becker and van Orstrand's "Hyperbolic Functions," Smithsonian Institution, 1909; Vassall's "Nouvelles Tables des Logarithmes," Paris, Gauthier-Villars, 1872; as well as other text-books. Works dealing with the applications of hyperbolic functions to electrical engineering are: "The Application of Hyperbolic Functions to Electrical Engineering Problems," by the author, The University of London Press, 1911, and Fleming's "The Propagation of Electrical Currents in Telephone and Telegraph Conductors," Constable & Co., London, 1911. A three-dimensional complex-angle geometrical model,f from which the hyperbolic sines and cosines of complex angles can be presented projectively, has 'been constructed and described. BRIEF BIBLIOGRAPHY or TABLES or HYPERBOLIC FUNCTIONS (i) "Tafeln der Hyperbelfunctionen und der Kreisfunctionen," by Dr. W. Ligowski, Berlin, 1890, Ernst and Korn, 104 pages, giving five-figure logarithms of sinh 6, cosh 6, and tanh up to 8 = 9, by steps of o.ooi up to 6 = 2 and from 2.0 to 9.0 by steps of * This compendium has, by permission, been included in this book at its second edition, as Table XXIH. f "A new geometrical model for the orthogonal projection of the cosines and sines of complex angles" by A. E. Kennelly, Proc. Am. Ac. of Arts and Sciences, Vol. 54, April, 1919, pp. 371-378. [210] EXPLANATORY TEXT o.oi; also the Gudermannian angle to two or more decimals of a second of arc, and other tables. (2) Smithsonian Mathematical Tables, "Hyperbolic Functions," by George F. Becker, and C. E. van Or strand, Smithsonian Institute, Washington, D.C., 1909, 321 pages, giving five-figure logarithms of sinh 6, cosh 0, and tanh 0, by steps of o.oooi up to o.i, by steps of o.ooi from o.i to 3.0, and by steps of o.oi from 3.0 to 6.0; also similar five-figure tables of natural real hyperbolic functions, and various other tables. (3) "Alternating-Current Phenomena in Parallel Conductors," Vol. I by F. E. Pernot, John Wiley, New York, 1918, containing a Table of six-decimal logarithms of hyperbolic functions, up to 2.0 by steps of o.ooi. These present a higher order of precision by one unit, than have been previously available for real hyperbolic functions. The following is a list of all the tables of Complex Hyperbolic Functions known to the present writer, in the order of date of publication: (4) ChrystaFs "Algebra," Edinburgh, 1889, briefly discusses the theory of sinh 6, cosh0, and tanh 6 where 6 is complex; or of the form x + iy. Graphs are given in out- line for these functions, from which a few numerical values may be read. (5) The paper on "Resonance in Alternating-Current Lines," by E. J. Houston and A. E. Kennelly, Transactions A. I. E. E., April, 1895, Vol. XII, pages 133-169, contains a Plate for the graphical evaluation of sinh 6 and of cosh 0, being a complex variable x + iq, between the limits of x = o and #=i.25;g = o and q = , by steps of 0.05 in x and q. The Plate is 40 cm. X 34 cm. and corresponds to Plates VII VIILv of the Atlas prepared from tables in this book, except that it gives the result in regular polar coordinates instead of regular rectangular coordinates. It was produced, by a graphical process, for a precision of the 2.5th order. (6) The first tables of complex hyperbolic functions were a short set published by Dr. James McMahon in his Chapter IV, entitled "Hyperbolic Functions," of a book by Merriam and Woodward on "Higher Mathematics," pages 107-168. The tables gave sinh (x + iy) and cosh (x + iy) from x = o to x = 1.5, by steps of o.i, and also from y = o to y = 1.5, by steps of o.i, Wiley & Sons, New York, 1896. The chapter has since been issued as a separate volume by the same publishers. (7) A table of hyperbolic functions of semi-Lmaginaries or sinh, cosh, tanh, coth, sech and cosech of x /45, by steps of o.i in x up to x = 20.5, was published by the present writer in a paper on "The Alternating-Current Theory of Transmission Speed over Submarine Telegraph Cables," in the Proceedings of The International Electrical Congress of St. Louis, Section A, Vol. I, pages 68-105, 1904. This table is reproduced in Table VI of this volume. (8) Some short tables of sinh, cosh, tanh, coth, sech, and cosech p/5 by steps of o.i in p, up to p = 1.5, for five particular values of 5, published by the present writer in an article on "The Distribution of Pressure and Current over Alternating-Current Cir- cuits," in the Harvard Engineering Journal, 1905-06. C"! 3 EXPLANATORY TEXT (9) Short three-digit tables of sinh and cosh (x + iy) up to x = i, and y = i, by W. E. Miller, in a paper "Formulae, Constants, and Hyperbolic Functions for Trans- mission-Line Problems" in the General Electrical Review, Schenectady, N. Y., May, 1910. Supplement. (10) "Tables of Hyperbolic Functions in Reference to Long Alternating-Current Transmission Lines," published by the present writer in the Transactions of the American Institute of Electrical Engineers, December 1911, pages 2495-2506. These give sinh, cosh, and tanh p /5 from p = o to p = 0.5, by steps of o.i, and from 5 = 60 to 5 = 90 by steps of i. These tables are incorporated in Tables I, II, and III of this volume. (n) "Tables of Sines, Cosines, Tangents, Cosecants, Secants, and Cotangents of Real and Complex Hyperbolic Angles," published by the present writer in The Har- vard Engineering Journal" 1912. These gave sinh, cosh, and tanh p /8 from p = o to p i by steps of o.i, and from 5 = 45 to 5 = 90 by steps of i; also corresponding tables of (sinh 0)/0 and of (tanh 0)/0. These tables are published in separate form by the Harvard Engineering Journal. They are incorporated in tables I, II, III, IV, and V of this volume. NEW TABLES INTRODUCED IN THE SECOND EDITION Tables I to V in this volume were computed for the range of 45 to 90 in the slope or argument 5 of the entering vector quantity; because at that time it did not appear that there would be any need for the range from o to 45. Alternating-current lines used for the transmission or distribution of power have linear hyperbolic angles a, the slope of which is commonly between 80 and 90, rarely falling as low as 45. It has been shown during recent years, however, that railway-signal engineers employ track- signaling circuits, formed of the rails. These are metallic circuits of low frequency, small linear capacitance and large distributed linear leakance. The linear hyperbolic angles a of such circuits develop slopes lying within the range 5 = o to 45. It has therefore become desirable to cover this range, at least as far as p = i. For that pur- pose, Tables XVII to XXI have been inserted. They run by steps of 0.05 in p, from o to i.o, and by steps of 5 in 5, from o to 45. This new tabulated material is available for use in track-signaling and similar computations. It is hoped to incorpo- rate it graphically into the associated Chart Atlas at the first opportunity. Table XVII presents sinh p/ as a polar planevector. It corresponds to and may be regarded as an extension of Table I. Similarly Tables XVIII, XLX, XX and XXI correspond respectively to Tables II, III, IV and V. Whereas, however, Tables I to V are carried to five decimal places in the sizes and three decimal places in the slopes of the evaluated quantities, the new tables are carried to six decimal places in sizes and four decimal places in slopes. They thus aim at one higher order of precision.* Table XXII is similar to Table XI and expands the region covered by the first six entries of the latter into a correspondingly magnified field. It has been found that in * The author desires to express his acknowledgment of the painstaking assistance, on these tables, by Miss Lillian L. Hodgdon, of the Harvard Observatory computing staff. [212] EXPLANATORY TEXT dealing with short lengths of alternating-current line, having negligible linear induc- tance as well as negligible linear leakance, and therefore having a semi-imaginary linear angle, there is frequent need for a magnified table of this kind. It may be noted that whereas Table VI expresses slopes in degrees and minutes, Table XXII expresses them as degrees, and four-place decimals of a degree. Table XXIII is a useful collection of 238 formulas, with a few insertions, taken from Becker and van Orstrand's book of Tables of real hyperbolic functions, referred to in the footnote on page 210. TABLE XVII. HYPERBOLIC SINES, sinh ( P /5) = r 0.05 0.000000 0.0000 0.050021 o.oooo 0.100167 o.oooo 5 o.oooooo 5.0000 0.050020 5.0042 0.100164 5.0167 10 o.oooooo IO.OOOO 0.050020 10.0081 0.100157 10.0325 IS o.oooooo 15.0000 0.050019 15.0119 0.100144 15.0478 20 0.000000 20.0000 0.050016 20.0153 0.100128 20.0614 25 o.oooooo 25.0000 0.050014 25.0183 0.100107 25.0730 30 o.oooooo 30.0000 0.050011 30.0208 0.100083 30.0828 35 o.oooooo 35.0000 0.050006 35.0222 0.100057 35-0897 40 o.oooooo 40.0000 0.050003 40.0236 0.100029 40.0942 45 o.oooooo 45.0000 0.05000 45.0236 O.IOOOO 45-0952 0-35 0.40 0-45 0-357190 o.oooo 0.410752 o.oooo 0.465342 o.oooo 5 0.357081 5.2014 0.410589 5-2625 0.465109 5-3314 10 0.35675? 10.3969 0.410105 10.5172 0.464420 10.6531 15 0.356228 15.5808 0.409316 15-7569 0.463294 15-9558 20 0-355512 20.7472 0.408245 20.9742 0.461767 21.2303 25 0.354628 25.8908 0.406925 26.1625 0.459886 26.4686 30 0.353606 31.0089 0.405397 31.3161 0.457709 31.6630 35 0.352475 36.0961 0.403709 36-4305 0.455303 36.8086 40 0.351270 41-1503 0.401910 41.5017 0.452741 41.8997 45 0.35003 46.1694 0.40006 46.5278 0.45010 46.9338 0.70 o o o 0.758584 o.oooo 5 0.757700 5-7875 10 0.755078 H.5533 15 0.750800 17-2755 20 0.745004 22.9344 25 0.737873 28.5103 30 0.729632 33.9861 35 0.720538 39.3467 40 0.710872 44.5800 45 0.700934 49-6767 0-75 o 0.822317 o.oooo 0.821227 5.9000 0.817993 11.7753 0.812719 17.6019 0.805576 23.3567 0.796787 29.0175 0.786636 34-5647 0-775439 39-98ii 0.763544 45-2525 0.75I3I7 50-3678 0.80 o 0.888106 o.oooo 0.886780 6.0192 0.882843 1 2.0108 0.876425 17.9481 0.867734 23.8044 0.857048 0.844710 0.831105 0.816660 0.801819 29-5561 35.1800 40.6567 45-9697 51.1064 Examples, sinh (0.35 /35) = 0.352475 /36.O96i. sinh (0.80 /} = 0.886780 /6.oi92. TABLE XVII. HYPERBOLIC SINES. O.I5 O.2O sinh ( P /5) = r /?. 0.25 CONTINUED 0.30 o O 0.150563 o.oooo 0.201336 O.OOOO 0.252612 0.0000 0.304520 O.OOOO 5 0.150554 5-0372 0.201316 5.0661 0-252573 5-1033 0.304452 5-1483 10 0.150529 10.0733 0.201256 10.1303 0-252455 10.2033 0.304248 10.2922 15 0.150488 15.0172 0.201157 15.1906 0.252263 15.2972 0-303915 15.4275 20 0.150432 20.1380 0.201024 20.2450 0.252002 20.3825 0.303465 20.5500 25 0.150362 25.1644 0.200859 25.2919 0.251680 25-4561 0.302909 25.6558 3 0.150282 30.1858 0.200669 30.3303 0.251308 30.5158 0.302265 30.7419 35 0.150193 35.2017 0.200458 35-3586 0.250896 35-5600 0.301553 35.8058 40 0.150098 40.2114 . 0.200233 40.3761 0.250457 40.5872 0-300795 40.8456 45 0.15000 45.2146 O.20OOO 45.3817 0.25001 45-5965 0.30001 45.8592 o. 50 0-55 0.60 o, 65 o o o 0.521095 O.OOOO 0.578152 O.OOOO 0.636654 O.OOOO 0.696748 O.OOOO 5 0.520776 5.4078 0.577726 5-4919 0.636100 5-5833 0.696042 5.6819 10 0.519829 10.8039 0.576463 10.9697 0.634456 11.1500 0.693947 "3477 15 0.518282 16.1767 0.574400 16.4197 0.631774 16.6842 0.690530 16.9694 20 0.516184 21.5153 0.571604 21.8286 0.628138 22.1700 0.685898 22.5386 25 0.513601 26.8094 0.568162 27.1844 0.623662 27.5933 0.680199 28.0355 3 0.510612 32.0503 0.564179 32.4764 0.618485 32.9414 0.673609 33-4450 35 0.507309 37-2305 0-559779 37-6956 0.612769 38.2036 0.666335 38.7539 40 0.503794 42.3439 0.555099 42.8344 0.606689 43-3705 0.658601 43-9525 45 0.50017 47.3872 0.550279 47.8880 0.600432 48.4369 0.650644 49-0330 0.85 0.90 0-95 I .00 o o o o O 0.956116 O.OOOO 1.026517 O.OOOO 1.099484 O.OOOO 1.175201 O.OOOO 5 0.954520 6.1447 1.024615 6.2767 1.097239 6.4147 1.172573 6.5592 IO 0.949784 12.2592 1.018975 12.5200 1.090583 12.7928 1.164779 13-0775 15 0.942064 18.3131 1.009783 18.6967 1.079736 19.0986 1.152083 19.5180 20 0.931612 24.2775 0-997344 24.7753 1.065062 25.2969 1-134913 25.8417 25 0.918768 30.1255 0.982060 30.7253 1.047043 31-3544 1.113841 32.0122 3 0.903942 35-8314 0.964429 36.5183 1.026266 37.2400 1.089558 37-9955 35 0.887604 41-3730 0.945011 42.1294 1.003398 42.9253 1.062847 43-7603 40 0.870267 46.7314 0.924414 47-5372 0.979156 48.3864 1.034550 49.2789 45 0.852463 51.8917 0.903276 52.7242 0.954292 53-6033 1.005545 54-5292 Examples, sinh (0.90/20) = 0.997344/2^ sinh (1.0/0) = 1.175201/0. TABLE XVIII. HYPERBOLIC COSINES. cosh(p/5) = r/7- 0.05 o-3S 0.40 0.70 0-7S O.IO I.OOOOOO o.oooo 1.001250 o.oooo 1.005004 o.oooo 5 I.OOOOOO o.oooo 1.001231 0.0125 1.004929 0.0497 10 I.OOOOOO o.oooo 1.001175 0.0244 1.004703 0.0978 15 I.OOOOOO o.oooo 1.001083 0.0358 1.004335 0.1428 20 I.OOOOOO o.oooo 1.000958 0.0461 1.003836 0.1836 25 I.OOOOOO o.oooo 1.000804 0.0547 1.003221 0.2189 3 I.OOOOOO o.oooo 1.000626 0.0619 1.002507 0.2478 35 I.OOOOOO o.oooo 1.000428 0.0672 1.001718 0.2689 40 I.OOOOOO o.oooo 1.000218 0.0705 1.000876 0.2820 45 I.OOOOOO o.oooo I.OOOOO 0.0716 I.OOOOI 0.2864 0.45 o 1.061878 O.OOOO 1.081072 o.oooo 1.102970 o.oooo 5 1.060965 0.5861 1.079886 0.7567 i. 101477 0-9453 10 1.058252 1.1561 1.076361 1-4933 1.097041 1.8669 15 1.053819 1.6950 1.070598 2.1911 1.089786 2.7411 20 1.047791 2.1872 1.062762 2.8303 1.079919 3-5450 25 1.040347 2.6189 1-053079 3-3936 1.067721 4-2567 30 1.031702 2.9772 1.041829 3-8639 1-053539 4-8550 35 I.O22II2 3.2506 1.029339 4.2264 1.037782 5.3208 40 I.OII859 3-4295 1.015974 4.4678 1.020908 5-6375 45 I.OOI25 3-5071 1.00213 4-5784 1.00341 5.7906 0.80 o 1.255169 o.oooo 1.294683 o.oooo 1-337435 o.oooo 5 1.251669 2.1083 1.290690 2-3744 1.332918 2.6483 10 1.241264 4-1750 1.278820 4.7044 1.319493 5-2500 15 1.224242 6.1581 1.259402 6-9456 1-297535 7-7578 20 1.201069 8.0155 1.232970 9.0520 1.267648 10.1236 25 1.172378 9.7039 1.200241 10.9775 1.230648 12.2978 30 1.138949 11.1803 1.162104 12.6747 1.187536 14.2294 35 1.101690 12.4000 1.119583 14.0942 I-I39459 15.8644 40 1.061615 13-3186 1.073820 15.1850 1.087691 17-1458 45 1.019823 13-8911 1.026048 15.8947 1.033602 18.0136 Examples, cosh (0.10/25) = 1.003221 /o.2i89. cosh (0.75/40) = i.Q7382o/i5.i850. TABLE XVIII. HYPERBOLIC COSINES, cosh (p/3) = r /?. CONTINUED O.I5 O.2O O.2S 0.30 o o o o o 1.011271 o.oooo 1.020067 o.oooo 1.031413 O.OOOO 1.045339 o.oooo 5 I.OIIIOI O.I I II 1.019765 0.1964 1.030943 0.3047 1.044665 0.4350 10 1.010595 0.2189 1.018868 0.3872 1.029547 0.6006 1.042663 0.8578 IS 1.009769 0.3203 1.017404 0.5664 1.027265 0.8794 1.039390 1.2567 20 1.008648 0.4119 1.015415 0.7292 1.024167 1.1328 1.034944 1.6200 25 1.007265 0.4914 1.012961 0.8703 1.020343 1-3533 1.029455 1-9375 3 1.005662 0.5561 1. 010116 0.9858 1.015908 1-5344 1.023084 2.J994 35 1.003887 0.6042 1.006966 1.0719 1.010992 1.6703 1. 01602 1 2-3975 40 1.001995 0.6339 1.003604 1.1258 1.005746 1-7567 1.008476 2-5253 45 1.00005 0.6445 1.00013 1.1458 1.00033 1.7901 1.00068 2-5773 0.50 0.55 0.60 0.65 o o o o O 1.127626 O.OOOO 1.155101 O.OOOO 1.185465 o.oooo 1.218793 o.oooo 5 1.125794 1.1506 1.152899 1.3711 1.182861 1.6053 1.215756 1.8514 IO 1.120349 2.2736 1.146351 2.7108 1.175118 3-1753 1.206728 3.6644 15 1. 111445 3-34" 1.135641 3.9872 1.162452 4-6750 1.191956 5.4000 20 1.099330 4.3264 1.121066 5.1694 1.145211 6.0689 1.171848 7.0194 25 1.084345 5.2028 1.103034 6.2267 1.123873 7.3222 1.146955 8.4836 30 1.066914 5-9453 1.082045 7.1292 1.099026 8.4006 1.117959 9-7533 35 1.047534 6.5297 1.058692 7.8481 1.071363 9.2700 1.085656 10.7894 40 1.026760 6-9350 1-033637 8.3566 1.041655 9.8980 1.050938 "5539 45 1.00520 7.1424 1.007598 8.6311 1.010745 10.2544 1.014773 12.0094 0.85 0.90 0-95 1. 00 o o O 1-383531 o.oooo 1.433086 o.oooo 1.486225 o.oooo 1.543081 0.0000 5 1.378460 2.9289 1.427430 3-2150 1.479951 3.5053 1.536155 3.7989 IO 1-363390 5.8092 1.410623 6-3794 1-461313 6.9589 1.515588 7-5475 15 1.338746 8.5914 1-383145 9.4428 1.430851 10.3086 1.481987 11.1858 20 1.305212 11.2253 1-345771 12.3525 1.389438 13.5005 1.436333 14-6653 25 1.263709 13-6589 299533 I5-0545 1-338233 16.4789 1.379926 17.9267 3 1-215358 15-8375 .245685 17.4914 1.278634 19.1844 1.314321 20.9097 35 1.161440 17.7030 -185653 19.6019 1.212219 21-5531 1.241261 23-5483 40 1.103367 19.1936 .120988 21.3192 1.140689 23-5142 1.162611 25.7689 45 1.042645 20.2411 053338 22.5686 1.065840 24.9872 1.080307 27.4870 Examples, cosh (0.25/30) = 1.015908/1" cosh (1.00/40) = 1.162611/2.5 [2173 TABLE XIX. HYPERBOLIC TANGENTS, tanh (p l&) = r 0.05 o o.oooooo o.oooo 0.049958 o.oooo 0.099668 o.oooo 5 o.oooooo 5.0000 0.049959 4.9917 0.099673 4.9670 10 o.oooooo IO.OOOO 0.049961 9-9837 0.099688 9-9347 IS o.oooooo 15.0000 0.049965 14.9761 0.099712 14.9050 20 o.oooooo 20.0000 0.049968 19.9692 0.099745 19.8778 25 o.oooooo 25.0000 0.049974 24.9636 0.099786 24.8541 3 o.oooooo 300000 0.049980 29.9589 0.099833 29.8350 35 O.OOOOOO 35.0000 0.040985 34-9550 0.099886 34.8208 40 o.oooooo 40.0000 0.049992 39-9531 0.099941 39.8122 45 o.oooooo 45.0000 0.05000 44.9520 O.IOOOO 44.8087 o-35 0.40 0.70 o-75 0-45 0.336375 o.oooo 0-379949 o.oooo 0.421899 o.oooo s 0.336562 4-6153 0.380215 4-5058 0.422260 4.3861 10 0.337H9 9.2408 0.381011 9.0239 0.423339 8.7862 15 0.338036 13-8858 0.382325 13.5658 0.425124 13-2147 20 0.339296 18.5600 0-384135 18.1439 0.427594 17.6853 25 0.340875 23.2719 0.386414 22.7689 0.430717 22.2119 3 0.342740 28.0317 0.389121 27.4522 0.434449 26.8080 35 0.344849 32.8455 0.392202 32.2041 0.438727 31.4878 40 0.347153 37.7208 o.39559i 37-0339 0.443469 36.2622 45 0-34959 42.6623 0.39921 41.9494 0.44857 41.1432 0.80 0.604367 o.oooo 0.635150 o.oooo 0.664037 o.oooo s 0-605352 3.6792 0.636270 3-5256 0.665292 3-3709 10 0.608313 7.3783 0.639647 7.0709 0.669078 6.7608 15 0.613278 11.1174 0.645321 10.6563 0.675454 10.1903 20 0.620284 14.9189 0.653363 14.3047 0.684523 13.6808 25 0.629382 18.8064 0.663856 18.0400 0.696420 17.2583 30 0.640619 22.8058 0.676907 21.8900 0.711313 20.9506 35 0.654029 26.9467 0.692614 25.8869 0.729386 24.7922 40 0.669614 31.2614 0.711054 30.0675 0.750820 28.8239 45 0.687309 35-7856 0.732244 34-4731 0.775752 33-0928 Examples, tanh (0.75/25) = 0.663856/18^.0400. tanh (0.40/20) = 0.384135 /i 8. 1 439^ O8] TABLE XIX. HYPERBOLIC TANGENTS, tanh (p /6) = r fa. CONTINUED 0.15 0.25 0.30 0.148885 o.oooo 0.197375 o.oooo 0.244919 0.0000 0.291312 0.0000 5 0.148901 4.9261 0.197414 4.8697 0.244992 4.7986 0.291435 4-7133 IO 0.148951 9.8544 0.197529 9-7431 0.245210 9.6027 0.291799 9-4344 IS 0.149032 14.7869 0.197718 14.6242 0.245567 14.4178 0.292398 14.1708 20 0.149142 19.7261 0.197972 19-5158 0.246055 19.2497 0.293219 18.9300 25 0.149277 24.6730 0.198289 24.4216 0.246663 24.1028 0.294242 23-7183 30 0.149436 29.6297 0.198659 29-3445 0.247373 28.9814 0.295445 28.5425 35 0.149611 34-5975 0.199071 34.2867 0.248168 33.8897 0.296798 33-4083 40 0.149799 39-5775 0.199514 39-2503 0.249027 38.8305 0.298267 38-3203 45 0.14999 44.5701 0.19997 44.2359 0.24993 43-8064 0.29981 43.2819 0.50 0.55 0.60 0.65 e o O 0.462117 0.0000 0.500521 0.0000 0.537049 o.oooo 0.571669 O.OOOO 5 0.462586 4.2572 0.501107 4.1208 0.537764 3-9780 0.572518 3-8305 IO 0.463988 8.5303 0.502867 8.2589 0.539908 7-9747 0.575065 7-6803 15 0.466314 12.8356 0.505793 12.4325 0.543484 12.0092 0.579325 11.5694 20 0.469545 17.1889 0.509875 16.6592 0.548491 16.1011 0.585313 15.5192 25 0.473651 21.6066 0.515090 20-9577 0.554922 20.2711 0.593047 19-5519 30 0.478587 26.1050 0.521400 2S-3472 0.562757 24.5408 0.602535 23.6917 35 0.484289 30.7008 0.528746 29.8475 0.571953 28.9336 0.613763 27-9645 40 0.490664 35.4089 0.537034 34-4778 0.582428 334725 0.626679 32.3986 45 0-49759 40.2448 0.546130 39-2569 0.594049 38.1825 0.641172 37.0236 o. 85 0.90 0.95 1. 00 o o e o o O 0.691070 o.oooo 0.716298 O.OOOO 0.739782 O.OOOO 0.761595 O.OOOO 5 0.692454 3-2158 0.717804 3.0617 0.741402 2.9094 0.763317 2.7603 10 0.696634 6.4500 0.722358 6.1406 0.746304 5.8339 0.768533 5.5300 15 0.703692 9.7217 0.730063 9.2539 0.754611 8.7900 0.777391 8.3322 20 0-713763 13.0522 0.741095 12.4228 0.766541 11.7964 0.790146 11.1764 2 5 0.727041 16.4666 0.755702 15.6708 0.782407 14.8755 0.807174 14.0855 30 0.743766 19-9939 0.774215 19.0269 0.802627 18.0556 0.828989 17.0858 35 0.764227 23.6700 0.797038 22.5275 0.827737 21.3722 0.856264 20. 2 1 2O 40 0.788737 27-5378 0.824642 26.2180 0.858389 24.8722 0.889851 23-5IOO 45 0.817596 31.6506 0.857537 30.1556 0.895343 28.6161 0.930795 27.O422 Examples, tanh (0.25/25) = o.246663/24.io28. tanh (0.90/30) = 0.774215 1 1 219 ] o 5 10 25 30 35 40 45 o 5 10 IS 20 25 30 35 40 45 o 5 10 IS 20 25 30 35 40 45 I.OOOOOO I.OOOOOO I.OOOOOO I.OOOOOO I.OOOOOO o.oooo o.oooo o.oooo 0.0000 o.oooo I.OOOOOO O.OOOO I.OOOOOO 0.0000 I.OOOOOO O.OOOO I.OOOOOO O.OOOO I.OOOOOO O.OOOO 0-35 o 1.020543 o.oooo 1.020230 0.2014 1.019305 0.3969 1.017796 0.5808 1.015748 0.7472 1.013224 0.8908 1.010302 1.0089 1.007070 1.0961 1.003630 1.1503 1.00008 1.1694 0.70 o 1.083691 o.oooo 1.082429 0.7875 1.078682 1.5533 .072572 2.2755 .064292 2.9344 .054105 3-5103 .042331 3.9861 .029340 4.3467 1.015532 4.5800 1.001334 4-6767 sinh TING FACTOR. e '& 0.05 O.IO o o 1.000420 o.oooo 1.001670 o.oooo 1.000404 0.0042 1.001638 0.0167 1.000398 0.0081 1.001570 0.0325 1.000373 0.0119 1.001442 0.0478 1.000317 0.0153 1.001279 0.0614 1.000279 0.0183 1.001071 0.0730 1.000216 0.0208 1.000834 0.0828 I.OOOI34 O.O222 1.000571 0.0897 1.000061 0.0236 1.000290 0.0942 i.ooooo 0.0236 i.ooooo 0.0952 0.40 0.45 o o .026880 o.oooo 1.034093 o.oooo .026473 0.2625 1-033577 0.3314 .025264 0.5172 1.032044 0.6531 .023290 0.7569 1.029542 0.9558 .020612 0.9742 1.026148 1.2303 .017312 1.1625 1.021969 1.4686 .013493 1.3161 1.017131 1.6630 1.009272 1.4305 1.011784 I. 8086 1.004775 1-5017 1.006092 1.8997 1.00014 1.5278 1.00023 1.9338 o-7S 0.80 o o 1.096423 o.oooo 1.110132 o.oooo 1.094969 0.9000 1.108475 1.0192 1.090658 1.7753 1.103554 2.0108 1.083625 2.6019 1.095532 2.9481 1.074102 3.3567 1.084668 3.8044 1.062383 4-0175 1.071311 4-5561 1.048848 4.5647 1.055881 5.1800 1.033919 4.9811 1.038881 5.6567 1.018059 5-2525 1.020825 5.9697 1.001756 5.3678 1.002274 6.1064 Example. sinh (0.40/25) 0-40/25 = 1.017312^ C220] TABLE XX. CORRECTING O.IS 0.20 FACTO D sinh = rr. CONTINUED 0.30 e 0.25 o o o O 1-003753 o.oooo 1.006680 o.oooo 1.010448 o.oooo 1.015067 o.oooo 5 1.003696 0.0372 1.006579 0.066 1 1.010291 0.1033 1,014839 0.1483 10 1.003527 0.0733 1.006278 0.1303 1.009820 0.2033 1.014161 0.2922 is 1.003253 0.0172 1.005784 0.1906 1.009051 0.2972 1.013051 0.4275 20 1.002878 0.1380 1.005119 0.2450 1.008007 0.3825 1.011549 0.5500 25 1.002412 0.1644 1.004296 0.2919 1.006722 0.4561 1.000696 0.6558 3 1.001878 0.1858 1.003343 0.3303 1.005233 0.5158 1.007550 0.7419 35 1.001287 0.2017 1.002290 0.3586 1.003584 0.5600 1.005177 0.8058 40 1.000653 0.2114 1.001167 0.3761 1.001830 0.5872 1.002651 0.8456 45 I.OOOOO 0.2146 I.OOOOI 0.3817 1.00003 0.5965 1.00005 0.8592 0.50 o-SS 0.60 0.65 o o e 1.042190 O.OOOO 1.051185 o.oooo 1.061090 o.oooo 1.071920 o.oooo 5 1.041552 0.4078 1.050411 0.4919 1.060167 0.5833 1.070833 0.6819 10 1.039657 0.8039 1.048114 0.9697 1.057426 1.1500 1.0^7611 1-3477 IS 1.036564 1.1767 1.044363 1.4197 1.052956 1.6842 1.062354 1.9694 20 1.032369 1-5153 1.039280 1.8286 1.046896 2.1700 1.055228 2.5386 25 1.027202 1.8094 1.033021 2.1844 1.039436 2-5933 1.046459 3-0355 30 1.021223 2.0503 1.025779 2.4764 1.030808 2.9414 1.036322 3-4450 35 1.014618 2.2305 1.017780 2.6956 1.021281 3.2036 1.025131 3-7539 40 1.007587 2.3439 1.009270 2.8344 1.011148 3-370S 1.013232 3.9525 45 1.00035 2.3872 1.000508 2.8880 1.000720 3-4369 1.000991 4-0330 0.85 0.00 0.95 1. 00 o o o o 1.124842 o.oooo 1.140574 o.oooo 1-157352 o.oooo 1.175201 O.OOOO 5 1.122964 1.1447 1.138461 1.2767 1.154989 1.4147 1-172573 1-5592 10 I.II7393 2.2592 1.132194 2.5200 1.147982 2.7928 1.164779 3-0775 IS 1.108311 3-3131 1.121981 3.6967 1.136564 4.0986 1.152083 4.5180 20 1.096014 4.2775 I. 108160 4-7753 I.I2III7 5.2969 1.134913 5-8417 25 1.080904 5-1255 1.091177 5-7253 1.102151 6.3544 1.113841 7.0122 3 1.063461 5.8314 1.071587 6.5183 1.080280 7.2400 1.089558 7-9955 35 1.044241 6.3730 1.050012 7.1294 1.056208 7.9253 1.062847 8.7603 40 1.023843 6.7314 1.027127 7-5372 1.030691 8.3864 1.034550 9.2789 45 1.002897 6.8917 1.003640 7.7242 1.004518 8.6033 1.005545 9.5292 Example. sinh (0.95/25) _ 0-95/25^ [221] TABLE XXI. CORRECTING FACTOR. taphg e 0.05 1. 000000 o.oooo 0.999160 o.oooo 0.996680 o.oooo S I.OOOOOO o.oooo 0.999177 0.0083 0.996732 0.0330 10 I.OOOOOO o.oooo 0.999223 0.0163 0.996882 0.0653 IS I.OOOOOO o.oooo 0.999291 0.0239 0.997119 0.0950 20 I.OOOOOO o.oooo 0.999359 0.0308 0.997448 0.1222 25 I.OOOOOO o.oooo 0.999475 0.0364 0.997858 0.1459 3 I.OOOOOO o.oooo 0.999591 0.0411 0.998331 0.1650 35 I.OOOOOO o.oooo 0.999707 0.0450 0.998856 0.1792 40 I.OOOOOO o.oooo 0.999844 0.0469 0.999414 0.1878 45 I.OOOOOO o.oooo I. 00000 0.0480 0.99999 0.1913 0.35 0.40 0.45 o 0.961071 o.oooo 0.949872 o.oooo 0.937553 o.oooo 5 0.961606 0.3847 0.950538 0.4942 0-938355 0.6139 10 0.963197 0.7592 0.952528 0.9761 0.940753 1.2138 i5 0.965817 1.1142 0.955812 1.4342 0.944719 1.7853 20 0.969419 1.4400 0.960338 1.8561 0.950209 2.3147 25 0.973929 1.7281 0.966036 2.2311 0.957150 2.7881 3 0.979258 1.9683 0.972801 2.5478 0.965442 3.1920 35 0.985284 2.1545 0.980505 2.7959 0.974949 3.5122 40 0.991867 2.2792 0.988977 2.9661 0.985487 3-7378 45 0.99883 2.3377 0.99800 3.0506 0.99684 3.8568. 0.70 0.75 0.80 o 0.863381 o.oooo 0.846867 o.oooo 0.830046 o.oooo 5 0.864789 1.3208 0.848360 1-4744 0.831615 1.6291 10 0.869019 2.6217 0.852862 2.9291 0.836347 3.2392 i5 0.876111 3.8826 0.860428 4-3437 0.844318 4.8097 20 0.886121 5.0811 0.871150 5-6953 0-855654 6.3192 25 0.899117 6.1936 0.885142 6.9600 0.870525 7-7417 30 0.915169 7.1942 0.902542 8.1100 0.889141 9.0494 35 0.934328 8.0533 0.923485 9.1131 0.911732 10.2078 40 0.956592 8.7386 0.948072 9.9325 0.938525 11.1761 45 0.981870 9.2144 0.976325 10.5269 0.969690 11.9072 Note. Negative quantities are in heavy type. Example. tanh (0.75/25) 0.75/25 [223] XAXAJV^ O.J.TI VJ 1 f\\-, J. V e r ti- ^UNTLNUED o.i 5 O.2O 0.25 0.30 o o O 0.992567 o.oooo 0.986875 o.oooo 0.979676 o.oooo 0.971040 o.oooo 5 0.992676 0.0739 0.987069 0.1303 0.979968 0.2014 0.971449 0.2867 10 0.993006 0.1456 0.987642 0.2569 0.980839 0.3973 0.972665 0.5656 15 0.993547 0.2131 0.988588 0.3758 0.982269 0.5822 0.974660 0.8292 20 0.994280 0.2739 0.989860 0.4842 0.984222 0.7503 0-977396 1.0700 25 0.995182 0.3270 0.991446 0.5784 0.986650 0.8972 0.980807 1.2817 30 0.996237 0.3703 0.993295 0.6555 0.989493 1.0186 0.984817 1.4575 35 0.997410 0.4025 0.995357 0.7133 0.992673 x.i 103 0.989327 1.5917 40 0.998660 0.4225 0.997571 0.7497 0.996106 1.1695 0.994224 1.6797 45 0.99995 0.4299 0.99987 0.7641 0.99971 1.1936 0.99937 1-7181 0.50 0.55 O.6o 0.65 o o o o 0.924234 o.oooo 0.910038 o.oooo 0.895082 o.oooo 0.879491 o.oooo 5 0.925172 0.7428 0.91 1 104 0.8792 0.896274 I.O22O 0.880796 1.1695 10 0.927976 1.4697 0.914304 1.7411 0.899847 2.0253 0.884716 2.3197 15 0.932627 2.1644 0.919624 2.5675 0.905806 2.9908 0.891269 3.4306 20 0.939089 2.8111 0.927046 3.3408 0.914151 3.8989 0.900482 4.4808 25 0.947302 3-3934 0.936527 4-0423 0.924870 4.2289 0.912380 5.4481 30 0.957174 3-8950 0.948001 4.6528 0.937928 5-4592 0.926976 6.3083 35 0.968577 4.2992 0.961356 5-1525 0.953254 6.0664 0.944250 7-0355 40 0.981327 4.59" 0.976426 5.5222 0.970713 6.5275 0.964122 7.6014 45 0.99516 4-7552 0.992963 5.7431 0.990081 6.8175 0.986419 7.9764 0.85 0.90 0.95 1.00 o o o O 0.813024 o.oooo 0.795887 o.oooo 0.778718 o.oooo 0.761595 o.oooo 5 0.814651 1.7842 0.797560 1.9383 0.780424 2.0906 0.763317 2.2397 IO 0.819569 3-5500 0.802620 3.8594 0-785583 4-1661 0-768533 4.4700 15 0.827873 5-2783 0.811181 5.7461 0.794327 6.2100 0.777391 6.6678 20 0.839722 6.9478 0.823438 7-5772 0.806886 8.2036 0.790146 8.8236 25 0.855342 8.5334 0.839669 9.3292 0.823586 10.1245 0.807174 10.9145 30 0.875019 10.0061 0.860239 10.9731 0.844870 11.9444 0.828989 12.9142 35 0.899091 11.3300 0.885598 12.4725 0.871302 13.6278 0.856264 14.7880 40 0.927926 12.4622 0.916269 13.7820 0.903567 15.1278 0.889851 16.4900 45 0.961877 13.3494 0.952819 14-8444 0.942466 16.3839 0.930795 17.9578 Note. Negative quantities are in heavy type. tanh (1.0/10) Eiample. : 5 = 0.768533X4^4700. [223] TABLE XXIL FUNCTIONS OF SEMI-IMAGINARIES COMPLEX VARIABLE 0/45 (slope constant). 9 Sinh0/45 Cosh 0/45 tanh 6 /45 Coseche/45 hyp. Size Slope Size ^ Slope Size _ Slope Size Slope rads. numeric degrees numeric degrees numeric degrees numeric degrees 0.00 o.ooooo '45.0000 I.OOOOO o.oooo o.ooooo 45.0000 oc 45.0000 O.OI O.OIOOO 45.0009 I.OOOOO 0.0028 O.OIOOO 44.9981 IOO.OOOOO 45.0009 O.O2 0.02OOO 45-0037 I.OOOOO 0.0114 O.O2OOO 44.9923 50.00000 45-0037 0.03 0.03000 45.0084 I.OOOOO 0.0257 0.03000 44.9827 33-33333 45.0084 0.04 0.04000 4S-oi5i I.OOOOO 0.0458 0.04000 44-9693 25.00000 45.0151 O.OS 0.05000 45-o 2 36 I.OOOOO 0.0716 0.05000 44.9520 20.00000 45-0236 O.O6 0.06000 45-0342 I.OOOOO 0.1031 0.06000 44.9310 16.66667 45-0342 O.O? 0.07000 45-0465 .00000 0.1403 0.07000 44.9062 14.28571 45-0465 0.08 0.08000 45.0608 .00000 0.1833 0.08000 44-8775 12.50000 45.0608 O.OQ 0.09000 45.0770 .00001 0.2320 O.OQOOO 44.8450 n. mil 45-0770 O.IO O.IOOOO 45-095 2 .00001 0.2864 O.IOOOO 44.8087 10.00000 45-0952 O.II O.IIOOO 45-"52 .00001 0.3466 O.IIOOO 44.7686 9.09091 45.52 O.I2 O.I2OOO 45-I37 2 .00002 0.4125 O.I2OOO 44.7247 8-33333 45.1372 0.13 0.13000 45.1611 1.00003 0.4841 0.13000 44.6770 7-69231 45.1611 0.14 0.14000 45.1869 1.00004 0.5614 0.13999 44-6255 7.14286 45-1869 0.15 0.15000 45.2146 1.00005 0.6445 0.14999 44-5701 6.66667 452146 0.16 0.16000 45.2442 1.00006 0-7333 0.15999 44.5109 6.25000 45.2442 0.17 0.17000 45-2757 1.00008 0.8278 0.16999 44-4479 5-88235 45-2757 0.18 0.18000 45-3092 1.00009 0.9281 0.17998 44.3811 5.55556 453092 0.19 0.19000 45-3445 I.OOOII 1.0341 0.18998 44.3104 5-26316 45-3445 O.20 O.20OOO 45-38I7 1.00013 1.1458 0.19997 44-2359 5.00000 45-38I7 O.2I O.2IOOO 45.4208 1. 00016 1.2632 0.20997 44.1576 4.76190 454208 0.22 O.22OOO 45.4619 I.OOO2O 1.3864 0.21996 44-0755 4-54545 45.4619 0.23 0.23000 45-5048 1.00024 1-5152 0.22995 43.9896 4.34783 45.5048 O.24 0.24000 45-5497 1.00028 1.6498 0.23994 43.8999 4.16667 45-5497 0.25 0.25001 45-5965 1.00033 1.7901 0.24993 43.8064 3.99984 45.5965 O.26 0.26001 45-6453 1.00038 1.9361 0.25991 43.7092 3.84601 45-6453 0.27 0.27001 45-6959 1.00044 2.0878 0.26989 43.6081 3.70357 45-6959 0.28 0.28001 45-7485 1.00051 2.2452 0.27987 43-5032 3-57I30 45.7485 O.29 0.29001 45.8029 1.00060 2.4084 0.28984 43-3945 3.44816 45.8029 0.30 0.30001 45-8592 1.00068 2-5773 0.29981 43.2819 3-33322 45-8592 0.31 0.31002 45-9I74 1.00077 2.7520 0.30978 43-1654 3.22560 45.9174 0.32 0.32002 45-9775 1.00087 2.9323 0.31974 43-0452 3.12481 45-9775 0-33 0.33002 46.0396 1.00098 3.83 0.32970 42-9213 3.03012 46.0396 0-34 0.34002 46.1036 I.OOIII 3-3099 0.33965 42-7937 2.94100 46.1036 0-35 0.35003 46.1694 1.00125 3-507I 0-34959 42.6623 2.85690 46.1694 0.36 0.36003 46.2372 1.00140 3-7ioo 0-35953 42.5272 2-77755 46.2372 0-37 0.37004 46.3069 1.00156 3-9i85 0.36946 42.3884 2.70241 46.3069 0.38 0.38004 46.3786 1.00174 4-1328 0-37939 42-2458 2.63130 46.3786 0-39 0.39005 46.4522 1.00193 4-3528 0.38930 42.0994 2.56377 46.4522 0.40 0.40006 46-5278 1.00213 4-5784 0.39921 41.9494 2.49963 46.5278 0.41 0.41006 46.6053 1.00235 4.8096 0.40910 41-7957 2.43867 46.6053 0.42 0.42007 46.6846 1.00259 5-0464 0.41899 41.6382 2.38056 46.6846 0.43 0.43008 46-7658 1.00285 5.2889 0.42886 41.4778 2-32515 46.7658 o-44 0.44009 46.8489 1.00312 5-5369 0.43872 41.3120 2.27226 46.8489 0-45 0.45010 46.9338 1.00341 5.7906 0.44857 41.1432 2.22173 46.9338 0.46 0.4601 2 47.0206 1.00372 6.0498 0.45841 40.9708 2-17335 47.0206 0.47 0.47013 47.1093 1.00406 6.3146 0.46823 40.7947 2.12707 47.1093 0.48 0.48014 47.2000 1.00442 6.5850 0.47803 40.6150 2.08273 47.2000 0.49 0.49016 47.2926 1.00480 6.8609 0.48782 40.4317 2.04015 47.2926 0.50 0.50017 47-3872 1.00520 7.1424 0-49759 40.2448 1.99932 47.3873 Note. Negative quantities are in heavy type. Examples, sinh 0.13 745 =* 0.13000 745. 1611. cosh 0.26/45 = 1.00038 /i.936i. [224] TABLE XXII. FUNCTIONS OF SEMI-IMAGINARIES. CONTINUED COMPLEX VARIABLE 6 /45 (slope constant). e Sech 6 /45 Coth /45 Sinh /45 / 9 / 4 5 tanh B / 4 S / /45 hyp. Size_ Slope Size Slope Size Slope Size Slope rads. numeric degrees numeric degrees numeric degrees numeric degrees 0.00 I.OOOOO o.oooo oc 45.0000 I.OOOOO o.oooo I.OOOOO o.oooo 0.01 I.OOOOO 0.0028 IOO.OOOOO 44.9981 I.OOOOO 0.0009 I.OOOOO 0.0019 O.02 I.OOOOO 0.0114 50.00000 449923 I.OOOOO 0.0037 I.OOOOO 0.0077 0.03 I.OOOOO 0.0257 33-33333 44.9827 I.OOOOO 0.0084 I.OOOOO 0.0173 O.O4 I.OOOOO 0.0458 25.00000 449693 I.OOOOO 0.0151 I.OOOOO 0.0307 0.05 I.OOOOO 0.0716 20.00000 449520 I.OOOOO 0.0236 I.OOOOO 0.0480 O.O6 I.OOOOO 0.1031 16.66667 44.9310 I.OOOOO 0.0342 I.OOOOO 0.0690 O.O7 I.OOOOO 0.1403 14.28571 44.9062 I.OOOOO 0.0465 0.99999 0.0938 0.08 I.OOOOO 0.1833 12.50000 44-8775 I.OOOOO 0.0608 0-99999 0.1225 O.OQ 0.99999 0.2320 ii.iiin 44.8450 I.OOOOO 0.0770 0.99999 0.1550 O.IO 0.99999 0.2864 IO.OOOOO 44-8087 I.OOOOO 0.0952 0-99999 0.1913 O.I I 0.99999 0.3466 9.09091 44.7686 I.OOOOO 0.1152 0.99998 0.2314 0.12 0.99998 0.4125 8-33333 44-7247 I.OOOOO 0.1372 0.99998 0.2753 0.13 0.99997 0.4841 7.69231 44.6770 I.OOOOO 0.1611 0.99997 0.3230 O.I4 0.99996 0.5614 7-14337 446255 I.OOOOO 0.1869 0.99996 0-3745 0.15 0.99995 0.6445 6.66712 445701 I.OOOOO 0.2146 099995 0.4299 O.l6 0.99994 0-7333 6.25039 445109 I.OOOOO 0.2442 0.99994 o 4891 O.I? 0.99992 0.8278 5.88271 444479 I.OOOOI 0.2757 0.99992 0.5521 0.18 0.99991 0.9281 S-SS6i8 44.3811 I.OOOOI 0.3092 0.99991 06189 O.I9 0.99989 0341 5.26371 44.3104 I.OOOOI 0-3445 0.99989 0.6896 O.2O 0.99987 .1458 5.00050 44-2359 I.OOOOI 0.3817 0.99987 0.7641 0.21 0.99984 .2632 4-76258 441576 I.OOOO2 0.4208 0.99984 0.8424 O.22 0.99980 .3864 4.54628 44-0755 I.OOOO2 0.4619 0.99981 0.9245 0.23 0.99976 .5152 4-34878 43.9896 I.OOOO2 0.5048 0.99978 1.0104 O.24 0.99972 1.6498 4.16771 438999 I.OOOO2 0-5497 0-99975 I.IOOI 0.25 0.99967 1.7901 4.OOI I 2 43.8064 1.00003 0-5965 0.99971 1.1936 0.26 0.99962 1.9361 3.84748 43.7092 1.00003 0-6453 0.99966 2.2908 0.27 0.99956 2.0878 3.70521 43.6081 1.00003 0-6959 0.99960 1.3919 0.28 0.99949 2.2452 3-57309 43.5032 1.00004 07485 0-99953 1.4968 O.29 0.99940 2.4084 3.45018 433945 1.00004 0.8029 0-99945 1.6055 0.30 0.99932 2-5773 3-33544 43 2819 1.00005 0.8592 0-99937 1.7181 0.31 0.99923 2.7520 3.22810 43.1654 1.00005 0.9174 0.99928 1.8346 0.32 0.99913 29323 3- I2 754 430452 1.00006 0-9775 0.99919 1.9548 0-33 0.99902 3.1183 3-0330S 42.9213 1.00006 1.0396 0.99908 2.0787 0-34 0.99889 3 3099 2.94421 42-7937 1.00007 1.1036 0.99896 2.2063 0-35 0.99875 3-5071 2.86049 42.6623 1.00008 1.1694 0.99883 2.3377 0.36 0.99860 3.7100 2.78141 42.5272 1.00009 1.2372 0.99869 2.4728 o-37 0.99844 39185 2.70665 42-3884 I.OOOIO 1.3069 0.99854 2.6116 0.38 0.99826 4-1328 2.63580 42.2458 I.OOOII 1-3786 0-99837 3.7543 o-39 0.99807 43528 2.56870 42.0994 1.00013 1.4522 0.09819 2.9006 0.40 0.99788 4-5784 2.50494 42.9494 1.00014 1.5278 0.99800 3.0506 0.41 0.99766 4.8096 2.44438 4i 7957 1.00016 1.6053 0.99780 3.2043 0.42 0.99742 5.0464 2.38668 41.6382 1.00017 1.6846 0-99759 3.3618 o-43 0.99716 5.2889 2.33174 41.4778 1.00019 1.7658 0.99736 3.5230 o-44 0.99689 5.5369 2-27935 41.3120 I.OOO2I 1.8489 0.99711 3.6880 0-45 0.99660 5.7906 2.22930 41 US* 1.00023 I.9338 0.99684 3.8568 0.46 0.99629 6.0498 2.18145 40.9708 1.00025 2.0206 0.99655 4.0292 0.47 0.99596 6.3146 2.I357I 40.7947 1.00027 2.1093 0.99624 4.2053 0.48 0.99560 6.5850 2.09190 40.6150 1.00030 2.2OOO 0.99591 4.3850 0-49 0.99522 6.8609 2.04995 40.4317 1.00032 2.2926 0-99555 4.5683 0.50 0^9483 7.1424 2.00969 40.2448 1.00035 2.3872 0.99516 4-7552 Note. Negative quantities are in heavy type. Examples. tanh 0.39 /45 o-39 coth 0.50 /45 TABLE XXIII. HYPERBOLIC FUNCTION FORMULAS (from Smithsonian Mathematical Tables No. 1871 of 1909, Becker and van Orstrand's "Hyperbolic Functions," by permission.) A. RELATIONS BETWEEN HYPERBOLIC AND CIRCULAR FUNCTIONS 1. sinh u = i sin iu = tan gd u. 2. cosh u = cos iu = sec gd u. 3. tanh u = i tan iu = sin gd u. 4. tanh \u tan \ gd u. 5. 0" = (i + sin gd u) -i- cos gd u, = [i - cos (J TT + gd tt)3 -i- sin ( TT + gd ), = tan (JTT + % gdu). 6. sinh iu = i sin u. 7. cosh ra = cos u. 8. tanh * = i tan w. 8a. sin u = i sinh iu = tanh (grf" 1 u). 8b. cos M = cosh iu = sech (gd"" 1 u). 8c. tan M = i tanh i = sinh (gd~ l u). 9. sinh (u w>) = i sin (z> =F i), = sinh u cos v i cosh w sin v. 10. cosh (w iv) = cos (v T iu), = cosh M cos v db t sinh w sin w. loa. sin (w =t *) = i sinh (t tw) = sin cosh v i cos sinh v. xob. cos ( =b iv) = cosh (v =F iu) = cos M cosh v =F i sin sinh . 1 1 . cosh (w TT) = cos m IT. (m is an integer.) 12. sinh (2 w + i) \ i IT = i sin (2 w + i) I T. (m is an integer.) B. RELATIONS AMONG THE HYPERBOLIC FUNCTIONS 13. sinh u= \(e u e~ u ) = sinh ( u) = (csch u) ~ l , = 2 tanh \ u -i- (i tanh 2 \ u) = tanh u -f- (i tanh 2 )*. 14. cosh = (e u + e~") = cosh ( u) = (sech u)~ l , = (i + tanh 2 i u) -5- (i - tanh 2 } M) = i -=- (i - tanh 2 )*. 15. tanh u = (e u - e~") -T- (e u + r) = - tanh (- w), = (coth u)~ l = sinh M -i- cosh M = (i sech 2 M)*. 16. sechtt = sech ( w) = (i tanh 2 )*. [226] FORMULAS 17. cschw = csch ( ) = (coth 2 w i)*. 1 8. coth u = - coth (- u) = (csch 2 w + i)*. 19. cosh 2 u sinh 2 u = i. 20. sinh = V i (cosh w i). 21. cosh % w = V i (cosh w + i). 22. tanh ^ = (cosh w i) -5- sinh u, = sinh u -T- (i + cosh w) = V (cosh u i) -r- (cosh M + i). 23. sinh 2 = 2 sinh w cosh u = 2 tanh M -f- (i tanh 2 u). 24. cosh 2 = cosh 2 u + sinh 2 w = 2 cosh 2 M i, = 1+2 sinh 2 = (i + tanh 2 ) -f- (i tanh 2 w). 25. tanh 2U = 2 tanh M -f- (i + tanh 2 ). 26. sinh 3 = 3 sinh w + 4 sinh 3 . 27. cosh 3 = 4 cosh 3 3 cosh u. 28. tanh 3 = (3 tanh u + tanh 3 w) -f- (i + 3 tanh 2 ). 28a> m cosh u + n sinh w = J (w + ) e u + \ (m ri) e~". a8b. m e u we~ u = (m ri) cosh w + (w =F w) sinh u. 29. sinh M = n cosh"" 1 u sinh w + ^-^ -4-J cosh"" 3 M sinh 3 +.... 6 /M ('M T ) 30. cosh nu = cosh" u -\ cosh n ~ 2 u sinh 2 u + 2 31. sinh w + sinh i> = 2 sinh \ (u -\- v) cosh (M z>). 32. sinh w sinh v = 2 cosh !( + ) sinh % (u v). 33. cosh u -f cosh = 2 cosh ( + v) cosh ( v). 34. cosh cosh v = 2 sinh %(u + v) sinh H M ~ *) 35. sinh u + cosh M = (i + tanh w) -H (i tanh i w). 36. (sinh w + cosh u) n = cosh w + sinh nu. 36a. a sinh u + b cosh u = V# 2 ~ ^ 2 sinh ( + tanh" 1 ^). a> b = V* 2 - a 2 cosh ( + coth" 1 1). 6 > a __^____^^ i 36b. a cosh i sinh u = Va 2 6 2 cosh (w tanh" 1 j). 37. tanh w + tanh z> = sinh (M + v) + cosh w cosh v. 38. tanh M tanh v = sinh (u v) -f- cosh w cosh . 39. coth u + coth t> = sinh ( + v) -i- sinh M sinh . [227] FORMULAS 40. coth u coth v = sinh (u v) -f- sinh u sinh v. 41. sinh (u v) = sinh w cosh z> =t cosh M sinh t>. 42. cosh (u =t z>) = cosh w cosh z; sinh M sinh z>. 43. tanh (u z) = (tanh w tanh v) -H (i tanh w tanh z>). 44. coth (u v) = (coth M coth z> i) -f- (coth z> coth M). 45. sinh (u + v) + sinh (w zi) = 2 sinh u cosh z>. 46. sinh (u + z>) sinh ( z>) = 2 cosh w sinh v. 47. cosh ( + )+ cosh (u v) = 2 cosh u cosh z>. 48. cosh (u + v) cosh (M ) = 2 sinh u sinh v. 49. tanh \ (u + z;) = (sinh M -f sinh z>) -f- (cosh M + cosh v). 50. tanh ^ (w v) = (sinh M sinh v) -f- (cosh w + cosh z>). 51. coth J (M + z>) = (sinh w sinh v) -t- (cosh M cosh ). 52. coth 5 ( ) = (sinh M + sinh z>) -f- (cosh cosh z>). tanh + tanh v _ sinh (w + v) tanh M tanh v sinh ( z;) * coth u + coth v _ _ sinh (w + ) coth M coth z sinh (w v) 55. sinh (w + z>) + cosh ( + v) = (cosh w + sinh M) (cosh + shih v). 56. shih (w + v) sinh ( ) = sinh 2 u sinh 2 , = cosh 2 u cosh 2 z>. 57. cosh (u + z>) cosh (u v) = cosh 2 u + sinh 2 , = sinh 2 M + cosh 2 v. 58. sinh (#w TT) = o. (m is an integer.) 59. cosh (miir) = ( i) m . 60. tanh (mi TT) = o. 61. sinh (u + miir) = ( i) m sinh. 62. cosh (u + mi IT) = ( i) m coshw. 63. sinh (2 m + i) *' TT = i. 64. cosh (2 w + i) 3 i TT = o. 65. sinh ( ) = i cosh M. \ 2 / . cosh ( w) = =t * sinh M. \2 / 66. [228] FORMULAS 66a. sinh { (u + iv) + i f \ = sinh { (u + iv) + i i \ i cosh (u + iv). 66b. cosh \ (u + iv) + i f | = cosh | (w + >) + T; \ = i.sinh (M + iv). 66c. tanh ^ (w + iv) + i f | = tanh | (w + iv) + i i_ \ = coth (u + iv). 66d. sinh | (w + iv) + i TT \ = sinh \ (u + iv) + is_\ = sinh (u + iv). 66e. cosh \ (u + iv) + i * \ = cosh \ (u + iv) + i 2 \ = cosh (u + iv). 66f. tanh \ (u + iv) + i T \ = tanh \ (u + iv) + i 2 \ = tanh (u + iv). 67. tanh (u + iir) = tanh u. 67a. If sinh ^ (u + t (i - q) \ = x + iy; then sinh \ u + i (i + q) \ =- x + iy. 670. If cosh \(u + i(i-q)\=x + iy: then cosh \ u + i (i + q) \ =- x + iy. 67C. If tanh \ u + i (i - q) \ = x + iy: then tanh \ u + i (i 4- q) I = x - iy. C. INVERSE HYPERBOLIC FUNCTIONS /T T-TI u * ^ + i; 69. cosh- 1 u = log (w + Vw 2 i) = sinh -1 V 2 - i = / , 2 _ ^ r du 70. tanh" 1 w = \ log (i + ) log (i u) = I 2 * %/ A ** 71. coth" 1 M = ^ log (i + u) i log (u i) = / j _ W 2 : 72. 73. csch" 1 u = 74. sin" 1 w = sinh" 1 iu = i log (I'M + V 1 2 )- 75. cos" 1 u = i cosh" 1 M = i log (M + i Vi ~ 2 )- 76. tan- 1 u= i tanh" 1 ZM = : log (i + iu) . log (i iu). t 1. 11: 21 77. cot' 1 u- i coth- 1 iu = . log (iu i) TJ Jg (* M + J )- 2 Z* ** 78. sin" 1 iu = i sinh" 1 u = i log (w C 22 9 .1 FORMULAS 1 7 80. tan" 1 iu = i tanh- 1 u = - log (i + ) - log (i ). 2 2 81. cot- 1 iu = i cotir 1 u = -- log (w + i) + log (u i). 2 2 82. cosh- 1 \(u + -} = sinh- 1 \(u - -} = \ / \ / tt i- i ~ = 2 tanh- 1 - = log M. + i 83. tanh -1 tan M = \gdzu. 84. tan" 1 tanh M = \ gd~ l 2 . 85. cosh" 1 esc 2 = sinh- 1 cot 2 u = tanh -1 cos 2 u = log tan u. 86. tanh- 1 tan 2 (J TT + i ) = J log esc w. 87. tankr 1 tan 2 1 = ^ log sec u. 88. cosh" 1 cosh" 1 v = cosh" 1 [ V (w 2 i) (v 2 i)]. 89. sinh" 1 sinh" 1 z> = sinh- 1 \ju V i + v 2 z; V i + w 2 ]. D. SERIES M 2 7< 3 M 4 90. e = I+M + - + - + -+ . .. (M 2 w > o) 93 . 94. log (i + M) = u -- w 2 H -- 3 - w 4 + . 234 95- log ferj) = 2 tt + W 3 + M 6 + W 7 + . . . 1 (M 2 e^-,tt + ife> v 7fe) < *;'^] M 3 5 M 7 97- smh = * + +++... ( 2 < a ) 2 [230] FORMULAS '' +JL--U 6! 98. cOSh=I+^-+- -+- f + . . . ( 2 , 5 6 1 101 . sech u = i ^ + 4 2 24 720 102. U CSCh U = I T-U 2 + -~ U 4 108. com" 1 = tanh" 1 - = I -. -\ r H 7 + . . M u 3 * 5 M 5 7 w 7 I . 2 IM 13** 135 6 360 15120 103. gdu = = u z~ w3 H w5 7 + . . . (M small) r i sech 3 w 13 sech 5 u = sech u ... (u large) 2 2 3 245 I - . I s . 6l - / T\ 104. tt = ga 1 < = i) 22 W 2 244 tt 4 2466 tt 6 III / 107. tanh" 1 u = u + - w -\ 6 H w + . . . ( < i) 3 5 7 i oo . seen w *~ \^V/JI.A AVCL M 22 244 2466 II II I^I 1351 / '\ no. csch-'* = sinh" 1 - ----_+--_- --- , + . . . = log f + I^_I3 4 + I3|^_ B-< (tt2<1 ) U 22 244 2466 FORMULAS E. DERIVATIVES de u in. -7- = e u . du . log e U I 112. d j = - (III U da v dv . 113. -r = a", -r- .log, a. du du 114. -3 = u u (i + log e ). J sinh M iis. : = coshtt. du d cosh M . , 116. : = smhtt. du d tanh u 117. -j = sech 2 . d coth u 118. ; = csch 2 M. du d sech u no. ; = sechw. tanhM. du d csch u 1 20. ; = csch u. coth u. du 121 d sinh" 1 u du ^#2 _|_ j J cosh" 1 M i 122. = 123. du d tanh" 1 u d coth" 1 u 124. -j = du d sech" 1 u 125. ; = du u 126. du u ^ + z dgd u 127. 3 = sechw. du d gd -1 128. ^3 = secw. du [232] FORMULAS F. INTEGRALS. (INTEGRATION CONSTANTS ARE OMITTED.) 1 29. I sinh u du = cosh u. 130. I cosh udu = sinh u. 131. I tanh udu = log cosh . 132. I coth udu = log sinh . X 33- / sech udu = 2 tan" 1 e u = gd u. 134. I csch udu = log tanh - - 135. / sinh" u du = - sinh "-' M. cosh u / sinh n ~ 2 u du, J w n J I w _|_ 2 /* = ; sinh " 4l M cosh w I sinh n+2 M rfw. + i w + i J 136. I cosh" udu = - sinh w. cosh n-1 M + / cosh n ~ ? w dw, J w w J sinh u cosh n+1 u -\ ^ / cosh n+2 du. n + i 137. I u sinh udu = u cosh M sinh M. 138. I u cosh udu = u sinh w cosh . 139. / w 2 sinh udu = (u 2 + 2) cosh u 2 u sinh . 140. I M" sinh udu = u n cosh ww"" 1 sinh u + n(n i) I n ~ s si 141. I sinh 2 w JM = \ (sinh M cosh u u). 142. I sinh . cosh u du = \ cosh (2 ). ^ 143. / cosh 2 u du = % (sinh u cosh + M). [233] FORMULAS 144. I tanh 2 u du = u tanh w. 145. I coth 2 u du = u coth u. 146. I sech 2 rf = tanhw. 147. I sech 3 M du = i sech w tanh u + \ gd u. 148. I csch 2 udu= coth . 149. / sinh- 1 u du = u sinh- 1 (i + w 2 )*. 150. I cosh- 1 u du = u cosh" 1 u (w 2 i)*. 151. / tanh- 1 udu = u tanh- 1 w + i log (i w 2 ). 152. T sinh- 1 u du = i (2 w 2 + i) sinh- 1 u - u (i + 2 )* . 153- T M cosh- 1 u du = l\ (2 u 2 - i) cosh- 1 u - u (u? - i) . 154 . i (cosh a + cosh w)" 1 du = 2 csch o. tanh" 1 (tanh ^ w. tanh a), = csch a log cosh (w + a) - log cosh \(u - a) . 155. I (cos a + cosh u)~ l du - 2 esc a. tan -1 ( tanh $ w. tan \ a). 15 6. / (i + cos a. cosh w)" 1 du = 2 esc a. tanh- 1 (tanh i w. tan a). 157. I sinh cos d = I (i + tan 2 )* J tan < = sec < tan + | gd -1 0, = | tan $ (i + tan 2 <)* + \ sinh- 1 (tan 0). Here = gd u. r du .u r du 168. I , , ,.. = sinh- 1 -- f -p-j - 5T7 = sin" 1 J (u 2 + a 2 )* a J (a 2 n ')* """ a' r du ,u r du .u l6o. I 7-S - ^T = COSh" 1 - I r-~ -- 5rr- = COS 1 J (u 2 a 2 )* a J (a 2 2 )* a /a a a J a 2 + 2 a a / d i . f du i u - - = - sech- 1 - I 7-5 - ^T- = sec * - M (a 2 2 )* a a J u (u 2 a 2 )* a a / d i , u f du i u u(a* + u 2 )* = a CSCh a" J ( 2 - a 2 ) = a C! a /du i , aw + b . , , TT = = sinh- 1 7 -- rjrr- , a positive, ac > b 2 ; (au 2 + 2bu + c)* v C - P) 1 i , au + & ... ^ ,, = -= cosh" 1 ^75 y , a positive, ac < b 2 ; = cos" 1 , ^ U X1 > a negative. (b 2 ac) [235] FORMULAS / du (au 2 tan au + b (ac - 6 2 )* (ac - i " v-i au + b (b* - a, tanlr (b 2 - ac)* ac > b*; ac < b 2 , au + b < (b 2 ac)*. ac < 6 2 , au + 6 > (6 2 - ac)*. 176. J (a- u) (u - 6)* (a - 6)* b a b or or 2 \u b tan -1 (b - a)* 2 (a - 6)i coth- 1 I77< f J (a- du b~=~a' u b a b tanh" 1 u) (b - w)* (b - a)* 2 \b u or or TT - rr (b a)* -. - TTJ (a b)* coth- 1 tan a - - a b (The real form is to be taken.) b u b a (The real form is to be taken.) 178. 179. 1 80. 181. 182. 183- 184. 185. = i w ( 2 a 2 )* | a 2 cosh" 1 a 2 )* JM = ^ u (a 2 - 2 )i + ^ a 2 sin" 1 - o> du= a 2 )* /<'- JV- JV + //flu ** u ~~ a '' /gOU utf* u du = -j (aw i). a / U m MU yy> f* u m -(P u du = I u m ~ l e? u du. a aj /e? u du _ i e" /^e u a 7 !* 1 u m m i [ ""^ / w*"" 1 J //T& a^rfw = 6 log a [236] FORMULAS C _a u u n _ na u u n ~ l n (n i) a u u n ~ 2 J log a (log a) 2 (log a) 3 n (n i) (n 2) 2. i a" (log a) n+1 l8? r a "^ M = a " r E iq g g _ (log a) 2 J u n n i [ "- 1 (w 2) w"" 2 (w 2) (w - 3) - ; (log a) "- 1 ra"du 1 (w- 2) (w- 3) .... 2 .iJ ~u~ \' Ca u du _ . (u log a) 2 ( log a) 3 J U 2.2 I 3.3 ! du e u du - log ( V a + be mu + V"a) . /ue u du e" (i + ) 2 ~TT " /e u log 195. I logudu = u log u u. 196. Ju m logudu = u m + l [^^ ~ (w + l)2 J- 197. I (log )" du = u (log w) n n I (log w)"- 1 rfw. r /i w>n+1 (lg M )" n C m n ^n- 198. / m (log u) n du = - I w m (log )" y m + i w -f- i7 20 - (log ) n o, = -T- (log M 2) V (& + bu) + 2 V a tan" 1 -vl - , if a < o. [238] FORMULAS 213. Ce~<**du = -^ = r(i). Jo 2 a 20, 214. rue-*du=T (n+ J = ^ Jo a n+l a n s . r-r~ ^= i Jo - +1 . 2 n a 2i6. I e "*~ ^r du = - - \ IT a > o. . I e~"*~ & du = - A| IT Jo 2 > /*" i [TT 217. ' e""" V M d = -- A - ' Jo 2n V n /*e- u 17 ==~du = \ i Vw > w f 00 ^M 7T 219. . , , - r = - J o sinh (w) 2 w f 00 M^M 7T 2 . . , ( - r- = - 5- J o smh (nu) 4 w j 220 221 f*it /IT sinh (mu) . sinh () du = I cosh (WM) . cosh () < */ o Jo = o, if m is different from n. /it (*i* i^ cosh 2 (mu) du = I sinh 2 (w)