EXCHANGE 
 
AUG 17 i 
 
 Observation and Reduction of Occultations of 
 Stars by the Moon 
 
 WITH A DETERMINATION OF THE RESULTING LONGITUDE OF 
 
 THE FLOWER OBSERVATORY, AND CORRECTIONS TO THE 
 
 RIGHT ASCENSION, DECLINATION AND SEMI-DIAMETER 
 
 OF THE MOON 
 
 BY 
 
 KRIKORIS GARABED BOHJELIAN 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE 
 
 UNIVERSITY OF PENNSYLVANIA IN PARTIAL FULFILMENT 
 
 OF THE REQUIREMENTS FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 
 UN!\ 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER, PA. 
 
 1915 
 
Observation and Reduction of Occupations of 
 Stars by the Moon 
 
 WITH A DETERMINATION OF THE RESULTING LONGITUDE OF 
 
 THE FLOWER OBSERVATORY, AND CORRECTIONS TO THE 
 
 RIGHT ASCENSION, DECLINATION AND SEMI-DIAMETER 
 
 OF THE MOON 
 
 BY 
 
 KRIKORIS GARABED BOHJELIAN 
 
 A THESIS 
 
 PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE 
 
 UNIVERSITY OF PENNSYLVANIA IN PARTIAL FULFILMENT 
 
 OF THE REQUIREMENTS FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER, PA. 
 
 1915 
 
INTRODUCTION. 
 
 It is well known that corrections to the coordinates, distance, 
 and size of the Moon can be determined from the observations 
 of occultations more accurately than from any other method. 
 
 If simultaneous observations of this kind are secured from two 
 stations on the Earth, which differ widely in latitude, the oblate- 
 ness of the Earth can also be found ; and whatever the situation 
 of the stations, their difference in longitude can be thus deter- 
 mined with a higher accuracy than by any other method, except 
 that of the Telegraph and Wireless. 
 
 As the Longitude of the Flower Observatory was accurately 
 determined, both by Telegraph and Wireless method, a com- 
 parison of these results with a value found from occultations 
 becomes of interest, and as the later observations have a special 
 value for improving our knowledge of the Moon's motion, the 
 following piece of work was undertaken with these objects. 
 
 The work was begun in the early summer of 1914, the obser- 
 vations being made with the 1 8-inch equatorial of the Flower 
 Observatory of the University of Pennsylvania. 
 
 Among the occultations observed it. was learned that 13 had 
 been simultaneously observed by Prof. Asaph Hall, with the 
 26-inch equatorial of the U. S. Naval Observatory. 
 
 Through the courtesy of the Director, Captain J. A. Hooge- 
 werff, these observations were forwarded to Prof. Eric Doolittle. 
 
 It is the results from these stars, which form the principal 
 basis of the following investigation. 
 
 OBSERVED TIMES OF OCCULTATIONS. 
 
 Date, 1914 
 
 Star 
 
 Phase 
 
 Phila. M. T. 
 
 Washington M. T. 
 
 July 17 
 
 q Tauri 
 
 I 
 
 I3 h 59 m 33 s -4 
 
 I3 h 5i m 6 s .o 
 
 
 
 
 20 
 
 
 I 
 
 14 n 18 .0 
 
 14 2 33 .6 
 
 
 
 
 16 
 
 
 E 
 
 H 38 57-8 
 
 14 29 4.1 
 
 
 
 
 q 
 
 
 E 
 
 14 57 4.2 
 
 H 47 37-5 
 
 
 
 
 20 
 
 
 E 
 
 15 6 57.8 
 
 14 57 47-1 
 
 
 
 
 21 
 
 
 E 
 
 15 19 3 -2 
 
 
 
 
 
 22 
 
 
 E 
 
 15 22 23 .2 
 
 
 A 
 
 g- 
 
 r 
 
 A Aquarii 
 
 I 
 
 - ii 7 52 .9 
 
 10 57 23 .6 
 
 
 
 
 11 
 
 E 
 
 12 28 19 .9 
 
 12 16 25 .6 
 
 
 
 
 78 
 
 I 
 
 12 46 42 .4 
 
 12 34 35-3 
 
 
 
 
 78 " 
 
 E 
 
 14 7 2 .7 
 
 13 55 52 .4 
 
 
 30 
 
 r Sagit 
 
 I 
 
 9 14 18 .2 
 
 9 3 50 .6 
 
 
 
 
 
 E 
 
 10 30 18 .9 ' 
 
 10 20 53 .3 
 
 Sept. 14 
 
 K Gemin. 
 
 I 
 
 14 16 3 .2 
 
 14 8 20 .9 
 
 
 
 E 
 
 14 50 o .4 
 
 H 39 3 -o 
 
 311753 
 
OBSERVATION AND REDUCTION OF 
 
 PRELIMINARY COMPUTATION. 
 
 The right ascension and declination of the above stars, reduced 
 to apparent place for the observed times are as follows: 
 
 Date, 1914 
 
 Name 
 
 Right Ascension, a 
 
 Declination, 8 
 
 July 17 
 
 
 
 
 Aug. 7 
 
 30 
 Sept. 14 
 
 q Tauri 
 
 20 " 
 
 16 " 
 
 21 " 
 
 22 " 
 
 X Aquarii 
 
 78 c ; 
 
 T Sagittarn 
 ic Geminorum 
 
 55 i' 39"-45 
 55 10 58 .50 
 54 55 42 .11 
 55 12 5 -68 
 55 14 12 .27 
 342 2 49 .65 
 342 32 15 .38 
 285 24 35 .99 
 114 49 28 .89 
 
 +24 12' 4 ".72 
 24 6 9. 90 
 
 24 I 21 .47 
 24 17 23 .16 
 
 +24 15 47 .93 
 8 i 56 .12 
 - 7 39 24 .26 
 -27 47 53 -78 
 +24 36 19 .74 
 
 The right ascension, declination and horizontal parallax of the 
 Moon for consecutive hours on the successive dates are found 
 to have the following values: 
 
 Date, 1914 
 
 
 A 
 
 D 
 
 7T 
 
 July 17 
 
 i8 h 
 
 53 
 
 29' 
 
 14 
 
 ".70 
 
 +24 
 
 27' 
 
 24 
 
 ".0 
 
 54' 
 
 4 8". 9 8 
 
 
 19 
 
 54 
 
 i 
 
 31 
 
 05 
 
 
 34 
 
 59 
 
 .8 
 
 
 50 .03 
 
 
 20 
 
 54 
 
 33 
 
 52 
 
 45 
 
 
 42 
 
 29 
 
 .0 
 
 
 51 .10 
 
 
 21 
 
 55 
 
 6 
 
 19 
 
 50 
 
 
 49 
 
 
 .6 
 
 
 52 .17 
 
 
 22 
 
 55 
 
 38 
 
 
 75 
 
 +24 
 
 57 
 
 7 
 
 4 
 
 54 
 
 53 .26 
 
 Aug. 7 
 
 15 
 
 340 
 
 49 
 
 21 
 
 .90 
 
 - 7 
 
 45 
 
 44 
 
 9 
 
 55 
 
 28 .89 
 
 
 16 
 
 341 
 
 17 
 
 20 
 
 .40 
 
 7 
 
 31 
 
 24 
 
 3 
 
 
 27 .56 
 
 
 17 
 
 341 
 
 45 
 
 15 
 
 .60 
 
 7 
 
 17 
 
 2 
 
 7 
 
 
 26 .28 
 
 
 18 
 
 342 
 
 13 
 
 7 
 
 65 
 
 7 
 
 2 
 
 40 
 
 .1 
 
 
 25 .01 
 
 
 19 
 
 342 
 
 40 
 
 56 
 
 .70 
 
 - 6 
 
 48 
 
 16 
 
 5 
 
 55 
 
 23 -74 
 
 Aug. 30 
 
 13 
 
 284 
 
 30 
 
 16 
 
 .95 
 
 -27 
 
 I 
 
 29 
 
 5 
 
 57 
 
 25 .78 
 
 
 H 
 
 285 
 
 7 
 
 12 
 
 .30 
 
 26 
 
 56 
 
 
 .7 
 
 
 24 -47 
 
 
 15 
 
 285 
 
 44 
 
 2 
 
 55 
 
 26 
 
 50 
 
 50 
 
 .6 
 
 
 23 .15 
 
 
 16 
 
 286 
 
 20 
 
 47 
 
 .70 
 
 -26 
 
 45 
 
 17 
 
 .2 
 
 57 
 
 21 .83 
 
 Sept. 14 
 
 18 
 
 H3 
 
 7 
 
 15 
 
 .90 
 
 +25 
 
 3 i 
 
 35 
 
 .6 
 
 56 
 
 58 -49 
 
 
 19 
 
 113 
 
 42 
 
 37 
 
 .05 
 
 
 24 
 
 18 
 
 4 
 
 57 
 
 o .71 
 
 
 20 | 114 
 
 17 
 
 56 
 
 .85 
 
 
 16 
 
 52 
 
 .1 
 
 57 
 
 2 -94 
 
 
 21 114 
 
 53 
 
 15 
 
 .00 
 
 +25 
 
 9 
 
 16 
 
 .8 
 
 57 
 
 5 .18 
 
 The coordinates of the Moon's center and their derivatives 
 for the above hours are computed from the formulae: 
 
 x 
 
 cos D sin (A a) 
 
 sin 
 
 sin (D - 5) cos 2 \(A - a) + sin (D - d) sin 2 \(A - a) 
 
 sin TT 
 
OCCULTATIONS OF STARS BY THE MOON 
 
 Date 
 
 
 X 
 
 x' 
 
 y 
 
 y' 
 
 July 17 
 
 
 
 
 
 
 q Tauri 
 
 i8 h 
 
 -I.53447I4 
 
 +.5370561 
 
 +0.2879661 
 
 +1335941 
 
 
 19 
 
 - .9973448 
 
 1910 
 
 .4215411 
 
 572 
 
 
 20 
 
 .4601054 
 
 2817 
 
 .5550831 
 
 282 
 
 
 21 
 
 + .0772044 
 
 3309 
 
 .6886000 
 
 .1335060 
 
 20 Tauri 
 
 18 
 
 1.6891416 
 
 2214 
 
 .3976049 
 
 .1329862 
 
 
 19 
 
 -I.I5I8438 
 
 3673 
 
 .5305682 
 
 409 
 
 
 20 
 
 O.6I442IO 
 
 4714 
 
 .6634894 
 
 031 
 
 
 21 
 
 +0.0769149 
 
 .5375338 
 
 .7663772 
 
 .1328740 
 
 1 6 Tauri 
 
 18 
 
 -1.4356003 
 
 .5369492 
 
 .4824448 
 
 .1339496 
 
 
 19 
 
 0.8985849 
 
 .5370746 
 
 .6163670 
 
 8963 
 
 
 20 
 
 0.3614650 
 
 1081 
 
 .7502065 
 
 8517 
 
 
 21 
 
 +O.I757I73 
 
 2065 
 
 .8840731 
 
 .1338142 
 
 21 Tauri 
 
 19 
 
 I.I70408l 
 
 4028 
 
 .3261183 
 
 .1328847 
 
 
 20 
 
 -0.6329631 
 
 4872 
 
 .4590122 
 
 8943 
 
 
 21 
 
 -0.0954338 
 
 57H 
 
 5918895 
 
 8515 
 
 
 22 
 
 + 0.4421798 
 
 6558 
 
 .7246927 
 
 7562 
 
 22 Tauri 
 
 19 
 
 -1.2053922 
 
 4409 
 
 3553572 
 
 7474 
 
 
 2O 
 
 0.6679063 
 
 5296 
 
 .4881108 
 
 7515 
 
 
 21 
 
 -0.1303335 
 
 6169 
 
 .6208439 
 
 7064 
 
 
 22 
 
 + 0.4073266 
 
 .5377030 
 
 .7535066 
 
 .1326120 
 
 Aug. 7 
 
 
 
 
 
 
 X Aquar. 
 
 15 
 
 I.3II9302 
 
 .4987673 
 
 .2898113 
 
 .2599905 
 
 
 16 
 
 -0.8I3I459 
 
 7982 
 
 .5497641 
 
 9144 
 
 
 17 
 
 -0.3I4340I 
 
 8103 
 
 .8096338 
 
 8292 
 
 
 18 
 
 0.1844686 
 
 8039 
 
 1.0694310 
 
 .2597498 
 
 78 Aquar. 
 
 16 
 
 -I-339I549 
 
 2319 
 
 0.1422995 
 
 .2602937 
 
 
 17 
 
 0.8408942 
 
 2868 
 
 .4025699 
 
 2468 
 
 
 18 
 
 -0.3425869 
 
 3251 
 
 .6627924 
 
 1979 
 
 
 19 
 
 +0.1557504 
 
 4983468 
 
 .9229651 
 
 .2601472 
 
 Aug. 30 
 
 
 
 
 
 
 T Sagit. 
 
 13 
 
 0.8425482 
 
 .5723855 
 
 .8049339 
 
 .0945414 
 
 
 14 
 
 0.2701396 
 
 4281 
 
 .8994167 
 
 4283 
 
 
 15 
 
 +0.3023008 
 
 4491 
 
 .9938027 
 
 35i8 
 
 
 16 
 
 +0.8747514 
 
 .5724485 
 
 1.0881326 
 
 +.0943142 
 
 Sept. 14 
 
 
 
 
 
 
 K Gemin. 
 
 18 
 
 I.6l88750 
 
 .5594691 
 
 .9800360 
 
 -.1344901 
 
 
 19 
 
 -1.0593263 
 
 5259 
 
 .8458102 
 
 2051 
 
 
 20 
 
 -0.4998271 
 
 5686 
 
 .7115758 
 
 2730 
 
 
 21 
 
 +0.0597563 
 
 +-559595I 
 
 +0.5772448 
 
 -.1343938 
 
 The coordinates and their derivatives for Philadelphia and 
 Washington are computed from the formulae: 
 
 p sin 
 
 b sin B 
 
 p cos <p f cos (M a) = b cos B 
 
 = p cos (p f sin (n a) 
 = b sin (B - 5) 
 
6 OBSERVATION AND REDUCTION OF 
 
 where for Phila. <p r = 39 46' 32". 9 and log p = 9.999400 
 long. = 5 h i m 6 s .5i 
 
 and for Wash. <?' = 38 55' I4".o and log p = 9.999431 and 
 long. = 5 h 8 m I5 S 78. 
 
 
 
 7 
 
 E 
 
 1 
 
 17 
 
 f 
 
 i? 
 
 q Tauri 
 
 Phila. 
 Wash. 
 
 -.7674918 
 -.7785898 
 
 +.5816687 
 +.5811267 
 
 -.7427171 
 .7634340 
 
 + .5034517 
 .5062870 
 
 20 " 
 
 Phila. 
 Wash. 
 
 .7664782 
 
 ; .7790062 
 
 .5669897 
 .5666500 
 
 .7342088 
 .7540638 
 
 .4918996 
 .4904268 
 
 16 " 
 
 Phila. 
 Wash. 
 
 .7556986 
 .7722021 
 
 .5289977 
 5287325 
 
 
 
 21 " 
 
 Phila. 
 
 
 
 .7215053 
 
 .4747025 
 
 22 " 
 
 Phila. 
 
 
 
 .7180547 
 
 47IIOI8 
 
 X Aqua. 
 
 Phila. 
 Wash. 
 
 .4707500 
 .5200407 
 
 .7173353 
 .6997824 
 
 .83796466 
 .2926039 
 
 7345912 
 .7196153 
 
 78 " 
 
 Phila. 
 Wash. 
 
 .1979788 
 .2406685 
 
 .7320044 
 .7292446 
 
 .0692538 
 .0323282 
 
 .7350480 
 .7230056 
 
 r Sagit. 
 
 Phila. 
 Wash. 
 
 +.1569340 
 +.1214520 
 
 .9155368 
 .9116275 
 
 + .3904763 
 + .3611283 
 
 .8737904 
 .8746794 
 
 K Gemin. 
 
 Phila. 
 Wash. 
 
 -.7668061 
 -.7790125 
 
 .5673483 
 +.5643084 
 
 -.7535621 
 -.7705400 
 
 .5202768 
 + .5202708 
 
 With the assumed value of the longitude of Flower Observatory 
 viz: 5 h 8 m 6 s .5i and of the U. S. Naval Observatory viz: 5 h 8 m 
 15 s . 78, we reduce Phila. and Wash, times to Greenwich times 
 and assuming the values of t 1 sufficiently near to these times, so 
 that x and y may be assumed to vary uniformly during the 
 interval we compute certain auxiliaries first introduced by Bessel 
 by the formulae: 
 
 m sin M = 
 m cos M = 
 
 n sin N = x' 
 n cos N = y' 
 
 sn 
 
 
 sin (M - N) log k 9435376o. 
 
 After obtaining the above auxiliaries viz: log m, log n, M, N, 
 
OCCULTATIONS OF STARS BY THE MOON 
 
 and ^ we next compute 12, T, x, v, from the formulae: 
 
 12 = h [-cos 4, - -cos (M - N) 1 - (/' - t) h = 3600 
 In n J 
 
 f-p __ J. J^ 
 
 n 
 
 sin N yo cos 
 #o cos N yo sin N 
 
 
 
 I 
 
 1 
 
 
 
 
 
 
 Phila. 
 
 Wash. 
 
 T 
 
 log x 
 
 V 
 
 q Tauri 
 
 I 
 
 5 h jm I9 s 79 
 
 5 h 8 m 4I 8 .89 
 
 +20.565 
 
 9.81273 
 
 + 1.98 
 
 
 E 
 
 1947 
 
 35-82 
 
 20.565 
 
 9.81269 
 
 i 
 
 20 " 
 
 I 
 
 19 .OO 
 
 33.84 
 
 2O.79O 
 
 9.89853 
 
 < 
 
 
 E 
 
 19 .82 
 
 42.42 
 
 2O.79O 
 
 9.89849 
 
 < 
 
 16 " 
 
 E 
 
 19-53 
 
 40 .11 
 
 20.315 
 
 9.9H35 
 
 ' 
 
 21 " 
 
 E 
 
 19.19 
 
 
 2O.9II 
 
 977634 
 
 ' 
 
 22 " 
 
 E 
 
 20.28 
 
 
 20.960 
 
 9.80208 
 
 1 
 
 \ Aquar. 
 
 I 
 
 19.13 
 
 32.51 
 
 I6.83I 
 
 9.93616 
 
 ; 92 
 
 
 E 
 
 17 .68 
 
 36.43 
 
 17.762 
 
 0.05258 
 
 
 78 " 
 
 I 
 
 18.99 
 
 15-63 
 
 17-995 
 
 9.87278 
 
 < . 
 
 
 E 
 
 18 .72 
 
 41 .60 
 
 17-995 
 
 9.87080 
 
 1 
 
 T Sagit. 
 
 I 
 
 1 8 .90 
 
 37.78 
 
 14.207 
 
 9.96912 
 
 1. 80 
 
 
 E 
 
 12 .67 
 
 31.96 
 
 I4.2O8 
 
 9.96912 
 
 
 
 K Gemin. 
 
 I 
 
 23-94 
 
 32.74 
 
 20.446 
 
 O.OOII2 
 
 1.82 
 
 
 E 
 
 5 i 23 .85 
 
 5 8 32.49 
 
 +20.506 
 
 9.93839 
 
 + 1.82 
 
 The coefficients for the final equations are obtained from the 
 expressions : 
 
 v tan \f/, v.E = v(n(t + W T) tan \l/x), v sec \f/ 
 
 
 
 Philadelphia 
 
 Washington 
 
 v tan \f/ 
 
 v-E 
 
 v sec ^ 
 
 v tan ^ 
 
 v-E 
 
 v sec \ff 
 
 q Tauri 
 
 I 
 
 +0.779 
 
 2.2OO 
 
 -2.13 
 
 +0.779 
 
 2.24O 
 
 2.130 
 
 
 E 
 
 +O.I3I 
 
 -0.732 
 
 + 1.98 
 
 0.188 
 
 -0.815 
 
 + 1.980 
 
 20 " 
 
 I 
 
 +0.423 
 
 2.140 
 
 2. 02 
 
 +0.385 
 
 2.070 
 
 2.O2 
 
 
 E 
 
 1.160 
 
 +O.2O7 
 
 + 2.29 
 
 1. 120 
 
 +0.134 
 
 +2.27 
 
 16 " 
 
 E 
 
 -0.963 
 
 +0.073 
 
 +2. 2O 
 
 O.926 
 
 +O.OO4 
 
 +2.18 
 
 21 " 
 
 E 
 
 +0.267 
 
 0.782 
 
 + 1-99 
 
 
 
 
 22 " 
 
 E 
 
 -0.033 
 
 -0.595 
 
 + 1.98 
 
 
 
 
 X Aquari 
 
 I 
 
 +0.068 
 
 -0.751 
 
 -1.92 
 
 O.OI7 
 
 0.8II 
 
 -1.92 
 
 
 E 
 
 -0.732 
 
 O.O92 
 
 +4.06 
 
 -0.673 
 
 + I.2IO 
 
 +2.04 
 
 78 " 
 
 I 
 
 +0.040 
 
 O.24O 
 
 -i-93 
 
 + 0.083 
 
 -P-365 
 
 -1.94 
 
 
 E 
 
 I.OIO 
 
 + 1.990 
 
 +2.18 
 
 -0.946 
 
 + 1.870 
 
 +2.05 
 
 r Sagit. 
 
 I 
 
 +0.361 
 
 O.27I 
 
 1.84 
 
 +0.342 
 
 -0-335 
 
 -1.83 
 
 
 E 
 
 I.OIO 
 
 +2.320 
 
 +2.06 
 
 -0.958 
 
 +2.230 
 
 +2.04 
 
 K Gemin. 
 
 I 
 
 +1.960 
 
 -3.180 
 
 -2.67 
 
 +2.090 
 
 -3.330 
 
 -2.77 
 
 
 E 
 
 -3.880 
 
 +2.970 
 
 +4.28 
 
 -4470 
 
 +3-130 
 
 +4.82 
 
8 OBSERVATION AND REDUCTION OF 
 
 FORMATION OF THE FINAL EQUATIONS OF 
 CONDITIONS. 
 
 Writing the results thus far obtained, we may now set up the 
 following equations, which we divide into four groups : 
 
 W = 5 h 
 W = 
 W = 
 W = 
 
 W = 
 W = 
 
 W = 5 
 W = 
 W = 
 W = 
 W = 5 
 
 July 17. 
 
 Group I 
 
 /I 
 
 
 
 i m 
 
 19* 
 
 '.79 - 
 
 1.987 
 
 + 0.7791? 
 
 2.I37TAK 
 
 2.2OOA?r 
 
 (2) 
 
 
 19 
 
 49 ~ 
 
 1.98 
 
 + 0.131 
 
 + I-98 
 
 - 0.732 
 
 (4) 
 
 
 19 
 
 .00 
 
 1.98 
 
 + 0.423 
 
 2. 02 
 
 2.140 
 
 (6) 
 
 
 19 
 
 .82 - 
 
 1.98 
 
 1.160 
 
 + 2.29 
 
 + 0.207 
 
 (8) 
 
 
 19 
 
 53 - 
 
 1.98 
 
 - 0.963 
 
 + 2. 2O 
 
 + 0.073 
 
 (10) 
 
 
 19 
 
 .19 - 
 
 1.98 
 
 + 0.267 
 
 + 1-99 
 
 0.782 
 
 (30) 
 
 i 
 
 20 .28 
 
 1.98 
 
 - 0.033 
 
 + 1.98 
 
 - 0.595 
 
 (40) 
 
 8 
 
 41 
 
 .89- 
 
 1.98 
 
 + 0.799 
 
 -2.I3 
 
 2.240 
 
 (i) 
 
 
 35 
 
 83- 
 
 1.98 
 
 + 0.188 
 
 + 1.99 
 
 - 0.815 
 
 (3) 
 
 
 33 
 
 .84- 
 
 1.98 
 
 + 0.385 
 
 2.OI 
 
 2.070 
 
 (5) 
 
 
 42 
 
 .42 - 
 
 1.98 
 
 1. 120 
 
 + 0.27 
 
 + 0.134 
 
 (7) 
 
 8 
 
 40 
 
 .11 
 
 1.98 
 
 0.926 
 
 + 2.18 
 
 + 0.004 
 
 (9) 
 
 Aug. 7. 
 
 Group II' i 
 
 W 
 
 = 5 b 
 
 i m I9 8 .I3 
 
 i 
 
 .927 + o.o68t? i.937rA/c O.75IA7T 
 
 (12) 
 
 W 
 
 = 
 
 17 
 
 .69 
 
 i 
 
 .92 
 
 + 0.732 
 
 + 2.06 
 
 0.092 
 
 (14) 
 
 W 
 
 = 
 
 18 
 
 .99 
 
 i 
 
 .92 
 
 + 0.040 
 
 i 
 
 93 
 
 0.240 
 
 (16) 
 
 w 
 
 = 5 
 
 i 18 
 
 .72 
 
 i 
 
 .92 
 
 I.OIO 
 
 + 2 
 
 .18 
 
 + 1.990 
 
 (18) 
 
 w 
 
 = 5 
 
 8 32 
 
 51 
 
 - 1.92 
 
 + 0.017 
 
 I 
 
 .92 
 
 0.811 
 
 (ii) 
 
 w 
 
 = 
 
 36 
 
 43 
 
 i 
 
 .92 
 
 - 0.673 
 
 + 2.O4 
 
 + I.2IO 
 
 (13) 
 
 w 
 
 = 
 
 15 
 
 63 
 
 i 
 
 .92 
 
 + 0.084 
 
 I 
 
 94 
 
 ~ 0.365 
 
 (15) 
 
 w 
 
 = 5 
 
 8 41 
 
 .60 
 
 i 
 
 .92 
 
 0.946 
 
 2 
 
 05 
 
 + 1.870 
 
 (17) 
 
 Aug. 30. Group III' i 
 
 W = 5 h i m i8 8 .90 1.807 + 0.361$ i.847rA/c o.27iA?r (20) 
 
 W = 5 i 12 .67 1.80 i.oio + 4.06 + 2.320 (22) 
 
 W = 5 8 37 .78 - i .80 + 0.342 - 1.83 - 0.335 (19) 
 
 W' = 5 8 31 .96 i. 80 0.958 + 2.04 + 2.230 (21) 
 
 Sept. 14. Group IV 7 i 
 
 W = 5 h i m 23 8 .94 1.827 + 1.96?? 2.67?rAK 3.i8A7r (24) 
 
 W = 5 i 23 .85 - 1.82 - 3.88 + 4.28 + 2.97 (26) 
 
 W = 5 8 32 .74 - 1.82 + 2.09 - 2.77 - 3.33 (23) 
 
 W = 5 8 32 .49 - 1.82 - 4.47 + 4-82 + 3.13 (25) 
 
 If we assume 7, &, ATT, and irAk to be the same in all of these 
 four groups an assumption which involves no appreciable error 
 we shall have 26 equations in four groups, between those 
 quantities and w and w'. The longitude of Washington (5 h 8 m 
 1 5 s . 78) however, will be considered to be correctly obtained. 
 
OCCULTATIONS OF STARS BY THE MOON 9 
 
 It is evident, however, that for various reasons a direct 
 solution of these equations in each group will not be expedient. 
 In the first place, the large terms involved would render the 
 operation very laborious, and furthermore, it will not be possible 
 to separate ATT from the remaining quantities, without assuming 
 both w and w r to be known. 
 
 We therefore proceed as follows: Assuming the equations of 
 equal weight, we subtract the first from the third, the third from 
 the fifth, etc.; and the fourth from the second, the sixth from the 
 fourth, etc.; continuing thus we obtain the following groups of 
 equations : 
 
 Group I' 2 
 
 4- 
 
 0.6481? 
 
 
 
 4.II07TA/C 
 
 = 4- 
 
 i. 
 
 47OA7T 
 
 - 0.32 
 
 4- 
 
 0.292 
 
 
 
 3.000 
 
 = 4- 1.410 
 
 4- 0.47 
 
 4- 
 
 1.580 
 
 - 
 
 4.310 
 
 = 4- 2.350 
 
 4- 0.82 
 
 4- 
 
 0.195 
 
 
 
 0.092 
 
 = 4- 0.134 
 
 4- 0.29 
 
 
 
 1.740 
 
 4- 
 
 4-320 
 
 = 2.280 
 
 4- 0.26 
 
 4- 
 
 0.611 
 
 
 
 4-I2O 
 
 = 4- 1.420 
 
 6.07 
 
 4- 
 
 0.197 
 
 
 
 4.OOO 
 
 = 4- 1.850 
 
 4- 1.98 
 
 4- 
 
 1.510 
 
 
 
 4.290 
 
 = 4- 2. 200 
 
 4- 8.58 
 
 4- 
 
 0.195 
 
 
 
 0.089 
 
 = + 
 
 o. 
 
 130 
 
 4- 2.31 
 
 + 
 
 1.730 
 
 + 
 
 4.320 
 
 = 2.210 
 
 4- 1.78 
 
 Group II' 2 
 
 4- 
 
 o.Sootf 
 
 4- 
 
 3-997rA 
 
 = 4- 
 
 0.659A7T 
 
 - 144 
 
 
 
 0.772 
 
 + 3-99 
 
 = 4- 
 
 0.148 
 
 4- 1.30 
 
 4- 
 
 1.050 
 
 
 
 4.10 
 
 = 4- 2.230 
 
 - 0.27 
 
 4- 
 
 1.080 
 
 4-4-io = - 
 
 2.740 
 
 4- 0.41 
 
 4- 
 
 0.690 
 
 
 
 3.96 
 
 = 4- 2.O2O 
 
 4- 3-92 
 
 - 
 
 0.756 
 
 4- 
 
 3-98 
 
 - - 1.580 
 
 20.80 
 
 4- 
 
 1.030 
 
 
 
 3-99 
 
 = 4- 2.240 
 
 4- 25.90 
 
 
 
 0.963 
 
 4- 
 
 2.97 
 
 = 2.680 
 
 - 9-09 
 
 Group III' 2 
 
 + 1.37$ 3.907rAc = + 2.59AX 6.23 
 + 1.30 - 3-88 = + 2.56 - 5.82 
 
 Group IV' 2 
 
 + 5.841? 6.96?rAK = + 6.I5A7T 0.09 
 + 6.56 - 7.60 = + 6.47 - 0.25 
 
 By means of these four groups of equations of condition, viz: 
 (I'2, II'2, IH'2, IV'2) we now determine the most probable 
 values of & and ATT. The value of ATT, however, cannot be 
 well determined, as we have before remarked, If it were not 
 
IO OBSERVATION AND REDUCTION OF 
 
 known a priori that such was the case, it would be shown from 
 the normal equations, which would be practically indeterminate 
 for this quantity. 
 
 We should, therefore, determine & and nAk in terms of ATT, 
 in order to see what effect an error in TT will have upon the 
 longitude. 
 
 We derive from the above equations, for groups I '2 and 1 1 '2, 
 only, the following two sets of normal equations; the last two 
 groups are solved as they stand, since there are two equations 
 in each group. 
 
 Normal to Group I' 2 (or I' 3) 
 
 + II.Slz? 2I.237TA/C = + 10.07 ATT + I4-5O 
 
 21.231? + i33.057rA/c = 62.5IA7T 14.89 
 
 Normal to Group IT 2 (or 11' 3) 
 + 6.54$ 12.47^ = + 7.27A?r + 51.9 
 
 12.40?? + I28.6TA/C = 48.3OA7T 186.6 
 
 From I '3 we obtain 
 
 & = + 1.428 + .ooSATT 
 ?rA& = + o.i 12 470A7T 
 
 To find 7 we now substitute these values in (i), (3), (5), (7), 
 and (9), and observing that w f 5 h 8 m i5 s -78, we find the mean 
 value of 7 to be 
 
 7 = + 6".52 - .639A7T 
 
 We now substitute these values of #, 7rA&, 7 in (2), (4), (6), 
 (8), (10), (30), (40), when we find the following values for the 
 difference of longitude between Greenwich and the Flower 
 Astronomical Observatory of the University of Pennsylvania: 
 
 W = 5*1 jm 78^7 _j_ .07A7T 
 
 w = 5 i 6 .99 .40 
 
 w = 5 i 6 .49 - .07 
 
 w = 5 i 6. 53 + .39 
 
 w = 5 i 6 .51 + .29 
 
 w = 5 i 6 .89 .48 
 
 = 5 i 7 -54 ~ -27 
 
 Mean w = 5 h i m 6 S .96 .o6A7r 
 
OCCULTATIONS OF STARS BY THE MOON 
 
 II 
 
 And the resulting longitude from Washington is 
 
 \ = 7 m 8 S .82 - .06A7T 
 
 With the above values of y and & we may now determine correc- 
 tions to the assumed right ascension and declination of the Moon. 
 We have the formulae : 
 
 sin N cos D Aa + cos N. A5 = 7 
 cos N cos D Aa sin N. A5 = # 
 and from these 
 
 Aa = + 6".58 A5 = +- 2".Q6 
 
 Assuming the errors of the star places to be inappreciable, these 
 will represent the errors in the computed right ascension and 
 declination of the Moon at a time corresponding to the mean of 
 times of the observations. 
 
 These corrections, it will be seen, are affected by any small 
 outstanding error in the parallax, as they have been derived by 
 assuming ATT = o. 
 
 In the same way, assuming Ax = o and taking the mean of the 
 values given above, viz: 329 1" we find from the above value of 
 
 -f- ."ii 2 
 
 A& = -f- .000034 
 
 we have assumed 
 therefore, 
 
 k = + .272506 
 k = + .272540 
 
 as shown from these observations. 
 
 In the same way by solving the other groups of equations, we 
 obtain the following results: 
 
 Group 
 
 *Lk 
 
 tf 
 
 7 
 
 Aa 
 
 I 
 
 II 
 III 
 
 IV 
 
 +".H2 O.470A7T 
 .8390.328 
 .5000.560 
 .3229.700 
 
 + l".428+00.008A7T 
 
 +6 .350+00.500 
 -5 .960+00.300 
 3 .820 10.500 
 
 + 6".52-.639A7r 
 + 7 .47 +.320 
 + u .59+440 
 + 9 -93 +.380 
 
 + 6".58 
 + 3 -72 
 
 +13 .92 
 
 + 11 .01 
 
 Group 
 
 A5 
 
 k 
 
 IV 
 
 X 
 
 I 
 II 
 III 
 
 IV 
 
 +2.96 
 +9.08 
 
 -3-99 
 
 1.22 
 
 .272540 
 .272408 
 .272386 
 .272415 
 
 5 h i 
 5 i 
 5 i 
 5 i 
 
 m 6 8 .96 o.o6A7T 
 6 .920.49 
 6.860.80 
 6 .89 I.48A7T 
 
 _ 7 m 
 
 -7 
 -7 
 -7 
 
 8 3 .82 0.o6Air 
 
 8 .860.49 
 8 .920.80 
 8 .89 I-48A7T 
 
 Mean 
 
 
 .272437 
 
 5 h i 
 
 m 6 8 .9i O.43A7T 
 
 _ 7 m 
 
 8 S .87 o.43A?r 
 
12 OBSERVATION AND REDUCTION OF OCCULTATIONS OF STARS 
 
 CONCLUSION. 
 
 The errors Aa and A5 in the Moon's position are somewhat 
 smaller than was to be expected, and indicate that this body is 
 following its computed path somewhat more closely than in 
 recent years. 
 
 The corrections A& to the apparent semi-diameter is markedly 
 negative, but it is possible that values of this quantity secured 
 from occultations may be influenced by the aperture of the 
 instrument employed. 
 
 The final mean value of longitude of Flower Observatory from 
 U. S. Naval Observatory, as shown above, from this work is 
 
 X = 7* 8 8 .87 
 
 The results previously obtained for the same quantity are: 
 By Telegraph X = - y m 8 s .9i 
 
 By Wireless X = - 7 m 8*74 
 
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