library) univtRSITY'OF I CALIFORNIA I ! SAN DIEGO j DR presented to the UNIVERSITY LIBRARY UNIVERSITY OF CALIFORNIA SAN DIEGO by Dr. George McEwen l*imi?llli'imini'ifiW'''h''*'' ^'EGO t 3 1822 01134 5659 lBARNlE'y^nfcNGUAGt„ BODKERYi NEW. USED, f 72579 Est] -oo&ht,sold 111 ■NEAR LIBRARyl ^ "-MED SANDIEC.0 LOANED, -- / : \ • V P A^. 7-1^ c^ THE 0;?, GEcr -■N THEORY AND PEACTICE OF INTERPOLATION: INCLUDING Mechanical Quadrature, and Other Important Problems Concerned with the Tabular Values of Functions. WITH THE REQUISITE TABLES. BY HERBERT L. RICE, M. S., Assistant in the Office of the American Ephemeris, AND Professor of astronomy in the Corcoran Scientific School, Washington, D.C. LYNN, MASS. The Nichols Press — Thos. P. Nichols. 1899. '-3 |4- Copyright, 1899, by HERBERT L. RICE, Washington, D.C. PREFACE. In preparing the following treatise the author has attempted no marked originality, either of subject matter or method. Indeed, sufficient has hitherto been written of Interpolation, Quadratures, etc., to firmly dissuade one from such an endeavor. Yet of the numerous contributions to these allied subjects, there has appeared tlius far no distinct treatise covering the entire ground. As a consequence the autlior has repeatedly felt the need of a work which would give — exclusive of other matter — a simple, practical, yet comprehensive discus- sion of all that is useful concerning Differences, Interpolation, Tabular Differ- entiation and Mechanical Quadrature; — a work, moreover, which would include all tables appertaining to the text which are required by a practical computer. To supply the want thus conceived, tlie autlitu- offei'S the present volume. But while viewing the matter in this practical sense, the writei' regards his work as no mere compilation. Many of the processes and developments are original, so far as he is concerned, and possibl}' altogether new ; while the same remark applies to a few of the minor results. In fact, if adverse criticism be forthcoming, it will probably result largely from the somewhat unusual or indi- vidual methods which in many instances iiave been emploj'ed in preference to the customary forms of analysis. On ' the other hand the author realizes fully the extent of his indebtedness to previous writers for valuable ideas and sug- gestions : and he desires especially to mention the works of Boole, Chauvenet, Encke, Loomis, Newcomb, and Sawitsch as most valuable sources of informa- tion, to which frequent reference has been made. Concerning the bibliographical list at the close of this volume (whicli includes the foregoing names), it is but proper to state that references to several of the earliest writers — sucli as Briggs, Wallis, Mouton, Cotes, Stirling, Mayer, Walmesley, Lalande — have purposely been omitted because of the general inaccessibility of tlieir works. As regards the writings of the present century, however, the author believes that all contributions of importance have been included, and trusts that any omissions of consequence hereafter detected will be regarded merely as oversights. IV PREFACE. Special care has been given to the preparation and printing of the tal)les, with the hope of securing absolute accuracy. At a considerable cost of lahoi-, and by wholly independent methods, the computations were all made in dupli- cate; and in every case the tabular values are true to the nearest unit of the last place. Though a few of these tables have appeared before, several are here published for the first time, and it is hoped they will prove useful to the computer. In conclusion, the author desires to express his cordial thanks and appre- ciation to Mr. E. C. RuEBSAJt, of the Nautical Almanac OfEce, and to Mr. M. E. PoKTER, of the Naval Observatory, for much valuable service and many useful suggestions received during the various phases of preparation of this treatise. Feelings of gratitude further inspire — simple justice even demands — a special word in commendation of the publishers, whose uniform courtesy, accuracy and skill have done much to enhance the general value of the work. H. L. R. W'a.suixgtox, D.C, December, 1899. CONTENTS. CHAPTEK I. OF DIFFERENCES. Section. Pajcc. 1. General remarks concerning tabular functions and the construction of mathematical tables, ......... 1 2. Fundamental definitions and notation. General schedule of functions and differences, .......... 2 3. Method of checking the numerical accuracy of differences. Theorem I, 4 4. N functions yield N—n wth differences. Theorem II, ... 5 5. Effect of inverting a given series. Theorem III, .... 5 6. Differences of two combined series. Theorem IV, .... 6 7. Irregularities in the differences of functions which aie not mathemati- cally exact, ........... 7 S. Detection of accidental errors by differences, 9 9. Numerical examples — in which only one function requires correction, 11 10. Numerical examples — involving two or three erroneous functions, . 13 11. General properties of differences. Expression of z/i"' in terms of the wth and higher derivatives of I'^(t); — equation (4), . . . ir> 12. Determination of the coeiBcients B, C, D, etc., in equation (4), . . 18 13. Remarkable formal relation between the expressions for //<,"* and ^'„, ■ 21 14. The //th differences of any rational integral function of the ii\h degree are constant. Theorem V, ........ 23 15. Converse of the foregoing proposition. Theorem VI, . . . . 24 16. 17. Convergency of differences. Ma;/nitude of talmlar Interral and rhar- acter of function, the principal elements involved. Numerical illus- trations, ............ 25 18. Expression of wF'(t), Examples, ............ 38 VI CONTEXTS. CHAPTEE II. OF INTERPOLATION. Section. Pai?e. 24. Statement of the problem, ......... 40 25. Eigorous jnoof of Newton's Formula, assuming that differences of some particular order are constant, ...... 41 26. Second demonstration of Newton's Formula, restricted as in §25, . 43 27. Formula for computing the interval n, . . . . . . . 43 28. Example of interpolation by Nkwton's Formula, the fourth differences being constant, .......... 44 29. Backward interpolation by Newton's Formula. Interpolation near the end of a series. Numerical example, ...... 44 30. General investigation proving that Newton'.s Formula is sensibly accu- rate as applied to series whose differences practically — though not absolutely — vanish beyond the 4th or 5th order, ... 46 31. Numerical example illustrating the foregoing discussion, ... 57 32. 33. Practical examples in the use of Newton's Formula, . . . . Gl 34. Transformations of Newton's Formula. Modification of the foregoing notation of differences. Stirling's Formula. Schedule of differ- ences referring to same. Example, ...... 62 35. Backivard interpolation by Stirling's Formula. Example, ... 65 36. Further example in the use of Stirling's Formula, .... 65 37. The al'jehrak mean. Practical precepts, ...... 66 38. Derivation of Bessel's Formula. Numerical application, ... 67 39. Second example of interpolation by Bessel's Formula, ... 68 40. Backward interpolation by Bessel's Formula. Example, ... 69 41. Property of Bessel's Coefficients, ........ 69 42. Comparison of the relative advantages of Newton's, Stirling's, and Bessel's Formulae, .......... 71 43. Simple interpolation. Magnitude of error arising from neglect of second differences, .......... 72 44. Interpolation by means of a corrected first difference. Example, . . 73 45. Backward interpolation by means of a corrected first difference. Examples, ........... 74 46. 47. Correction of erroneous tabular functions by direct interpolation. Example, ............ 76 48. Systematic interpolation of series. Reduction of a given tabular in- terval, ............ 78 49. Interpolation to halves. Practical rule, ...... SO 50. Precepts for systematic interpolation to halves. Schedule showing arrangement of quantities. Numeiical example, .... 81 51. Derivation of general formulae for reducing the tabular interval from o) to Htu), m being the reciprocal of a positive odd integer, . . 83 CONTENTS. VU Section. Page. 52. Systematic interpolation to thirds. Example, 88 53. Systematic interpolation to ffths. Example, ..... 89 54. On the best order of performing successive interpolations to halves, thirds, etc., ........... 91 55. Interpolation, with a constant interval ii, of an entire series of functions. Example, ............ 91 Examples, ............ 94 CHAPTER III. DEKIVATIVES OF TABULAR FUNCTIONS. 66. Concerning the close relation between differences and differential co efficients, . . . . . . . . . 57. Practical applications of formulae resulting from this relation. Impor tance of tabular derivatives in Astronomy, .... 58. Derivation of the required formulae in general terms, . 59. Formulae for computing derivatives at or near the beginning of a series Examples, .......... 60. Formulae apijlicable at or near the end of a series. Examples, . 61. Derivatives from Stirling's Formula. Rule for computing F'(t) Examples, .......... 62. Derivatives from Bessel's Formula. Simple expression for F'(t-\-^u>) Applications and examples, ....... 63. Interpolation by means of tabular first derivatives. Example, 64. Application of preceding method when second differences are nearly constant. Practical rule for this case. Examples, 65. Regarding a choice of formulae in any given case. Examples, 97 97 98 101 105 109 115 121 124 127 128 CHAPTER IV. OF MECHANICAL QUADRATURE. 66. Statement of the problem. Important applications of the method, . 130 67. Derivation of formulae for single integration from Newton's Formula. The auxiliary series 'F. Schedule of functions and differences, 131 68. Numerical applications illustrating two of the foregoing formulae, . 137 69. Precepts for computing a definite integral when either or both limits are other than tabular values of the argument T. Necessity of interpolation in this case, . , . . . . . .138 Vlll CONTENTS. Section. Page. 70, 71. Transformation and extension of tlie fundamental relations of §67, such that integrals whose limits are non-tahulaf values of T are expressed directly in terms of Interjiolated values of '/', F, J', zl", z/'", etc. Formulae and examples, ........ 140 72. Formulae for single integration as derived from Stirlixo's Formula. Schedule of functions and differences. Examples, . . . 14G 73. Generalization of preceding formulae to include integrals of ani/ limits. Example, 151 74. Formulae for single integration from Eessel's Formula. Extension to (1711/ limits. Examples, ......... 153 75. Double integration. The conditions involved, ..... IGO 76. Derivation of formulae for double integration from Newton's Formula. Introduction of the series "F. Schedule of functions and differ- ences. General fornndae and relations, ..... IGO 77. Value of the frst integral at the lower limit. Introduction and defi- nition of tlie quantity //„. Collection of formulae for double integration covering all possible cases. Examples, . . . IGG 78. Derivation of formulae for double integration from Stirling's and Bessel's Formulae. Schedule referring to same. Precepts and examples, . . . . . . . . . . . .173 79. Change in value of the double integral Y, due to an arbitrary change in the constant If, ......... . 188 Examples, 189 CHAFTER V. miscellaneous problems and applications. 80. Introductory statement, .......... 191 81. Problem I. — To find the sum of the Ath powers of the first ;• inte- gers. Application to >S'= 1^-1-2^ + 3^ + ^-r^ .... 191 82. 83. Problem II. — Given the series F_„, F_^, F„, F^, F„, etc., and an as- signed value of i''„; to find the corresponding interval n. Two solutions. Examples, ......... 192 84. Problem III. — To solve any numerical ccpiation containing but one unknown quantity. Example, ....... 195 85. Problem IV. — To find the value of the argument corresponding to a maximum or minimum function. Example, . . . . .196 86. Problem V. — Given a series of values, F_^, F_^, F„, F^, F^, etc., of some function F {T) analytically unknown; to find an approxi- mate algebraic exjiression for /'(?'). Examples, .... 198 87. Geometrical problem, .......... 200 88. Concluding remarks, . ^ 202 Examples, 203 CONTENTS. IX APPENDIX. ON THE SYMBOLIC METHOD OK DEVKLOPMENT. Section. 89. Introductory remarks, ......... 90. Definition and operation of the symbols A, A'-, A", etc., 91. Definition and operation of D, D^, D', etc., ..... 92. 93. Proof that the foregoing symbols of operation obey, in general, the fundamental laws of algebraic combination, .... 94. Consideration of negative powers of A and D, . 95. Remark concerning results established in the preceding sections, 96. Demonstration of Theorem III, ....... 97. Fundamental relation between A and D, . . . . . 98. Expression of A, A^ A', etc., in terms of ascending powers of D, Demonstration of Theorem V, . 99. Expression of D, D'-', D', etc., in terms of ascending powers of A, 100. Reduction of the tabular interval o). Expression of F, at K "3 t + 6a) F, «5 where a^^F,—F„, n^^=^ F^ — i^i, . . . ; &o = «i — «o, ^>i = «.,— «i, • • •; Cf,^bi — b„, Ci^Ih — />, , . . . ; and so on. We shall also find it convenient to represent a^, a^, a.,, .... by J/, 4', J,', .... , respectively; \, &i, Z>.,, .... by 4,", 4", 4"> > etc., Thus, generally, ./;"' denotes the (.■^-j-l)"' value in the column of «"' differences. As an example, we tabulate and difference several successive values of F i^T) = T'— 10 T-— 20, thus : THE THEORY AND PRACTICE OF IKTERPOLATION. T F(T) J' J" J'" Jiv Jv - 20 — 9 1 - 29 - 15 - 6 + 36 2 - 44 + 15 + 30 + 60 + 24 3 - 29 + 105 + 90 + 84 + 24 4 + 7f) + 279 + 174 + 108 + 24 5 (i + 355 + 91G + 561 + 282 The difterences are in all cases formed by subtracting (algebrai- cally) downwards, as in the above examijles. It will be noted that the even differences (j", J", . . . . ) always fall on the same lines with the argument and function, Avhile the odd differences ( J', //'", /P, . . .) lie between the lines. 3. Method of Checking the Numerical Accuracy of the Differ- ences. — If, in the numerical example of the last section, we take the algebraic sum of the six given values of J', we find _9 _ 15 4- 15 + 105 + 279 + 561 = +936 Subtracting the first value of F{T) from the last, we have + 916 _ (-20) = +936 which agrees with the first result. Again, in like manner, we find J,"'+Ji"'+J2"' = +36 + 60 + 84 = +180 = +174 -(-6) = 4"--^o" These relations may be expressed generally as follows : Theorem I. — The algehraic sum of any s consecutive values of /]("\ is equal to the last, minus the first, of the s-\-l consecutive j(„-i) terms used in forming the s values of /P"K To prove this proposition, let the differences be as below : //'»-" : ;,, h, h, //._, r,, h,^, ^'"' : /'^ h '^, ^.-1 f^. THE TIIEOKY AND PRACTICE OF IKTERPOLATION. 5 Then, from the (h'finition ol" differences, we have A-, = A,-A,, k, = l,,-h,, , k,_, = h,-h,_,, k, = h,^^-h. Hence, by additiuii, we find A-, + k, + k, + + k,_, + /■. = h,^, - A, which is the algebraic statement of Theoi-em I. This theorem may obvionsly be applied as an independent check upon the numerical accuracy of the differencing. 4. Theorem II. — //" the differences of N values of F{T) are taken, N — n values of z/<"* are derived ; it being assumed that jV>n. For, iV functions evidently yield JV — 1 vahies of y, JST- — 2 values of J", iV— 3 values of j"', etc. ; hence JV values of F{T) yield iV— ?i values of j<'". 5. Inversion of a Series of Functions. — It is sometimes necessary or convenient to invert a given column of functions, thus bringing the last value into the position of the first, the next to the last into the position of the second, etc. For example, let us invert the series given in §2, and observe the effect of this inversion upon the differ- ences. Thus we find : T F(T) J' J" J'" Jiv Jv 6 5 4 3 2 1 + 916 +365 + 76 - 29 - 44 - 29 - 20 -661 -279 -105 - 15 + 15 + 9 + 282 + 174 + 90 + 30 - 6 -108 - 84 - 60 - 36 + 24 + 24 + 24 Comparing this table with the original, we first observe that each column of differences is inverted, like the column of functions itself. Further, having regard to signs, we see that the first and third differ- ences (the odd orders) have changed signs throughout ; while J" and //'^ (the even orders) remain unaltered in sign. 6 TlIK TUEOKY AM) PRACTICE OF IXTKKPOLATION. To prove that such an eifect is true generally, we consider the two series below, the second series being an invi'ision of tlic lirst : F{T) J' J" J'" Jiv F(T) J' J" J'" Jlv «o *o F. F, . «o /80 F, a. ^ '•0 d„ F, «1 /8, Vo K Fs ffj K ''i d. F„ P'l Vi s, F. F:, "3 "4 K Cs ^0 «3 «4 A y-.- Comparing- the first differences, we find «i = F, - F, = _(//_/.;) = — a.. Hence, for the second differences, we obtain Po = "1 — «o = —0;! - (-«4) = "4 - «:! /3| = «2 — «i = — ".J — ( — a.,) = a.^ — a„ = /, Thus, the inversion of the functions inverts J', and changes its signs throughout ; whereas j" is inverted, but does not change in sign, Further, since j"' and .7'^ have the same relation to J", that j' and J" have to F{T), it is manifest that J>" inverts and changes signs, Avhile J'^ inverts with signs unaltered. Extending this i-easoning, we have the following proposition : Theokem III. — Inverting a series of functiotis inverts each column of differences and cha7iges the signs of the odd orders (//', J'", z/^, . . . .), uihile the signs of the even orders (J", zl", . . . .) remain unchanged. In pi-actice it is seldom necessary to re-tabulate tlic function in the inverted order, since we may I'cadily conceive the inversion to be made, merely allowing for the changes of sign in J', J'", J", etc. G. Theorem IV. — The v"' differences of the sitnis of two series of functions are equal to the sums of the corresiwnding n"' differences of the two component series. THE THEORY AND PRACTICE OF INTERPOLATION'. 7 To prove genenilly, Ic-f I'\. F^,J'\, , and f„,f^,f,, . . . . , denote tlie two series of i'uiictioiis ; then the sums of the two series will be K-\-fu, i^i-f:/i, P'-i-i-f,- • ■ • • Al.so, let us desig-uate the first differences of these three series by ./', 8', and D', respectively ; their values are hence as follows : J' F,-F, 8' A-L .t\~-A D' w +/,)-(/•; +./;,) (/';+/=)-(/';+/,) We therefoi'e have A' = (^\+A) - (^0+./;) i'\+A - K-fo = it'\-K) + (f-fo) ^\+f, - ^;-/; = {F,-F,) + (/,-r\) J/ + 8/ These relations prove the theorem directly for /i := 1 ; jjut since the second differences are formed fi-om the first difierences in the same manner that the latter are derived from the given functions, the theorem is also true for n = 2. Similarly with the following differences, each order being the first difference of the order just jireceding. Hence the theorem is true generally. As an example we write : F J' J" J'" - 5 - 4 + 9 + 40 + 95 + 1 + 13 + 31 + 55 + 12 + 18 + 24 + 6 + 6 / (5' (5" ,i"' + 14 + 1G + 19 + 19 + 13 + 2 + 3 -6 + 1 -3 -G -4 -3 F+f D' D" D'" + 9 + 12 + 28 + 59 + 108 + 3 + 16 + 31 + 49 + 13 + 15 + 18 + 2 + 3 It will be observed that the values of D', D" and D" are in accord with the theorem. 7. Irregularities in thr Differences. — In the numerical example of §2, the values of J" are all zero. In such a case, we say that the differences are perfectly smooth or regular. In practice, however, the 8 THE THEORY AND PKACTICE OF IKTEKl'OLATION. differences frequently exhibit a .small degree of irregularity, owing to the omission of decimals in the approximate values of the functions employed. As an example, we take the following \alues of T*, true to the neai-est unit of the second decimal : r F{T)^T* J' J" J'" Jlv 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 16.00 19.45 23.43 27.98 33.18 39.06 45.70 53.14 61.47 + 3.45 3.98 4.55 5.20 5.88 6.64 7.44 + 8.33 + 0.53 .57 .65 .68 .76 .80 + 0.89 + 0.04 .08 .03 .08 .04 + 0.09 + 0.04 - .05 + .05 - .04 + 0.05 That the irregularity here manifest in the outer differences is due to the fact that the tabular values are only approximate (not the true mathematical values of the function), may easily be shown l)y Theorem IV, thus : let F denote the true value of the function ; F, its approximate value as above ; / = F— F, the difference of these values. Then, since F is given to the nearest nnit of the second place, / may have any value from — 0.5 to -j-O.o, in terms of the same unit. Moreovei', the values of / do not follow any law of progression, but proceed at random, with arbitrary changes of sign. Hence, the differ- ences of / will be irregular. The differences of F must proceed regu- larly, however, since F is the true mathematical value of a continuous function. Now, since _F= F-\-f\ it follows from Theorem IV that the differences of F must equal tlie sums of the corresponding dif- erences of F and f ; therefore, the differences of F must contain just such irregularities as are inevitable in the differences of f. To illustrate this principle, we tabulate below the values of F, along with the given series, F ; Avhcnce _/' follows, in units of the second decimal, and also the differences of/ to the fourth order : THE THEORY AND PRACTICE OF INTERPOLATION". r F{T) F{T) f=F-F J' J" J'" Jlv 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 16.00,00 19.44,81 23.42,56 27.98,41 33.17,76 39.06,25 45.69,76 53.14,41 61.46,56 16.00 19.45 23.43 27.98 33.18 39.0(; 45.70 53.14 61.47 0.00 + 0.19 + 0.44 -0.41 + 0.24 -0.25 + 0.24 -0.41 + 0.44 + 0.19 + 0.25 -0.85 + 0.65 -0.49 + 0.49 -0.65 + 0.85 + 0.06 -1.10 + 1..-.0 -1.14 + 0.98 -1.14 + 1..-.0 -1.10 + 2.60 -2.64 + 2.12 -2.12 + 2.(;4 ■ + 3.76 -5.24 + 4.76 -4.24 + 4.76 We now bring together, iroxn the above tables, the fourth differ- ences of i''' and /' denoting these quantities by (.7'^)/'' and {J")f, resj^ec- tively. The fourth differences of F then follow, since we have shown that {J")F = (J''')F+ (J'")/; thus we form the table below : (J'v)F (J"')/ (Jiv)^ + 0.04 -0.05 + 0.05 -0.04 + 0.05 4 0.03,76 -0.0.5,24 + 0,04,76 -0.04,24 + 0.04,76 + 0.0024 + 0.0024 + 0.0024 + 0.0024 + 0.0024 It will be observed that the fourth differences of F'(T) ai-e absolutely uniform, — that is, the ii'regularities in (j'^)/' and (./'^')/ ex- actly correspond, or Ijalance. The slight irregularity in the outer differences of the series F{T) is therefore due entirely to the omis- sion of decimals, since it wholly disappears when we employ the true mathematical values, F(T). As a valual)le exercise, the student should now difference the function F directly, and Hud the foui'th differences exactly as above deduced. 8. Detection of Accidental Errors. — We have just seen hoAv a slight deviation from the true value of a tabular function will mani- fest itself by means of irregularities in the differences. If, then, some one value of a series is in error by an ajjpreciable quantity, an in- spection of the differences will indicate definitely the location and magnitude of the error sought. 10 THE THEOKY AND PRACTICE OF INTERPOLATION. To investigate tlie principle that underlies the method, let F F F F F F denote the correct values of any function F' {T) (tabulated for equi- distant vahies of T), and let the diftei'ences be as shown in the schedule below : F{T) J' J" J'" Jlv Jv F, F, a„ /> F., «i ''. C,l '4 F, F. F. "2 k. C] (/, <'o a.. h., f„ (i e. "4 1^4 '3 (/, «-2 F, F. h, ''4 ''4 '^4 F, F, F^o Fu a- (ho h. '■(1 '4 (/- F,, "u Let us now assume that some one function, say F^, is in eri'or by the quantity e, so that i^g-j-e is tabiilated in place of the true value F^ ; the differences of the incorrect sci'ies will therefore be found as follows : F(T)+t J' J" J"' Jiv Jv Fo F, «„ ^'0 F, «i '\ c'„ F. F, F, F, + ^ F, Fs F, F:„ Fu "2 «3 «, + £ "7 «8 O9 h.. 64+ € ^1(1 '■1 Cj-3€ 6-5 + 3e C9 '^1 ff2+« (/,-4£ d, + 6£ d.-4£ Fn "u Now, because the differences of the correct table contain no irregularities, we see that the differences of the incorrect table consist of series of regular values, to which are alternately added and sub- tracted the terms in e, shown in the above schedule. The law oi' progression and increase in the coetiicients of e, along the successive THE THEORY AND PRACTICE OF INTERPOLATION. 11 orders of differences, is easily seen to he that of the hinomial coef- ficients, with alternate signs. Hence, in practice, we have only to carry the difterencing' to that order at which the differences of the correct functions would vanish, or scnsihly so ; the location and mag- nitude of the error will then be clearly shown by a succession of -|- and — terms, following the binomial law. Thus, if the values of J' vanish in the correct table above, the fifth differences of the ineori'ect sei'ies will be 0, -{-e, — 5e, -(-lOe, — lOe, -|-5e, — e, 0; the initial value, -|-e, is therefore the error sought, both as to magnitude and sign. The required function is found by tracing backwards and downwai'ds along the line of heavy type from gj-j-e to J^\-\-e, which is the incorrect function ; and since the cor- rection is the negative of the error, we have {I^\-\-e) — e, or i^;, for the true value of .the function in question. 9. We shall now consider several examples, in order that the process may be fully understood. Example I. — Find the error in the following table of F{T) = T^: T F(T) = T^ J' J" Jill Jiv c Jiv+C 1 2 3 1 8 27 + 7 19 'i'7 + 12 18 + 6 + 6 - 4 + 36 -24 + 16 + 6 4 64 61 81 137 169 217 + 271 24 -10 + 10 5 125 20 + 40 -40 6 206 56 -60 + «;o 7 8 9 343 512 729 32 48 + 54 + 40 -10 -40 + 10 10 1000 The differencing is continued until we find a complete alternation of signs, as in J'^'. Now the binomial coefficients of the fourth order are 1, 4, 6, 4, 1 ; it is also seen that the values of J'^ are just these numbers multiplied by 10. Hence, an error of 10 units exists some- where in the function F ; its location is easily determined by ti-aeing backwards and downwards along the line of — 10, — 4, +20, 4-81, to the number 206, which is the quantity sought. The required function is also found by tracing backwards and upwards along the line of — 10, -j-lG, +32, -f-loT, to 206 ; in practice, both lines should be followed, to guard against mistake. 12 THE THEORY AND PRACTICE OF INTEHl'OI.ATIOX. Finally, the number l!()() is too sDiall by 10 unit.s, since the sign of the ei"i-oi" is shown by the leading or initial value of the binomial series in y, namely, — 10. A correction of -|-10 is therefore to be applied to the incoi-rcct function, giving 216 foi- its true valne. In the column c, following J'^ in the al)ove table, are given the corrections to j'^, dne to the correction of -j-lO to the function. The column J"-\-c therefore gives tiie 1th ditferences of the true or corrected sei'ies. It is alwa3^s well to re-ditference the sei'ies aftei- a correction has been applied, to check the accuracy of the work. Example II. — Find the error in the following table of logai'ithms: T logT A> J" J'" c J"'+c 45 50 65 60 65 70 75 80 1.6532 1.6990 1.7404 1.7787 1.8129 1.8451 1.8751 1.9031 + 458 414 383 342 322 300 + 280 -44 31 41 20 22 -20 + 13 -10 + 21 - 2 + 2 — 5 + 15 -15 + 5 + 8 5 6 + 2 The third differences are here sufficient to point out the erroi' ; the correction given under c appears to improve J'" in the best man- ner, thus indicating that log 60 should be 1.7782 instead of 1.7787. It will Ije observed that a correction oi" — (5 is nearly as efficient as — .3 in the above case, and that — 5.5 is better than either ; this is becau.se the value of log GO to five places is 1.77815. Example III. — Correct the error in tiie following ephemeris of the moon's latitude : Date 1898 Moon's Lat. /3 J' J" J'" Jiv Jv c A^+c May 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 -1 59 54.2 1 22 44.2 44 27.0 -0 5 45.3 + 32 39.9 1 10 23.4 1 46 12.4 2 20 14.7 + 2 51 51.2 / II + 37 10.0 38 17 2 38 41.7 38 25.2 37 43.5 35 49.0 34 2.3 + 31 36.5 + 1 7.2 + 24.5 -0 16.5 41.7 1 54.5 146.7 -2 25.8 II -42.7 41.0 25.2 -72.8 + 7.8 -39.1 It + 1.7 + 15.8 -47.6 + 80.6 -46.9 • • ■ • II + 14.1 - 63.4 + 128.2 -127.5 ir - 12.8 + 64.0 -128.0 + 128.0 + 1.3 0.6 0.2 + 0.5 . . . In this example the error is readily indicated in .d", for Avhich order the binomial coefficients are 1, 5, 10, 10, 5, 1. Although but THE TIIKORY AND PRACTICE OF INTERPOLATION. 13 four values of r are available, these are here sufficient. A slig-ht inspection shows that a correction of — 13".(), as applied to the latitude for May 11.0, will very nearly serve the purpose; — 13".() Ijeiug a trifle too great numerically, we soon find by trial that — 12".8 produces the best result. Hence, the moon's latitude for May 11.0 should read, +1° 10' 10".6. 10. Correction of Errors when More than One Function is Affected. — Thus far we have considered examjDles of an error in one function only. When two or more consecutive or neigh])oring- vahies are in error, the pi'oblem of correction becomes moi'c complicated and difl^cult. It may even become indeterminate in some cases, since only accidental errors can be detected by the differences. Several succes- sive functions, and possibly all, may contain systematic errors which do not affect the regularity of the differences. In general, the correction of a group of errors by diffei-ences may be considered practicable only when the law of the function is not obscured or altered l)y the presence of those eri-ors. More definitely, the method may be regarded as available in the case of two or per- haps thi-ee neighboring functions, provided the errors be accidental in character, and of suiticient magnitude to produce a distinct and defini- tive irregularity in the differences. Example I. — Correct the errors in the foUoAving tabulation of F{T) = 2T'— 25T— 40 : T F{T) J' J" J'" Cl J"'+e, c. J"'+Ci+C-2 -4 3 2 -J + 1 2 3 4 5 6 7 + 8 - 68 ^ 19 - 6 - 17 40 63 79 61 - 4 + 85 242 471 + 784 + 49 + 13 - 11 23 23 - 16 + IS 57 89 157 229 + 313 -36 - 24 -12 + 7 34 39 32 68 72 + 84 + 12 12 12 7 27 + 5 ( + 36 4 + 12 + 5 -15 + 15 — 5 + 12 12 12 12 12 + 20 -12 + 36 4 + 12 - 8 + 24 -24 + S + 12 12 12 12 12 12 12 12 12 + 12 We carry the difterences to the third order, and note that the first three values of /'" are constant, and equal to -|-12 ; hence, in u THE TIlKOliV AXr» PKACTIOE OF INTERPOLATION. column c, , we place tlu' convction of -j-"). Thi.'< gives a corrected series for J'", shown under ./'" + (•,. The latter column clearly indicates a correction of — 8, as applied in Cj ; this gives a final corrected column of third differences, witli llic constant value of -\-V2. Hence, ihe value F(7') for 7'=+L\ should read —74 in.stead of —7!); for T = -j-^' ^^'^ should have — 12 instead of — 4. Example II. — Correct the errors which occur in the following ephemeris of the sun's declination : Date Sun's Dec!. 1S98 8 Jan. 28 -18 6 34.7 30 17 34 4.0 Feb. 1 17 19.0 3 16 25 22.9 5 15 49 18.8 7 15 12 6.6 9 14 33 54,0 11 13 54 52.8 13 13 14 45.0 15 12 33 48.1 17 11 52 2.4 19 -11 9 31.4 + 32 30.7 33 45.0 34 56.1 36 4.1 37 12 2 38 12.6 39 1.2 40 7.8 40 56.9 41 45.7 + 42 31.0 J" J'" fl * C-2 J"'+c,+c„ fa J"'+Ci It + 74.3 71.1 68.0 68.1 60.4 48.6 66.6 49.1 48.8 + 45.3 - 3.2 II - 3.2 II -3.2 - 3.1 - 3.1 3.1 + 0.1 -3.2 - 3.1 3.1 1.1 + 9.6 + 1.9 - 5.1 3.2 -11.8 -9.6 + 3.0 -18.4 + 15.3 3.1 + 1S.0 + 3.2-9.0 + 12.2 -15.3 3.1 -17.5 + 9.0 - 8.5 + 5.1 3.4 - 0.3 - 3.5 -3.0 - 3.3 - 3.5 3.3 -3.5 In this case, the first, second, and last values of A"' are — 3.2, — 3.1 and — 3.."), respectively, thus indicating a decided uniform tend- ency in the third differences. The first function in ei-ror is clearly the value for Feb. 7, and the last, that for Feb. 11. There may be an uncertainty of a unit or two in the values of their corrections at the outset; a few trials, however, will indicate that — 3.2 is the best value to apply to -|-0.1 in j'", and -|-3.0 to the term — 11.8. By means of these corrections, the first three and the last two values of .;'" are brought into pi-actical uniformity. In the column c, & Cg are given the corrections to J'", according to the binomial numbers, 1, 3, 3, 1. In the next column, the sum J"' + i\ + r.^ i.s written, which evidently requires a third correction, tabulated under Cg . The differences are now sufficiently smooth. Since c^ coi-responds to a correction of — 5".l to 8 for Feb. 9, we conclude that the correct values of 8 for Feb. 7, 9, and 11, should read, —15° 12' 9".8, —14° 33' 59".l, and —13° .'34' 49".8, respectively. It occasionally happens that some ordei- of difference clearly THE TIIEOIJV AXn PUACTK'F, OF INTERPOLATION. 15 indicates a correction coiTesponding to tlic hiiioinial cocfficifnts of a lower order tlian tliat of tlic differenrc question. '^IMiis means tlie existence of an eiror in some earlier oi'der of dljf'crrvce, rather than an error in the cohnnn of functions. For example, if p i-equires a correction of the order 1, 3, .'{, 1, it follows that an error exists in /I", since ,P is the third difl'erence of /I". More generally, when //<"' requires a correction according to the binomial coefficients of the »/;"' order, an error exists in ./'"-"". These remarks inipiy the necessity of some caution on the part of the beginner. It will be observed that when either the first oi' last function of a series is in error, only the first or the last term in each order of difference will be affected, and only by an amount numerically equal to the erroi-. Hence, in such cases, the method above explained is of little value. In general, it may be stated that when errors have been dis- covered by differencing, it is advisable to re-comjiute the values in question, when the data for the calculation are available. General Properties of Differences. 11. Let F{i), F{t-\-w), F{t^2,oj), represent any series of tabular functions, whose diff'erences are taken as in the schedule below : Function, F(T) J' J" J'" Jfn) J) zlj v ..7/" /''('+ 3) = /XO + <"/'"(0 + ^ F" (t) + '^ F'" (0 + I F^'' (0 + F(f + 2.>) = F(f) + -2 (t) + 8l^i^-(0 + /'(^ + 4a>) = Fit) + io>F(f) + 16'^F"{f) + CA'^F"'(t)+256^F'^-{t) + Differencing these valnes of the functions in the usual manner, we obtain successively the expressions for J', J", .5 -^ F'Ut)+ . ■•' .. co^ J/' = o,= 2^"(0 + 2a.'i?""(0 + f§^F"'(t)+ Rw' F'-'-it) + .... J (3) It will be observed that all terms of the expansions (0) are of the general form, K(x/'F'^^\t) ; where K denotes a numerical factor, and r an integer which increases by unity as we proceed from any term to the next term following. Hence, the diferences will contain THE TIIKORY AND PRACTICE OF INTERPOLATION. 17 only terms of tliis form. We thus see, n priori, that aii}^ difference of the //"' order must he of tlic form Let us now as.'^ione what appeal's fi'om (1), (2), and (.'3) to he the general law; that is A = 1 r = n leaving- the coefficients B, C, D, We therefore assume undetermined for the present. Ji"' = a,"F'"' (t) + B«."+'i^' "+"(;) + Cw"+-F"'-^" (t) + Du,"+'F'"+'" (t) + (4) Since the value of f is arbitrary, we may write f -\- co for t; by making this substitution iji the right-hand member of (4), we evidently get the expression for the »"' difference inunediately following Ji"', — that is, the value of Ji'j^^. Hence we have .^i;\ „"F<"Ut + ,o)+Bo,"+'F^"+''(f + m)+Cw"+-F"'+-'(f + w) + I)m"+'F"^--'\f + „,) + . . Developing the functions of the right-hand member by Taylor's Theorem, we find fo) - •+1 " ^ w" ^.,,„ 7) + a,i^'"+''(0 +~F "-(0+1 F"'+^l {f)+ . . ■ ■ + Bo,"+' F'"+"(f) + o,F '"+"-' (f)+ '^V'"+'"(<)+ .... -1 + 0,,,"+'- F'-'+-\t) + ,.,F'"+'' (t) + . . . . + Z>(0"+' + . . . . Collecting the coefficients of F'^^f^), F"'+"{t), . . . ., we obtain //(;>, = a," F<"' (t) + (B + l)u>"+'F"'+'\f)+[ C +/i +- )o,"+-i^'"+2' (f) (5) + ( />+ C+ I +1 ) a,"+'i^'"+^' (0 + Siibtracting (4) from (5), and observing that /i:;.\ — ./i"' = ./i"+", we get j(n+i) _ ^"+^F>"+'\t}+fB +^\ oi"+^F<"+'''{f) + f <:'-{-?-+ ^^ o,"+'F^''+''(f) + (^+1+1+1)-"" ^"■"'(')+ • • • • ^^^ 18 THK THEORY AND PRACTICE OF INTERPOLATION. If, tlu'i'ofoi'i'. we ])iit li' = ^+^ ^'=^+1+1 ) (C) - = -444 we have J^;''^^'^ u>''*^F^"+'\t) + B'm''+-F'"^-'{t:)+C',.>"+^F'"^''\f) + D'm"^F^''*''{f)+ . ... (7) Hence, if the general form of expression assumed in (4) is true for the index //, it follows from (7) that it is also true for y/-|-l; but we see by equations (1), (2), and (3), that the law obtains for V ^1, 2, 3, respectivel}' ; hence it holds for n = 4; and so on indefinitely. The expression (4) is therefore true for all positive inte- gral values of n. 12. AVe have now to detenuini' the coefficients B, C, J), .... , of equation (4). These quantities are evidently functions of n and s, and will be determined in the following manner : First, we take .-! = 0. and determine the constants for ./y', which we shall denote for this purpose by B„, C„, D„, .... These values are found by induction, thus: the relations (H) give B„+^, Cn+x, A+i ill t^'i'i"*^ f>l" J^n, C„, n,,, . . . . Making n = l, we take By, Ci, D,, . . . . directly from the first of the equations (1); a continued apphcation of ((>) therefore gives succes- sively the values of B.,, B-^, B^, . . . . -S„_, , B„. Similarly, we derive C'„, //„, .... Hence, the coefficients of (4) become known for s = 0. Second, the coefficients of /;"' easily follow from those of j;,"'; for it is cleai- from the schedule of ^^11 that //j"' is related to F{t-\-S(o) in precisely the manner that /),"' is related to F{t)- Hence, if Ibr l)revity we write we shall have, since the value of / is arbitrary, THT5 THEORY AND PRACTICE OF INTERPOLATION. 1!) Then, expanding ^(/-f- s"w) in i' series of powers of sw, we arrive at an expression of the form (-4), in whicli the coefficients are fnlly determined functions of 71 and s. To perform the steps indicated, we take from tiie first of tlie equations (1) the following values: ^1 = i C\ = i I\ = ,V ■ ■ • • (8) To find B„: By repeated application of the first of (6), we have B, = 5, + i B, = B,+\ Hence, by the addition of these n — 1 equations, we get 7?„ = 7?, + *(//-l) = i + i(M-l) To find C„: Using the second of ((j), we obtain c, = c, + i /;, + -■ n ('.. = (',..-, + i B„_, + I whence, by addition, we find C„ = C, + i(5, + /i„+ . . . . +5„_,) + 1 (w_l) Since Ci = i, this gives r=ii— 1 C„ = ^{B, + B, + ....+ /?„^0 +1= \^B,+ l r= 1 But, from (9), we have B^ =z ', ; hence we get c..i2,. + ? = t + S-24»» + l) To find Z>„: Again, from ((>), we derive D, = A + i C + ,\ B., + ,->, (9) (10) ^,. = iA.-i + i C,._, + 1 2}„_i + jV 20 THE THEOKV AM) rKACTlCJh; OF IKTEUP0LAT10^•. whence r=l r= 1 r= 1 From (10), we have 24 I) = .v2;''^+-S''+S or = ,v ''Ijn-l)i2n- -1)" + A n(« — 1)"] ^ n'-' 2 24 ^;„ = £("+!) ( ai) In like manner, the process might be extended to the vakies of En, F„, . . . . ; but the results already obtained are here sufficient. Substituting- in equation (4) the values of B^, C,,, and />„, given by (9), (10), (11). (remembering that these values suppose .s r= 0), we have (12) Ji"' = o>"F-" (t) + "■ cu"+>F'"+>' (0 + ^(3" + 1) «;"«/'>"+=' (0 + ^ (« + l) o."+^'7^'"+»'(0+ . . We now obtain from (12) the expression for ./i"\ As already pi'oposed, we write Then, as shown above, we shall have ,2 2 ,a^8 Jl") = ^(t + S,n) = * CO + .50, *' (0 + ^ *"(0 + ^— *'"(')+ .... [2 (|i_ = U''F'''' (t) + Jiy+'F'"+'' (f)+ C,,,o"+-F'"+-' (t) + J),,o,"+-'F^"^''' (t) + + SW (o,"F-+''{f) + n„io"+'F'"+--{t)+c,y*U''"'+^'(f)+ . . . . + LZ. oy"F^"+-' (t) + 7.'„<„"+i F'"+»' (0 + [2 V + f^'(..^-(.)+. . . .) +. . . . Upon arranging this expression according to ascending powers of o), we get j(... = ^n;^,.„ ^f^ + (y,'„+.s-) ..."+'7^' ■■+>'(/■)+ (c'„+ n„s + ^) ^•'+-F'"^"-'(f) (13) Hence, substitnting the foregoing values of B„, C„, and B„, we DR. C ;--. McEWEN THE THEOHY AND PRACTICE OF INTEKPOLATION. 21 find tliat the values of i?, (\ I) I'ollows: ill e([ii:iti()ii (I) are as I> = L'4 n + 'ls (14) 'n{ii -f- 1 ) + ■-•( " + ■-•) These results are easily verified by substituting special values of n and s, and comparing Avith the coefficients in equations (1), (2), (H); thus, putting s = 1, and taking ?i =: 1,2, 3, successively, we ob- tain the numerical coefficients in the expansions of J,', ./",, and /,"', i-espeetively. 13. lieviarlablc Formal Relation hetween the Expressions for ./;"' and Jj. — The coefficients 7?„, C„, D„ , in the expression for /„", may also be determined by the following method, which not only is shortei- than the above, but also possesses the advantage of showing a direct relation between the expressions for j;" and z/„', respectively. Retaining the above notation, we write (12) in the form //„'■" = u,"r""{i) + «„ 0,"+' /■'"•+" (i!) + c„/"+'' is the coefficient of aj"+'-i^'"+'' (0 iu(lo). Writing n + 1 for n in(13aj, and using the relations (6), we have <^,.+i(2/) = ,'/■'+' + ( J^. + ir ] t/"^-' + { (-■„ + ^" + 11 (2_ 13 !/ (16) ^(^'" + ^'-^f4j^"^^^- ■ • Again, since the coefficients of q, (//) are those of j^', we ob- tain from (1), '■' Jt J /.'* /y* //' (17) 'h {!/) i;- !/■' ip 11 E li 22 THE THEORY AXD PRACTICE OF IXTERI'OI.ATION. By re-ai-ranging the terms of (16), we iiiul ?/"+- //"+■' »"+■' \ E li li //"+< y"+^ */"+'■' / ,^..+5 ,,«+« ,/n+7 + + = (.'/ + -!^+f +f + • • • • )(^"+^"y"^'+^".'/"^'+^'..y"^'+ • • 11 li li Hence, by (15«) and (17), avc have 'r„+i = 'fi • •(„ Taking /^ = 1, 2, 3, .... w — 1, successively, we iind r/.i = ', .... I)", respectively; then we shall ha\e //- n^ i>* !)''■ / , I)' U' D' = ir + '^: 7r+' + A , ;{,, + 1 , D"+"- + ^ ( « + 1 ) l>"^' + 14. Theorem Y. — The n"' differences of amj rationed iideiiral expression of the «"' degree are constant. If the f/eneral form of the function is F(T) = aT" + /8T"-' + yT"-'-+ . . . . , the constant value of ./"" is w"('. [n . For, from the nature of the function, we have, evidently, and F"'+"(t} = F'''+"-'{t) = .... =0 Hence, from (4), we have J',"' = io" F^"'(f) = o)"«[» (22) The theorem is therefoi'e true, whatever the value of the constant interval a.. Several examples have already occurred: in §2 we have 24 THE THEORY AXD PRACTICE OF INTERPOLATION. the differences of F (T) = T'—10T'—20; here ?i = 4, a = 1, 0; = 1. Hence, by (22), we get — the vahie. ah-eady found by differencing. In Example I of §9, F (T) ~ T\ m = 1; we there obtained for the value of the' third ditference J'" = (i which agrees with the theorem. Again, in Example I of §10, F{T) = 2T'—25T—iO, w — l; whence the theorem requii-es J'" = „[3_ = 21i = 12 which is the I'esult already obtained. 15. Theorem \1. — //' the n"' differences of a series of quantities (tabulated for equidistant values of T ) are constant, the given quanti- ties are the tabular values of a rational integral function of the form F{T)=aT"^liT"-'^yT"-'-\- .... This proposition is the converse of Theorem V, and is pi'oved as follows: Let F(T) denote the function whose true matheinatical values, tabulated for the given values of T, form the given series of quantities. From (4) and (5), we see that the expressions for z/^"' and .j;;'i agree only in their first terra, ai"F'"'{t); the remaining terms of like order in u) having unlike coefficients. Hence, the conditions necessary in order that J""' shall be constant throughout are as follows: First, that a)"F"'\t) does not vanish; Second, that w"+'F'"+"{t) = w"+'F"'+-\t) = .... = 0; But, since w cannot vanish, these conditions reduce to the form — ^"" (*)i' I (23) If now we put 7' = f + T (24) then, by Taylor's Theorem, we have THE THEORY AND PRACTICE OF INTERPOLATION. 25 r" ■^"+' By (23), this gives F{T) = F(t) + TF'(t) -\- . . . .+ /.'•■-ii(o+^ /''""(C) (25) In— 1 In in whicli, \vc obsi-rve, tlie cofltifieiit of t" caiiiiot vanish. Substituting" in (25) the value of r given hy (24), we obtain F" it) F""i.f) F(T) = F(t) + {T~t]F'(t)+ (T-t)' ^+. . . . + (7-/)" ^ Sinee t has a fixed value, the right-hand menibei' of this equation is an expression of the 7i"' degree in the variable 1\ and hence may be wi'itten in the form F(T)^aT" + liT-' + yT"--+ .... (•_'(;) Avhieh establishes the theorem. 16. Convergence of the Differences in Practice. — In the discussion of Theorems \ and VI, we were concerned with the true mathematical values of the quantities involved. In practice, however, the absolute or true mathematical values of functions are seldom employed; fre- quently, as previously noted, a function is tabulated only to a certain degree of approximation, enough decimals being retained to give the desired accuracy. We observe that in such cases there is a tendency of the differences to decrease numerically, and usually to vanish sensibly, as the oi'der of difference progresses. The explanation of this tendency follows readily from equation (4), thus: for any given function, the derivatives i^""(0, F"'+'\t), F'"^-' (f) have definite values; hence, the value of w may be chosen sufficiently small to render all the terms in the second member of (4) insensible, excejjt the first. When this condition obtains, the value of J*"' is sensibly constant, and equal to oj"F""(t). The differences of F{T) are thus practi- cally brought to a termination at the /t"' order, whether the function is algebraic or transcendental. In many cases the values of the successive derivatives converge rapidly; the chosen value of w may then be quite large, and yet allow the differences to sensibly vanish at an early order. This is equivalent 26 THE THEORV A^■D PRACTICE OF INTERPOLATION. to the obvious statement that, when a function is to be tabuhited so as to ditterence readily, the interval of the argument ujust be decided by the manner in which the given function varies. To exemplify these principles, we take the following table of seven-figure logarithms : T Log T J' J" J"' 1.00 1.01 1.02 1.03 1.04 1.05 1.06 0.0000000 .0043214 .0086002 .0128372 .0170333 .0211893 0.0253059 + 43214 42788 42370 41961 41560 + 41166 -426 418 409 401 -394 + 8 9 8 + 7 In this case, w = 0.01, t = 1.00, ^ + w = 1.01, t-\-2(o — 1.02, etc. To serve our present jiurpose, we here transcribe from (1), (2). and (3), the following expressions: ^o' = -^'"(0 + 'J^"W+^'^""(«J +^^'"^0+ • ■ • • j;' = „/^i*"'(o + a,'i^"'((;) + T-vu)^i'"^(0+ .... [ (27) j;" = <,/7^""(^)+|co*i?"^(0+ .... \ ^;v _ „4;riv,Y)+ .... / Since F{T) — log T, we have V<{t] = + -Vr' , F"(t) = -Mt-" , F"'{t) = +2.)^-' , F'--(t) = -MIt-' , . . . . where Jf is the modulus of the connnon system of logarithms, = 0.434294. Hence, wnth t = 1 and w = 0.01, we find ^F' {t) = +0.0043429,4 o'^F"(f} = -0.0000434,;! u>^F"\t) = +0.0000008,7 „r/r'v (t) = -0.0000000,3 Substituting these numerical values in (27), we obtain, in units of the 7th decimal, j; = +43214 /„" = -426 J/" = +8 J^'^' = which agree substantially with the results obtained above by direct differencing. The rapid convergence of the differences is thus seen THE THEORY AND PRACTICE OF INTERPOLATION. 27 to be due to the small value of the interval w, which makes the term w'F"'{f) appreciable, but renders aj'F^'(f), a/'F^it), .... (|uite insensible; accordingly. /'" is the last difterence which we need take into account, the remaining diffei'ences being practically zero. We may add that if the values of T in the jjresent table were 100, 10], etc., instead of the givi-n values, the intei'val o) would become 1 instead of 0.01, and hence oi, or, «/', <ractice, the differences generally terminate (either absolutely or approximately) at some ordei* eai'lier than would occur if the true mathematical values of the function Avere emploj'cd. It may be added that the above example affords an illustration of Theorem VI. For, since the second differences are here absolutely constant, it follows from this theorem that the tabular quantities are the true mathematical values (corresponding to the given values of T) of some function of the form F{T) E uT- + (iT+ y Thus, in i)articular, if the student tabulates the function F{T) E 16 (T''- 5.3325 2'+ 9.476975) foi- T ^ 5.16, 5.21, .... 5.4(5, and retains all decimals involved, he will find his tabular numbers identical with the above series. 18. To Exjiress co''F^"\t) in Terms of J\r, 4',"+'', Ji"+", etc. — The j)roblem consists in reversing the series (15), w^hich expresses z/i- in terms of o,"F"''{t), <«"+'F"'+"(/), .... Let us denote <-/^^'^'(/) by Xr\ then, Avriting successively, n, n + l, n-\-2, .... for u in (15), we have THE THEORY AND PRACTICE OF INTERPOLATION. 29 from which we obtain, by transposition, a-,, = Ji"' -/>'„.'•„« -C,,r„^,- A, .T„^,- . . . . \ ^).+l =" ^o" -'^ .1+1 •'•■.1+2 ^n+l^n+3 -^.i+l*^;.+4 . . . . f a;„+o= 4V'+-'-^„+...T„+3 - C„+„.x„+< -Z)„+„.-B„+. - . . . . / " -' The second of the equations (29) g-ives^ a vah\e of «„_(.,, which, substituted in the tirst equation, gives a?„ in terms of j,v", J{,"+", j„^„, .i„+,, . . . ; substituting in tlu' latter expression the vahie of x„^2 given hj tlie tliird of (29), we find «„ in terms of A*,"', Jy^"", z/i""^"', .r,,^.,, a-,,^.,, .... Con- tinuing this process of elimination indefinitely, we ari-ive at an expi-es- sion of tlie form - The coefficients b,, , c„ , d„, .... must now be determined. From (15a) we obtain the following- group of equations : fn+2 J • • • -, the proccss of reversion will be identical with that which gives ,r„ in terms of ./j"', J,V+", . . . . ; hence Ave must find !/" = 'I..+ ''..ff,.+i + '',.aring' coefficients in (3o) and ('^8), we find h, = -i '•, = + :\ 'fi^'- 3§<" + 2)(n + 3)g.r»+ .... (42) Equating coefficients in (33) and (42), we find K = -| ' '•.. = +y4(3" + ">) , d„ = _^(«, + 2)(«, + 3) , .... (43) These values being substituted in (30), the lattei- becomes x„E o>"F'"'(f) = 4V''-^./V'+''+ J4(3"+-"^). '/;,"+=' -^(w + ii)(« + 3)4V'+''+ . . (44) Using the symbolic uotation adopted in (21), we have the follow- ing expressions : D = J.-iJ^+lJ^-iJ^ + lJ^- . . . (45) i)"= (.4-i.^„^+A-'o^-iC+ ■•■)■■ = 4^-|4';+i + ^(3«, + r,)../''+^^-^(,, + 2)(«+3). /;■+»+ . . 19. Effect of a Change in the Argument Interval m, upon the Magnitude of the Several Orders of Differences. — Let us now suppose THF TilKOKY AND I'KAf'TICE OF INTERPOLATION. 31 that a second tabulation of F(T) has hciMi inado, differing from the first only in the value of the interval, to. Let o> = uuo be the in- terval of the argument in the second table; denoting the differences by 8', 8", 8'", . . . ., the new table will run as follows: T F(T) 8' 8" 8'" 8iv .... t F(t) So' 8r,' i + III 01 F(t+ 1,10,) K" 8„"' 8'" 8„"' t + 2»/(u F(t + 2iiiu,) 8," So"- t + 3w/o) F(f+3ino,) 8.," 8i'^ f + 4'/H(i) F(t + ivio,) We proceed to investigate the relations between 8', 8", 8 ", .... and //'. J", z/'", .... No restriction is placed upon the value of m; in the applications of the resulting formulae, however, in will usually be regarded as a positive proper fraction. The second tabulation will then give the function for closer values of T than the first. Since the value of o) is arbitrary, we may write niM for to in the right-hand member of (lo), and thus obtain the expression for 8!,"'; making this substitution, we find 8,V" = iii\,"F f)+ n„m"+'o,"+''F'"+'Uf)+ C„m"+"-o,"+-'F'"+"{f)+ . . . . (46) If, as above, we write ,;■,. for «)''F"'\t), this equation becomes 8J,'" = »'".-/•„+ £,.»("+» a-„+i + C„m"+^r„^.^ + .... (47) From (30) we obtain, in siiccession. (48) Eliminating .i„? ■^■«+ij • • • • from (47), by means of (48), there results an equation of the form 8-' = ///",/r' + /3„.J,V+>' + y„./r-^'+ . . . . (49) which, for « =r 1, becomes 8„' = .«j,/ + A.y„" + y..V"+ .... (50) 32 THE THEORY AM) PRACTICE OF IXTERPOLATION. IS^ow let «„ = w" //"+/?„'"""*"'//"+'+ C'„ »'""•"-.'/"""+ . . . . (51) be ail auxiliary expression, sucli that tlie coefficient of y"+'' is tlie coefficient of .»■„_,_,. in (47). From (33) Ave obtain, in succession, .'/" = gi+ /'„ „ (//(//)"+'+ f,'„( ■/«//) "+-+ .... Whence, by (15a), we have «„ = (f„ (m.)/) (55) and hence, also, or, by (17), 12. li 11 ,-. 1 + Sj = e'"' (56) Also, from (3(5), we have 1 + (f), = fl* the combination of which with (5(5) gives H , ., , . , , w(ot — 1) ., i)i(m — l) . . (»i — )• + !) [i IT. or 7/1 (m — 1) . ml'/n — l) . . . tm — r + 1) THE THEORY AND PRACTICE OF INTERPOLATION. 'sn (58) Comparing (54) and (~u), we liiid _ m(m — l) m{iii — \){m — 2) A - -^ ' V. = - Substituting tiicsc values in (oO), we obtain tlic following fuiida mental relation : tn(iii — 1 ) . (V ; ^ / m(iii — I) _, „ ■ + r + Again, using the relation (/„ = (/;, we obtain from (55) ■~,. = T., ('".'/) = ^Ti ('".'/) I" = -;' Hence, from (57), we find / m (m — 1 ) . m(m — l) (m —2) 2.. = (""/i+-^; — - + ij »i" (?» - 1) J,V'+" + ^ «t" (;;( - 1) (3M + l)«i-(3» + 5) + _„,"(„, _1) i{n + l)m"--2(:n"- + Zn + l)m + {n-\-2){n + ^) .... (G2) Ji»+^> (63) .-'ll ' . . . , Finally, we may symbolize these results by the following expres- sions : (64) 8 - viJ I "<"'-^) j2| H^-m''"'-^) ji^ M»^-l)--(^-3) j4 I mim-l). .{m-ij) ^ 11 li li V = ( iiij, + ^^^^j^+ -(— y— -) ^„3^ 15. 'o^+ K = ».H=+ „r\m-l)J^'+'^(m-l) (7..-ll)4,^+g(;»,-l)(».-2)(3w,-5) J„s^. = (;»J„+ "^^"'~^^ J/+ J= ».''J/+ i- m^(;«-l)j; + ^(».-l)(5,«-7) J/+ 8; = (»/ J„+ "'^"'^ ^^ J;^+ J= /»U;+ 2».^(;«-]) J/+ '^^ (m-1) (13m -17) j;+ o = / . iii.(m — l) ^ ., = in'J„'+ f m\in-l)J„'+im'(m-l)(_4iH-5)JJ-\- 34 THE THEORY AND PKACTICE OF INTERPOLATION. 20. Theorem YII. — If the n"' differences of a given series of functions are tiumerically large as comjyared xvith all the following differences, then, if the series he re-tahulated ivith the argument interval m times its original value, the n"' differences of the new series will he apj)roximatelii m" times the corresponding n"' differences of the original series. The theorem is a direct interpretation of equation (G3). For, if ■^o""^"* 4)'"^"' , • ■ • . are all small in comparison with z/i">, then the ap- proxmiate value of 8,/"' is J/i"//o'"'. Corollary. — If the n"' differences of the given series are con- stant, then the n*^ differences of the new series are also constant, and equal to m" times the original «-'* differences. For, if J<" is constant, j<"+i>, z/<"+2>, .... are all zero, and hence (63) gives, rigorously, gC) /«».//'"' 21. To illustrate the foregoing results, we take the following table of cubes: T F(T)^T^ J' J" J'" 100 103 106 109 112 115 1000000 1092727 1191016 1295029 1404928 1620875 + 92727 98289 104013 109899 + 115947 + 5562 5724 5886 + 6048 + 162 162 + 162 Here the interval w = 3. If we take m :== ^, the interval is reduced to 1, and hence the new table is as follows: T J3 8' 8" 8'" 100 101 102 103 104 105 1000000 1030301 1061208 1092727 1124864 1157625 + 30301 30907 31519 32137 + 32761 +006 612 618 + 624 O CD CD + + We now test the first three of the equations (Gi) ; substituting THE THEORY AND PRACTICE OF INTERPOLATION. 35 in tlic latter m=:}, and observing that the differences beyond J'" vanish, we find S / I /I I X /f " + r, ,1 III S: II I // " 2 / '" Si III \ /I '" /R-'^i From the first of" the above tables, Ave take //; = +92727 z//' = +5502 JJ" = +162 Whence, from (65), we derive 8^' = 30909-618 + 10 = 30301 8,/' = 618 -12 = 606 8^" = 6 which agree exactly with the values found in the second table above. It will be observed that So' and S(," come within s'o l^ail of equaling ^JJ and '//„", respectively; while 8/" = ^\A,"', exactly. These i-ela- tions are in accord with Theorem YII. 22. To Exjiress the Differences of F{T) in Terms of the (riven Functions only. — Let the given series be Fq, F^, F.^, F^, . . . . ; then the first differences are i^i — F^, F^ — F^, F^ — F.^, . . . . ; the second difi'erences, F., — 'IF,^F,, F^ — 2F^^F,,....; the third difter- ences, F^ — 'dF,^'dF,—F,,F, — 'iF.,^dF^—F,,....; and so on. The coefficients evidently follow the binomial law. Thus we have generally (66) Ji"' = F„-nF„_,+ '.^^^^F„_,- . . + (_iy„C,.i^„_, ± . . + (_l)"-i„i^,+(_l)'.i^„ in which, according to the usual notation, we put „CV for the co- efficient of x'' in the expansion of (l-\-x)'\ To prove (66), let us assume it true for the index n; then the expression for the ?i* difference immediately folloMnng /Ji;" (i.e., j[">) will be obtained by increasing the subscripts of F„, i^„_,, .... in (66) by unity. We therefore have A"' = ^„+,-«^„+'-^^^^^„-i- ■ . + (-iy+\,c,.+i^;-,-± • . + {~iyF, (67) Subtracting (66) from (67), we find + {-ir'(„c,.^,+,A)K-r± ■ . . + (-i)"(«+i) /;+(-!)"+'/'„ jr" = A" - A" = ^,.+1 - (« + 1) ^,. + ^^^^ ^,,-1 36 THE TIIEOEY AND PRACTICE OF INTERPOLATION, But, as proved in AlgL'l)ra, wo have r.C'r+l + ,flr — „+lC',._^i and hence the preceding equation becomes (68) ^r" = ^,,+1 - (" + l)^,.+ ^^^^,.-i- • • + (-l)^+'„+iC',.+,i^„..,.± . . + (-!)■■+■ 2^„ It follows from ((38) that if the law expressed in (GG) holds for n, it also holds for n-\-l. But we have seen above that the expres- sion is true for n ^ 1, 2 and 3. Hence it is true for n r= 4, and so on indefinitely; the equation (6G) is therefore true for all positive integral values of n. 23. To Express Any Function of a Given Series in Terms of Some Particular Function (Fq), and of the Differences (f/o, })„, c^, . . . .) which Follow that Function. — As before, let F^, F^, Fn, F-j, . . . . denote the given series, the differences being taken as in the schedule below : F{T) J' J" J'" Jiv Jv J VI K «0 a, «3 "4 h.. '■l 'o /o K K '■;! (L 1 ^'l K 1 K " F «„-I i' , ^„+> ff„ Let it be required to express F^ in terms of F^, a^, \, Co, d^, From the nature of the differences, we have ^1 = F. = F. = K + % J'\ + a, F., + a„ (F„ + 2a^ + h^) + (a, + 2b, + c^) = F, + 3a, + 3lr + r^ THE THEORY AXD I'KAOTICE OF IXTEKPOT.ATION. 37 and so on. The coefficients a<^aiii follow the binomial law, which sug- gests for the form of tlie general term — n(n-1) , n{n-l)(u-2) K = /•; + ««„ + -^^i„+^ ^ > r^+ .... (69) 'Vo })rove (()9) by induction, we assume that it is true for the index ii. Moreover, we evidently have We may now find o„ in terms o^f a^, h^,, c^, (Iq, . . . . from (G9), — since the relation is here the same as the relation of F,^ to i^oj «oj K, Co, • • • • ; thus we obtain , u(n — l) a„ = a^+7ih^+ c„+ .... Adding this value of a„ to that of i^„ given by (69), we find* Thus, having assumed the relation (09) to be true for the index 71, we find by (70) that it is also true when n-\-l is written for n; but we have shown directly that (69) holds for n == 1, 2 and 3. The formula (69) is therefore true for all positive integral values of n. *We here omit the proof for the general term, since the process is the same as in §22. \ 38 THE THEORY AND PRACTICE OF IKTBRPOLATION. EXAMPLES. 1. Tabulate the five-place logarithms of 25, 30, 35, .... 65, 70, and take the differences to the fifth order inclusive. Retain a copy of the table for further use. 2. Tabulate F{T) = log cosT, to five decimals, for T= 50°, 53°, 56°, .... 74°, 77°; difference to the fifth order, as in Example 1. Retain a copy of tlie table. 3. Verify the accuracy of both the functions and their ditferences in Examples 1 and 2, by noting the degree of regularity in l\ accord- ing to the method of §8. 4. Also, rigorously check the differencing in the above" examples, by taking the algebraic siim of each separate order, as explained in §3. 5. Add the two series of functions tabulated in Examples 1 and 2 ; difference the new series as before, and see that the resulting values of .r are the sums of the fifth diff"erence8 of the other series, according to Theorem IV. 6. Correct the errors in the following tables by the method of differences: {a) (V) (S) T F(T) = \, 0.21 4.7G2 .23 4.348 .25 4.000 .27 3.704 .29 3.465 .31 3.226 .33 3.030 .35 2.857 .37 2.703 0.39 2.564 Appa. Alt. Mean of Star Refraction o 10 5 19.2 12 4 27.5 14 3 49.5 16 3 18.4 18 2 57.5 20 2 38.8 22 2 23.3 24 2 10.2 26 1 58.9 Latitude Reduction o 1 ir 0,00 2 4S.(»2 4 1 35.80 6 2 23.12 8 3 9.75 10 3 55.11 12 4 40.05 14 5 23.28 16 6 4.95 18 6 44.86 THE THEOKY AND PRACTICE OF INTERPOLATION. 39 (cZ) («) (/) T F(T)=. Tsinr 0.48 0.7125 .50 .7173 .52 .7226 .54 .7273 .56 .7349 .58 .7419 .60 .7494 .62 .7568 .64 .7660 .66 .7751 .68 .7847 .70 .7947 0.72 0.8052 Date Log. Dist. of 1898 Mars from Earth Sept. 17 0.139162 21 .130819 25 .122145 29 .113130 Oct. 3 .103759 7 .094015 11 .083857 15 .073360 19 .062478 23 .051135 27 .039438 31 .027351 Nov. 4 0.014875 Date Lunar Dist. of 1898 Jupiter Dec. 1.0 105 5 59 1.5 99 18 28 2.0 93 31 31 2.5 87 44 46 3.0 81 57 48 3.5 76 10 17 4.0 70 21 14 4.5 64 30 37 5.0 68 39 44 5.5 52 42 5 6.0 46 43 12 6.5 40 40 43 7.0 34 34 29 7. Tabulate the following rational integral functions for the as- signed values of the argument. Before taking the differences, state at which order the latter become constant, and compute the constant value in each case, by Theorem V. Then take the differences, and see that the results agree with the comjjuted values. (a) F{T) E T''- 50 T' + 100 T-. (Tabulate for T = -8, -6, -4, -2, 0, +2, +4, +6, +8.) (b) F(T) E ST'' -72' -400. (T = 8.0, 8.3, 8.6, . . . .9.8.) (e) F{T) E 0.16r<-0.3r-'. (T = 2, 3, 4, 5, 6, 7, 8.) 8. By means of the first of equations (1), compute the value of //' which immediately follows log cos 56° in the table of Example 2. The value of w (=3°) must be expressed in circular measure. Com- pai"e the computed with the tabular value. 9. Tabulate F{T) = log T, to five places of decimals, for T = .30, 40, 50, 60, 70; denote this table by B, and that of Example 1 by A. A and B then diifer only in w, the interval having now been doubled. Then, in the second of the equations (64), put in = 2, and substitute from A the values of JJ', ,]^"', zl^'\ and j/, which correspond to T ^ 40. Whence, compute the value of 8/ corresponding to T ^ 40 in B, and compare computed with actual value. 10. In Example 1, compute the quantities J„''' and i^5(=log50), by (66) and (69) respectively; compare the results with the values found in the tabic. CITAPTER II. OF INTEKPOLATION. 24. Statement of the Problem. — Given a series of munerioal values of a function, for equidistant values of the argannent, it is required to find the value of the function for any intermediate value of the argu- ment, independently of the analytical foi'ni of the function, Avhich may or may not be given. Interjiolation is the process or method by which the rc(|uired values are found. Without certain resti'ictions or assumptions as to the character of the function and the interval of its tabulation, the problem of inter- polation is an indeterminate one. Thus it is evident, a 'priori, that from a series of temperatures recorded for every noon at a given station, it would be impossible to obtain by interpolation the tempera- ture at 8.00 P.M., for a given ^?ij. If, per contra, the thermometric readings were recorded for 7.00, 7.10, 7.20, 7.30, .... p.m., it is highly probable that the temperature at 7.14 p.m. could be intei'polated with accuracy. The Nautical Almanac gives the heliocentrjc longitude of Jupiter for eveiy 4th day; but, because of the slow, continuous, and syste- matic character of Juinter^s orbital motion, it is found sufficient to compute the longitudes from the tables direct for every 40th day only. The intermediate places are then readily interpolated with an accuracy which equals, if indeed it does not exceed, that of direct computa- tion. The moon's longitude is given in the Nautical Aluuoiac for every twelve houi-s; for the moon's orbital motion is so rapid and compli- cated that it would prove inexpedient to attempt the interpolation of accurate values of the longitude from an ephemeris given for whole day intervals. THE THEORY AND PRACTICE OF INTERPOLATION. 41 It therefore appear.s that, to render the problem of interpolation determinate, the tabnlar interval (w) nms^t be ssnthciently small that the nature or law of the funetion will be definitively shown by the tabnlar values in question. The eondition thus imposed will be satisfied when, in a given table, the differences become either rigoroui^} ij w sensUAij constant at some paiticular order.* This follows from the fact, soon to be proved, that for all such cases a formula of interpo- lation can be established, either rigorously or sensihly true, according to the foregoing distinction. 25. Extension of Formula (69) to Fractional and Negative Values of a, Provided the Differences of Some Particular Order are Constant. — We have shown (Theorem V) that the differences of a rational inte- gral function vanish beyond a certain order. We j^roceed to prove that, for any such function, the formula (69) is rigorously ti-ue for all values of n. Let F(T) denote any function whose differences become con- stant at the order i, and let j"> = Ig-, F(T) and its differences are then shown in the schedule on following page. * Excepting, of course, any periodic function whose tabular interval (m) differs but little from some multiple of its period, P. An example of such a series is the following : Date, 1S98 Day of the Year Heliocentric Longitude of Mercury J' J" Jill Jan. 4 Apr. 4 July 3 Oct. 1 Dec. 30 4 94 184 274 364 o / 93 105 33 117 40 129 14 140 10 +12 33 12 7 11 34 +10 56 —26 33 -38 1 —7 —5 where P (the time of one revolution of Mercury) = 87.97 days; and hence M = 90'' = P + 2''.03. The differences J' therefore correspond to a tabular interval of 2.03 days, and not to the interval 90 days, as the table itself would indicate. Now, the actual value of Mercury's longitude for Jan. 14 is found from the Nautical Almanac to be I = 149° 40'; if, however, we fail to account for the periodic character of this function, and argue solely from the numerical data at hand, we find by a rough interpolation, for Jan. 14, ; = 93°.0 + (U X 12°.6) = 94°.4 which bears no relation to the truth. The pos.sibility of thus committing serious error through fail- ing to account for completed periods or revolutions, suggests the necessity of caution in this direction. 42 TIIE THEORY AND PRACTICE OF INTERPOLATION. T F(T) J' J" j(i) t K t+u, F. % Ik t + 2u, t "T OOJ «1 a.. h b. 'o 'o h \ 1 'o i;+(i + 2))- F(t) = F„,-F^ .: F„ E F^t + mw) = -f'o + So' ^ ,, m(»i—l) ^ ,, m(m~l) . . (vi—i+1) , ., \L Li which is the same as formula (73), except that m is written for n. 27. To Find n, the Interval of Interjjolation. — The binomial co- efficients of Newton's Foi-mula are given in Table I, for every hun- dredth part of a unit in the argument n. The quantity ?t is called the interval of Interjiolation, and in practice is always less than unity. To obtain an expression for n, suppose that we are to interpolate the value of the fi;nction corresponding to the ai'gument T, whose value lies between t and t-\-w; then we shall have F„ E F{t + ni^) = F{T) , or t + no, = T and therefore T — f n = "—^ (74) (0 which determines the interval n. u THE THEORY AXD PKACTICE OF INTERPOLATION. 28. Example. — From the following tal)le of 2'\ find tlic \i\\ul- of {2.8y by Newton's Formula: T F(T) = T* J' J" J'" Jiv Jv 2 4 6 8 10 12 14 16 256 1296 4096 10000 20736 38416 + 240 1040 2800 5904 10736 + 17680 + 800 1 , „..., ._„„ 1 + 960 1'60 .„,, 3104 4832 , .0119 + 6944 ^--^J- + 384 384 + 384 Here avc have T = 2.8 r,„ = +240 f = 2 /,^, = +800 <„ = 2 r^ = +960 .„ ^ 2.8--J _ 0.4 r/„ = +384 /': = 16 r = It will be convenient to denote the coefHeients of a„, b(,, c^,, . . . . in (73) by A, B, C, . . . . , respectively. Then, fi-om Table I (with argmnent n =z 0.40), or by direct computation, we find We therefore obtain A = + 0.40 C = +0.0G40 B = -0.12 D = -0.0416 ^0 ^ + 16.00 Aa^ = + 96.00 BK = -96.00 Co, = + 61.44 Dd, = -15.9744 .-. (2.8)^ = 7*;.4 = +61.4656 This resnlt is easily verified, and found exact to the last figure. However, since Table I does not in general give the exact mathe- matical values of the interpolating coefficients, it follows that functions interpolated in this manner cannot ahvays be ahsolutehj correct. The results may be, as in logarithmic computation, but close approximations to the truth. 29. Backward Interj)olatio7i: — When the interval of interpolation approaches unity, it is usually more convenient to jjroceed backwards from the function which follows the value sought. The problem. THE THEORY AND PRACTICE OF INTERPOLATION. therefore, is to find /^_„; for this purpose, let F (T) he differenced :is in tlie schedule below — the values of //<" being supposed constant as before ; T F{T) J' J" J"' Jtv ■ ■ ■ z](0 t — 3, = 2 = 0.3* which is the interval counted hachwards from F = 38416. Hence, from Table I, we obtain A = +0.37 C = +0.06333 B = -0.11655 D = -0.04164 And for the differences required by (75), we have a_i = +17680 c_3 = +2112 b_„ = + 6944 d_, = + 384 Therefore, by (75), we derive F^ = +38416.00 -Aa_^ = - 6541.60 + Bh_^ = - 809.32 -Cc_^ = - 133.75 + Dd_^ = - 15.99 .-. F„ = (13.26)' = +30915.34 By direct calculation, we find (13.26)' = 30915.34492 + 30. Application of Newton's Formula, wlien the Differences Be- come only Approximately Constant. — We have proved (§§25 and 26) that (73) is true for all values of n, provided the differences of some particular order are rigorously constant. We now propose to show that, if the value of n lies between and -|-1, the formula is very approximately true for the more frequent case in which the differences of some oi'der become approximately, but not absolutely constant. The example given on page 8 is typical of this case; the numbers involved are not the true mathematical values of the quantities represented, and hence the irregularities, as already explained. Let Fo, F^, F^, Fi, .... Fr , ... . denote a series of approxi- mate tabular values of any fimction F(T), given for equidistant THE THEORY AND PEACTICE OF INTERPOLATION. 47 values of T, and true to the nearest unit of their last figure; let F(^, Fi, F^i F3, .... Fr, . . . . denote the corresponding true mathe- matical values of the series, which we shall designate generally as F; also, let Fr ^ F,.-\-f^; /". being the difference between the true and approximate values, due to the omission of decimals in the tabular quantities. The differences of F, and those of the series fo, f\ , f-i, fa, • • • • ■, are now defined by the two schedules l)elow: T F(T) J' J' J'" ■ ■ JiO Jc+ii • ■ t «+ (0 t + 2u> f + 3(0 t + 4(0 t + 5(0 Fi F, Fs F, F, "2 a. ''0 h ^0 c„ "3 "'0 "'-2 . (A) T / J' J" J'" • • j(i) JC+l) • • t /o t + (0 /, «0 A, t + 2 JC+l) • • t t + (0 it + 2(0 t + 3(0 i + 4(o « + 5(0 ^0 = ^o+/o ^^2 = -F'2+/2 ao + «o «i + «i «o + (C, as + «3 a, + «, ^'„ + ft /', + ft L + /3, i3 + ft /'. + ft '•u + yo ''1 + yi C„ + y„ <-s + ys "'0 + Mo W, + fXi W., + /i„ ' Let us now suppose that the differences j<'+i' in Table (C) are either alternately -|- and — , or that -|- and — signs follow each other 48 THE THEORY AND PRACTICE OF INTERPOLATION. irregiilnrly. ^Moreover, tlu' lorcgoing definition of F requires that the ♦ terms in //<■+" are sufficiently small to indicate that no ei-rors exceed- ing half a unit in the last place exist in the functions F(^T). The values of //'' are then approximately constant, and thei-efore Table (C) represents the typical case in practice. We pi-oceed to investigate the accm'acy of Newton's Foi-nuila as applied in this case; assuming that a is always taken within the limits and -|-1, and that terms beyond J<" are neglected. Applying (7)}) to find F,^ from Table (C), and omitting the terms beyond //<•>, we have F„ = (^,+/o) +^K + «„) + i?(/>o + /?„) + (7(o,+y„)+ . . . .+i(/„ + \,) (76) in which A, B, C, . . . . L denote the binomial coefficients of the //th order. Let us now examine the approximate formula (76), to dis- cover its maximum ei-ror when all conditions conspire to that end. The formula (7(5) may be written F. = {F, + Aa,+ Bh,+ .... +LI„) + Q',+Aa,+Bp,+ .... + LK) (77) For brevity, let us put Q E F^ + Aa,-^Bh^+ .... +LI, Ji E fo + Aa„ + BI3^+ . . . . +7>A„ }■ (77«) .-. F„ = Q + li It will be observed that Q is the value obtained for i^„ when (73) is applied to Table (A), terms beyond j"' being neglected. We leave the discussion of Q for the present, to consider the quantity M, which evidently expresses the error of interi)olation due to the un- avoidal)le eri'ors, f, contained in the tabular functions F. Applying tlio fornnilac of §22 to the differences of Table (B), we have «o = /i — /o A = A-2A+f„ y<> =/3 — 3/3+ 3/, — /„ , 80 =/,-4,/;+ Gf,- iA+f„ I ^' > *o = /o - 5/i + 10/3 - 10/, + 5/; - /„ THE THEOEY AND PRACTICE OP INTERPOLATION. 49 Hence, from (77a), we obtain B = /; + Aa^ + Blio+ Cy„ + m„ + Ec„+ . . . . +L\„ = /o + A (/;-/;,) + i?(/,-2A+/„) + C(/,-3/;+3/,-/„) + 7;(/,-4/3+6/;-4/,+/„)+ . . . . +f„(B-3C+6D-10E+ . . . .) +f^(C-AD + 10E~ . . . .) \ (79) +f,(D-5E+ . . . . ) +ME- ....)+.... J Now the binomial coefficients A, B, C, . . . . are connected by the following relations: Hence, since we have assumed that n lies between and -(-1, it follows that A, B, C, . . . . are alternately positive and negative, thus: ABODE .... + - + - + .... We therefore draw the following conclusions respecting (79) : The coefficient of /; is + ; « (I < which, for all values of x numerically less than tmity, may be expanded into the form (l+a;+a-2+.r^+ . . +x<+ . . )(1 -Ax + Bx^- Cx^+ . . ±Lx':f ■ ■) = (1 -«)""' 50 THE THEORY AJSTD PIU.CTICE OF INTERPOLATION. Upon equating the coefficients of .«' in the two members of this identity, we find 1-A+B- C+ ±L = f-lV {n-l)in-2){n-Z) . . . (n-{) = (i-i) 1-lVi-'^ . 1 ^ Now, the first member of this equation is the coefficient of /o iii (79) ; and since the final member contains only positive factors, it follows that the coefficient of Jl in (79) is a jjositi.ce quantity. Ac- cordingly, if we take fo = -{-0.5, in conjunction with the values of f\i fti fii designated in (80), the value of R given by (79) will then be the greatest possible under the assigned conditions. We now append a table of the quantities fo, f\, fi, fz-, as above determined, with their differences : T / J' J" J'" Jiv Jv Jvl Jvil t t + w )! + 2a. t + 3a) !' + 4«, < + 5(u +0.5 + 0.5 -0.5 + 0.5 -0.5 + 0.5 0.0 -1.0 + 1.0 -1.0 + 1.0 -1 + 2 —2 + 2 _2 + 3 -4 + 4 -4 -7 + 8 -8 48 + 15 -16 + 16 -31 + 32 -32 + 63 -64 (B') The special values which must be assigned to the quantities /o, ao, /8o, 7o, • • • • of Table (B) are, therefore. /o «o ^0 7(1 +0.5 0.0 -1 +3 + 15 in units of the last place of the tabular quantities F. Substituting these values in the original expression for li given in (77a), namely. we obtain B = f, + Aa, + Z?/3„ + (7y„ + . . . . B = +0.5-/i + 3C-7Z> + 15JS'-31i^+63G - (81) which gives the maximum value possible to R for j^^J. THE THEOKY AND PEACTICE OP INTERPOLATION. 51 To evaluate (81) for different values of n between and -f-l, we make use of the following abridged table: n = A B c D E F G + — + — + — + O.OO O.IO 0.20 0.30 0.40 0.50 0.60 O.70 0.80 0.90 l.OO .0000 .0450 .0800 .1050 .1200 .1250 .1200 .1050 .0800 .0450 .0000 .0000 .0285 .0480 .0595 .0640 .0625 .0560 .0455 .0320 .0165 .0000 .0000 .0207 .0336 .0402 .0416 .0391 .0336 .0262 .0176 .0087 .0000 0000 0161 0255 0297 0300 0273 0228 0173 0113 0054 0000 .0000 .0132 .0204 .0233 .0230 .0205 .0168 .0124 .0079 .0037 .0000 .0000 .0111 .0169 .0190 .0184 .0161 .0129 .0094 .0059 .0027 .0000 + — + — + — + (D) From these values we tabulate as follows : n 0.00 0.10 0.20 0.30 0.40 0.60 0.60 0.70 0.80 0.90 1.00 + + + + + + + + + - B .000 .045 .080 .105 .120 .125 .120 .105 .080 .045 .000 + 3C .000 .085 .144 .178 .192 .187 .168 .136 .096 .049 .000 - ID .000 .145 .235 .281 .291 .274 .235 .183 .123 .061 .000 + 15^ .000 .241 .382 .445 .450 .409 .342 .259 .169 .081 .000 -31i^ .000 .409 .632 .722 .713 .635 .521 .384 .245 .115 .000 + 63 (? .000 .699 1.065 1.197 1.159 1.014 .813 ..592 .372 .170 .000 If, now, we let R^i Rzi -^4? • differences beyond the 2d, 3d, 4th, lected, then, from (81), we find R, = 0.5 - i? denote the values of B when . order respectively are neg- i?3 = 0.5-5 + ZC B, = 0.5-B + 3C-7D (82) From the last table we obtain, by successive additions, the values of R.2, R3, Ri, . . . . as defined by (82); these values are tabulated below : n 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 R-i 0.50 0.55 0.58 0.60 0.62 0.63 0.62 0.60 0.58 0.55 0.50 B, 0.50 0.63 0.72 0.78 0.81 0.81 0.79 0.74 0.68 0.59 0.50 B, 0.50 0.78 0.96 1.06 1.10 1.09 1.02 0.92 0.80 0.66 0.50 ^5 0.50 1.02 1.34 1.51 1.55 1.50 1.37 1.18 0.97 0.74 0.50 •Be 0.50 1.42 1.97 2.23 2.27 2.13 1.89 1.57 1.21 0.85 0.50 B, 0.50 2.12 3.04 3.43 3.43 3.14 2.70 2.16 1.59 1.02 0.50 52 TIIE THEORY AIS^D PRACTICE OF INTERPOLATION. Whence it is seen that the greatest jwssible vahies of B, under the assumed conditions, are — 'J Ii„ i?3 R, ^5 ^s i?, 0.6 0.8 1.1 l.G 2.3 3.4 (83) While it is obvious that the combination of accidental errors /, shown in Table (B'), is very improbable, yet approximations to such combination will occur occasionally in jiractice. In such cases the errors (H) in functions interpolated by Newton's Formula may be a considerable part of the values given by (83). These values show that when the diflferences beyond J^ are neglected, the error i? cannot be greater than 1.6, in units of the last place in J^. In all probability this error will not exceed one unit ; and when it is considered that the results of an average logarithmic computation are uncertain with- in this amoimt, we are justified in neglecting the error I^, provided that fifth differences are practically constant. Beyond ^5, the limiting values of i? increase rapidl}-. AVe there- fore conclude that, aside from the inconvenience involved, it is im- practicable to interpolate by Newton's Formula when the differences beyond J" are too large to be neglected.* We now consider the expression Q of (77a), that is — Q= F,+ Aa, + Bh,+ . . . . +L1, (84) Now, because the differences of F in Table (C) become approxi- mately constant at J'", notwithstanding the irregularities they contain; so, a fortiori, must the differences of F in Table (A) become sensibly constant at J"', the quantities of this table being mathematically exact. Hence the differences //<'+" in Table (A), namely, tn„ , m, , 111, , in„ will form a series of continuous^ but very small terms, whose values are nearly equal to each other. Per contra, we have assumed that the differences ♦Excepting the case where F{T) is a rational integral function of T, whose tabular values are mathematically exact. TIIE THEORY AND PRACTICE OF INTERPOLATION. 53 of Table (C) either are alternately -|- and — , or that -f- and — terms succeed each other irregularly. It follows that the quantities in must be numerically less than the maximum value of /u, in the series /^o> /^i> /^a) /^a; .... For, otherwise, if the quantities m exceeded the greatest of the quantities fx, the former would mask the effect of the latter in the com- bined series ?M-)-/i,; hence there would be no general alternation of signs in the series + /i„ , /// , + jUi , //;„ + ^2 But this is contrary to our assumption that the differencing in Table (C) has been carried to an order z:/('+" which does exhibit a general alternation of signs. We therefore conclude that -m^ is numeri- cally less than the maximum value of jx. I^ow, from Table (B'), we observe that under the conditions assumed, The maximum value of a(=J') is 1 = (2)"; /8 (=-'") " 2 = (2y; y(=J"') « 4 = (2/; Hence, itia is numerically less than 2'. We have observed above that, as a consequence of the conditions herein assumed, the differences of F in Table (A) are converging, being practically insensible beyond z/'"; hence the fundamental expansions (0), and all relations deduced from these, are valid in this case. The formula (59) is therefore applicable to the series F'(T); hence, writing u for m in (59), we have 8„' = Aa„ -\- Bl>o + Ccg + .... +Ll^ + 31>ii„ + Nn^+ .... in which as many terms should be retained as accuracy requires. But we also have* and therefore F„ = F, + Aa„+Bb^+Cc^,+ .... +Ll^ + Mm^,+ Nn,+ .... * See §26, where the same relations were similarly employed. 54 THE THEORY AXD PRACTICE OF INTERPOLATION, Now, by (84), this equation may be written F„ = Q+ Mm, + Nn, + . . . . or F„ - Q = Mm, + iV?io + . . . . (85) The series 3//»|, -|- JVy^^ -]-.... therefore expresses the diffei-- ence between the true mathematical vahie of the interpolated function and its approximate value Q. But since, as above observed, the differ- ences 7)1 are nearly constant, it follows that the differences n are small in comparison. Hence, iV/?o is small as compared Avith 3I))i, ; in brief, ^Im^ represents, very nearly, the value of the rapidly converging series 3Imo -\- J\lio -)-.... in the right-hand member of (85) . The latter equation may therefore be wi'itten, Avithout sensible error, F,,-Q ^ Mm., (8(0 From (82) we dei'ive R^-R„ = + ZC = (2=-l)(+C) B^-R^ = - ID = (2^-1) l-D) R,-R,^ +15E = (2^-1) { + E) \ (87) Ri+,-B, = {2'-l){-iyM From the last of these, we obtain ± 2'M = R,+, - Ri ± M (88) "We have shown above that m^ is numerically less than 2'; tliis condition may be expressed in the form m, = 2' sin 6 where 6 may have aiuj value between and "lit. From this relation we obtain Mm., = 2'J/sme or, by (88), Mm., = (7i',+i-i?,±Ji) sine (89) Substituting this value of Miuq in (86), we get F„- Q == {R,^^-R,±M) sine (90) THE THEORY ANT) PRACTICE OF INTERPOLATION. 55 From (77rt), we have* K~Q = i?. (91) which, subtracted from (90), gives F,, — -f; = A'i+i sin6' - (1+ siu 6) A\±]H siufii From Table (D) above we see that beyond J'" the coefficient M cannot exceed O.O-i, which is an inappreciable quantity in the present discussion; we therefore write the last equation F„-F„ = ^,+1 sin e- (1 + sin 6) E, (92) The quantity i?i_^.i is numerically greater than 7?;, and both are alike in sign; this condition may be expressed by the relation in which xp has a definite value depending upon the value of i. Sub- stituting this expression for i?j in (92), the latter becomes K-Fn = i^i+i[sin6l- sinV(l + sine)] or K-Fn = ii',+i(sin0 cosV-sin'-.//) (93) Since cos^i/; is necessai'ily positive, and — sin^i// negative, it fol- lows that the coefficient of i?;+, in (93) will be numerically a maximum when sin ^ attains its greatest negative value; that is, when d^'^ir. Taking 6 := % tt in (93), we have F„ — F^ = ^;+i(-eos-f-sin-./^) = --K,+i (94) which is the maximum numerical value possible to F^ — F',i, all con- ditions favoring. If^j, is the true mathematical value of the required function. JT^ is the approximate value of this quantity which is obtained by applying Newton's Formula to Table (C), neglecting differences beyond ^''>: it being assumed, (1) that the given functions i^'o? -^n -^25 F'sj • • • • are true to the nearest unit of their last digit; (2) that n is positive * The quantity R defined in (77a) is not distinguislied by a subscript in tlie earlier part of this discussion. Considered as a particular term of the series ^2,-^3,-^4, • • • ■ , however, it is evi- dent that R should be designated as Ri . 56 THE THEORY AND PRACTICE OF INTERPOLATION. and less than unity; (3) that tlie differences //"• are approximately constant; and (4) that the dift'erences J"+'> are quite small, Avitli -|- and — signs following irregularly. Under these conditions, it follows from (94) that the computed value F^ t'l" never differ from the true value F„ by more than the quantity /i*,.!., . One point further, however, must be considered. In computing Fn by (TG), we shoidd, in practice, obtain the values of tlic several terms to one or two decimals further than are given in F, to avoid accumulation of errors in the final addition. But in writing the sum, J^„, the extra decimals are drojjped, the result being taken to the neared tn/if, as in F. Thus we actually use, not the quantity F„ ob- tained rigorously by (76), but a close approximation to that value, which Ave may denote by {F„). Accordingly, the relation F„~ (i-;) = ±0.5 expresses the maximum discrepancy between F„ and (F„). Combining this expression with (94), we finally obtain F,_ _(/.;) = -i?,^, ±0.5 (95) The quantity ^,^.1 + 0.5 therefore represents the final limit of error in the value of an interpolated function, in units of the last decimal of F. From the value of i?o given in (88), we find that when j^- is nearly constant, the limiting error is +2.8 units. Since it is highl}' impi'obable that all the necessary conditions will conspire to produce this inaxhnvm error, we may add that when the. differences practically terminate at the fifth order, interpolated functions will occasionally be in error by one unit, only rarely in error by two units, and never by three. With sixth, seventh, or higher differences employed, the results become subject to errors which in most cases would be intolerable, and which would probably be obviated by a direct calculation of the function. From the foregoing investigation it therefore appears that, for purposes of interpolation, tabular functions should always be given with an interval sufficiently small that differences beyond p may be THE THEORY AND PRACTICE OF LNTERPOLATION. 57 neglected. This condition is generally fulfilled in practice. As already stated in §24, the longitude and latitude of the moon are given in the Nautical Almanac for every twelve hours; from the values thus given, intermediate positions can always be safely interpolated by using diffei- ences no higher than the fourth or fifth order. On the other hand, a table of the moon's longitude for every 24 hours would yield diifer- ences of the eighth or even ninth order; the use of which in Newton's Formula might produce an error of several units in an interpolated position. In all that follows, we shall assume that differences beyond the fifth order may be neglected. This assumption made, it follows from the preceding investigation that the fundamental formulae, (73) and (75), may be applied in all cases without sensible error, provided that n is taken less than unity. 31. We shall now solve an example which illustrates the main points of the foregoing discussion. If we tabulate the function F{T) _ 75555 606607.920 199841.772 T + 50804.968 T'- + 5645.715 T^- 2169.395 T^ + 116.817 T^ + 1.507 T^ (96) for T = 0, 1, 2, 3, ... . 9, we find that the true mathematical values terminate in the fifth decimal. These values of F {T) are given in the table below, with their differences: F{T) 8.42511 1 6.40508 o 5.89492 3 6.53508 4 7.66492 5 8.55508 6 8.65492 7 7.85503 8 6.76481 9 7.00512 -2.02003 -0.51016 + 0.64016 1.12984 0.89016 + 0.09984 -0.79989 -1.09022 + 0.24031 + 1.50987 1.150.32 + 0.48968 -0.23968 0.79032 0.89973 -0.29033 + 1.33053 J"' -0.35955 0.66064 0.72936 0.55064 -0.10941 + 0.60940 + 1.62086 Jiv -0.30109 -0.06872 + 0.17872 0.44123 0.71881 + 1.01146 J^ + .23237 .24744 .26251 .27758 + .29265 Jvi + .01507 .01507 .01507 + .01507 (A') This table corresponds to Table (A) of the last section. It will be observed that the values of F are jjeculiar fi-om the fact that the 58 THE THEORY A^T5 PRACTICE OF rSTTERPOLATION. last three decimals of each differ only allglMii from the ([uantity 0.00500, or half a unit in the second decimal place; and, moreover, that the actual difference is, excepting the first function, alternately in excess and defect. This condition will rarely obtain, and is here selected only to illustrate the limiting case. If now we drop the last three decimals of F, we obtain a series of approximate values, denoted by -F. The following table gives F, true to the nearest unit of the second decimal, together vpith its differences : T F(T) J' J" jii, Jiv Jv Jvl 1 2 3 4 5 6 7 8 9 8.43 6.41 5.89 6.54 7.66 8.56 8.65 7.86 6.76 7.01 -2.02 -0.52 + 0.65 1.12 0.90 + 0.09 -0.79 —1.10 + 0.25 + 1.50 1.17 + 0.47 -0.22 0.81 0.88 -0.31 + 1.35 -0.33 0.70 0.69 0.59 -0.07 + 0.57 + 1.66 -0.37 + 0.01 0.10 0.52 0.64 + 1.09 + 0.38 0.09 0.42 0.12 + 0.45 -0.29 + 0.33 -0.30 + 0.33 (C) Table (C) corresponds to Table (C) of §30. It will be observed that r and J^', in (C), represent z/<" and J"+'^, of Table (C). The differencing in (C) is not carried beyond J^' , because of the alterna- tion of -\- and — terms. The above values of F may be written as follows: F = F + / 8.43 = 8.42511 + 0.00489 6.41 = 6.40508 + 0.00492 5.89 = 5.89492 - 0.00492 The quantities in the last colunni therelbre represent the residual terms denoted by f in the preceding section. Expressing these values in units of the second decimal, we have the following table of / and its differences: THE THEORY AND PRACTICE OF INTERPOLATION. 59 T / J' J" J'" Jiv jv J" 1 2 3 4 5 6 7 8 9 + 0.489 + 0.492 -0.492 + 0.492 -0.492 + 0.492 -0.492 + 0.497 -0.481 + 0.488 + 0.003 -0.984 + 0.984 -0.984 + 0.984 -0.984 + 0.989 -0.978 + 0.969 -0.987 + 1.968 -1.968 + 1.968 -1.968 + 1.973 -1.967 + 1.947 + 2.9.^5 -3.936 + 3.936 -3.936 + 3.941 -3.940 + 3.914 -6.891 + 7.872 -7.872 + 7.877 -7.881 + 7.8u4 + 14.763 -15.744 + 15.749 -15.758 + 15.735 -30.507 + 31.493 -31.507 + 31.493 (B") It will be observed that the quantities of Table (B") are close approximations to the (limiting) valnes given in Table (B'), of §30. Let ns now apply JS^ewton's Formula to interpolate the valne of F which corresponds to T = 0.40, in Table (C) . Neglecting differ- ences beyond J^, we take from Table I (for n = OAO) , and from Table (C), the quantities to be employed. The result is as follows: J'o = + 8.43 A = +0.40 a = -2.02 Aa = -0.8080 B = -0.12 b = +1.50 Bb = -0.1800 C = +0.064 e = -0.33 Cc = -0.0211 D = -0.0416 d = -0.37 Dd = + 0.0154 JS = +0.02995 e = +0.38 Ee = + 0.0114 • •• K = + 7.4477 Whence, we write for the value of the interpolated function, (K) = 7.45 = 7.44,77 + 0.00,23 = i^„ +0.00,23 (97) Computing the true value F„ from (96), we obtain F., = 7.4320416 + (98) Hence the value (i^„) ^7.45, interpolated from Table (C), is in error by 1.8 units of its last place. The value of Q is the result obtained by interpolating F^ from Table (A'), neglecting differences after J^. Thus we determine Q as follows : 60 THE THEORY AXB PRACTICE OF INTERPOLATION. /; = -1-8.4201 10 A = +0.40 «o = -2.02003 Aa, = -0.S0S012 B = -0.12 b, = +1.50987 />'/.„ = -0.181184 + C = +0.0()4 c„ = -0.3o'.)55 6V„ = -0.023011 + D = -0.0416 do = -0.30109 /^(/„ = +0.012525 + E = +0.02995 eo = +0.23237 A\ = +0.006959 + .-. Q = +7.432387 + The value of 7?g is computed from Table (B") in the same man- ner that Q has just been obtained from (A'). Thus we find /o = + 0.489 A = +0.40 «„ = + 0.003 A'a„ = + 0.001 B = -0.12 /?„ = - 0.987 ^A, = + 0.118 C = +0.064 y„ = + 2.955 Cyo = + 0.189 D = -0.0416 S, = - 6.891 i)8o = + 0.287 E = +0.02995 £„ = +14.763 &„ = + 0.442 .-. (In units of the second decimal) Ii\ = +1.526 [Cf. (83)] Now, from (91) we have K = (,> + B, (99) Substituting the above values of Q and Br,, we find F„ = 7.4324 + 0.01,53 = 7.4477 which agrees with the result obtained directly from Table (C). Since the sixth differences in Table (A') are constant, it follows that the true value F„ diffei-s fi-om the above value of Q only by the term in J" of Newton's Formula. Now, the coefficient of J" is found from Table (D) of the last section to be approximately — 0.0230. Hence, with z/" = -j-O-OlSOT, we derive F„ = Q- (0.0230 X 0.01507) = Q- 0.000346 = 7.432387 - 0.000346 = 7.432041 (nearly) which agrees with (98). The second of these equations gives Q = ^„ + 0.000340 + Substituting this value of Q in (99), we have F„ = 7^„ + iis + 0.0346 THE THEORY AND PRACTICE OF ENTERPOLATION. 61 where tlie numerical term is now expressed in llie same unit as R^. With the above determined value of ^5(^ -]-1.526), the last equation becomes F^ = F„ + 1.56 Finally, since we were obliged to write (i^„) greater than F^ by 0.23 units, it follows that the actual error of interpolation in this instance is 1.56 -|- 0.23, or approximately 1.8 units in the second deci- mal place; which agrees with the result previously obtained. 32. As a more practical application of Newton's Formula, we take the following Example. — Fi-om the appended table, find the sun's right-ascension for April 20'^ O''. Date 1898 Sun's R.A. J' J" J'" J'v April 1 6 11 16 21 26 May 1 6 h m 8 43 20.30 1 1 34.07 1 19 52.99 1 38 19.59 1 56 55.84 2 15 43.08 2 34 42.36 2 53 54.74 III 8 + 18 13.77 18 18.92 18 26.60 18 36.25 18 47.24 18 59.28 + 19 12.38 S + 5.15 7.68 9.65 10.99 12.04 + 13.10 8 + 2.53 1.97 1.34 1.05 + 1.06 S -0.56 0.63 -0.29 + 0.01 Letting t April 16, we have 30-16 5 0.80 Then, from Table I, and the above differences, we find m 6 F^ = 1 38 19.59 A = +0.80 a„ = +18 36.25 Aa„ = +0 14 53.000 B = -0.08 io = + 10.99 Bb^ = - 0.879 C = +0.032 c„ = + 1.05 C<'„ = + 0.034 D = -0.0176 d^ = + 0.01 Dd^ = 0.000 •. Suu's R.A., April 20'» 0" = 1 53 11.75 which is the value given in the American Fjjhemeris for 1898. 33. Since the value of n in the preceding example is only 0.2 less than unity, it is more convenient to intei'polate backivards from 02 THE THEORY AND PRACTICE OF INTERPOLATION. April 21, by means of (75). Thus, from Table I (for ?; = 0.20), and the tabular diiferences, we find n» 8 F, = 1 56 55.84 A = +0.20 a_, = +18 36.25 -A(,_, = -0 3 43.250 B = -0.08 b^„ = + 9.65 + £i_„ = - 0.772 C = +0.048 c_3 = + 1.97 -Cc_3 = - 0.095 Z> = -0.0336 d^ = - 0.56 + Dd_^ = + 0.019 .: Sun's E.A. , Apri' 20" O^ = 1 63 11.74 which agrees within 0^01 of the first result. Whenever a check is considered necessary, the interj^olation may be performed by both methods. Transformations or Newton's Formula. 34. Modification of the Foregoing Notation of Differences: Stir- llng's Formula. — In Newton's Formula of interpolation we use differences which depend only upon the functions F^, F-^, Fo, . . . .; the functions pi'cceding F^, whether given or not, are in no way in- volved. We shall now transform Newton's Formula in such a man- ner as to involve differences both preceding and following the function from which we set out. The resulting formulae will in general be more convenient, rapidly convergent, and accurate than Newton's Formula. In the schedule below, the pi-eceding notation of differences is modified: the even differences which fall on the horizontal line through Ff, are now denoted by the subscript zero, as ?>o and i ''i di «! it + 2a) a.. b„ <■-. (L e.. t + 3u, Fs «3 C3 e. THE THEORY AND PRACTICE OF INTERPOLATION. 63 To derive Stirling's Formula: Applying IN^ewton's Formula to the above schedule, we find for the value of F„, F„ = F^ + na, + lU,, + Cc^ + Dd^+ Ee,+ . . . . (100) where, as before, B, C, D, E, . . . . represent the binomial coefficients of z/", J'", J'^, J^, . . . . , respectively. Let us now put a = H"' + «i) ' « = i(«' + '"i) ' e = +(e'+«i) (101) from which, with the relations rti — a' = h^ , Ci — c' ^ d^ , ?! = e' + . . . we obtain flj = a + \b^ , c' =. c-\d^ , c, = c + it^, , e, = e+ . . . (102) Using the equations (102), together with the relations given in §23, we find a^ = a + ^h^ f'l ~ *o "*" "i = *o "''''"'" "2" '^o c„ = c' + 2d^ + e,= c+^d^+ e+ . . . } (103) d„ = d„+'2e,+ . . . =d„ + 2e+ . . . eg = Cj + . . . = e + . . . Upon substituting these values of (h, hi, C2, .... in (100), the latter becomes F^=::t\ + n{a + U;) + D(J>^ + c + kd,) + C{c + id^ + e+ . .) + D{d.^ + 2e+ . .) + Ee+. . = F^ + na + {B + ^)h^ + {C+B)c + {D + lC+\B)da+{E+2D+C)e+ . . . . Substituting in the last equation the values of B, C, D, E, namely, ^ ^ n{n-V) ^ ^ n{n-\)..{n-Z) 11 ' li ^ ^ n{ri-V){n-2) ^ ^ n(n~l) . . (n-4) li ' 11 we finally obtain n- w(a=-l) reV«=-l), w (w=-l)(ra''-4) ,,„,, F„ = Fo+na+-^b,+ -^ '-0+ ^ ^^ X + -^ j^ 'e+ . . (104) which is known as Stirling's Formula. The even differences em- ployed in this formula are those falling on the hoiizontal line through 64 THE THEORY AND PRACTICE OF INTERPOLATION. i^oj the odd differences are the means of those which fall immediately above and below this line, as defined by (101). Table II gives the values of Stirling's coefficients for the argu- ment n. A glance at this table shows how much more rapidly these coefficients converge than those of Newton's Formula. Example. — From the table below, find the K.A. of the sun for April 20'' 0". Date 1898 Sun's R.A. J' J" J'" Jiv April 1 6 11 h m 8 43 20.30 1 1 34.07 1 19 52.99 m 8 + 18 13.77 18 18.92 18 26.60 S + 5.15 7.68 s + 2.63 1.97 s -0.50 16 1 38 19.59 9.65 0.63 IS 36.25 18 47.24 18 59.28 + 19 12.38 1.34 1.05 + 1.06 21 26 May 1 6 1 56 55.84 2 15 43.08 2 34 42.36 2 53 54.74 10.99 12.04 + 13.10 -0.29 + 0.01 Taking t = April IG (as in §32), we have n = 52zl? = 0.80 The horizontal lines drawn in the body of the table indicate the differences to be employed in (104), as follows: (1) The required values of Fq, J", and J" are those included between two lines; (2) The required values of J' and J'" are the means of the quantities separated hy a single line. As before, we shall denote the coefficients of /I', J", J'", .... by A, B, C, . . . . Taking their values from Table II, with n = 0.80, and forming the required differences as indicated, we obtain /; = 1 38 19.59 A = +0.80 a = +18 31.425 Aa = + 14 49.140 i = +0.32 i„ = + 9.05 Bb„ = + 3.088 C = -0.048 c = + 1.00 Cc = - 0.080 D = -0.0096 d„ = - 0.03 Dd^ = + 0.006 Sun's R.A., April 20'' 0" = 1 53 11.74 which agrees exactly with the result foimd in §33. THE THEORY AND PRACTICE OF INTERPOLATION. 65 8;"). Bdcl-irard Tnterpolation hy Stirling's Formula. — When the forward interval approaches unity, it will be more convenient t(» pro- ceed hacliranl^ from the folloAviiig function by the formula „ n- , «. (rt^ — 1) w-(7i^ — 1) /'_„= F, - na + 2 l>« - ^ g ' <• + ^24 ' 120 e + (105) the coefficients of which are taken from Table II with the argument n, as before. It will be observed that (10;')) is derived from (104) by merely writing — // ibr ?? in the latter; or, by supposing the given series to be inverted, and hence (Theorem III) changing the signs of a, c, and <\ Example. — Solve the preceding example by (lOo) ; that is, find the sun's K.A. for April 20'' O** by backward interpolation. Taking t = April 21, we have M = ?lz^ = 0.20 r> The differences are formed for the date April 21 in the same manner as found above foi* April 20; thence, taking the coefficients from Table II, with u — 0.20, we find m J F, = 1 56 55.84 A = +0.20 a = +18 41.745 — Aa = — 3 44.349 B = +0.02 K = + 10.99 + BK= + 0.220 C = -0.032 c = + 1.20 -Cc = + 0.038 D = -0.0016 do = - 0.29 + Dd,= 0.000 .-. Sun's R.A. , April 20'" 0" = 1 53 11.75 36. Example. — Use Stirling's Formula to compute log sin 9° 22' fi'om the following table: T Log sin T J' J" J"' Jiv Jv O 6 7 8 9.01923 9.08589 9.14356 + 6666 5767 5077 -899 690 + 209 147 -62 + 17 9 9.19433 543 45 4534 4093 + 3728 102 + 76 + 19 10 11 12 9.23967 9.28060 9.31788 441 -365 -26 Here we have t = 9° n = ". ^ 0.36667 60 G6 THE THEORY AND PRACTICE OF INTEKPOLATION. and we therefore obtain /''„ = 9.19433 A = +0.3GGG7 a = +4805.5 Aa = + 17G2.0 £ = +0.0G722 /,,, = _ 543 Bl'o = - 3G.5 C = -0.05289 '■ = + 124.5 Cr = - C.G /; = -0.00485 d, = - 45 Dd, = + 0.2 E = +0.01022 e = + 18 Ee = + 0.2 .-. Log sin 9° OO' = 9.21152.3 The ti'ue vahie to six decimals is 9.211526. 37. T/ic Algehraic Mean. — It may be Avell to observe that in taking the mean of two quantities having Hke signs, and of nearly the same magnitude, it is easier to add one-half their dijferefice to the lesser number, than to take one-half the sum of the two quantities. That is, we proceed according to the identity iix + i/) = a- + i(i/—x) in which we suppose y numerically greater than .c. Thus, in the last example, instead of taking a = ^ («' + «,) = ^(5077 + 4534) = ^9611) = +4805.5 it is easier to follow the equivalent formula a = f,^-i(a,-a') = a,~h_K = 4534 + ^(543) = +4805.5 Similarly, we find c = 102 + 22.5 = +124.5 Per contra, to form the mean of two quantities having unlike signs, and differing but little in magnitude, it is easier to take their algebraic sum and then divide by two. For example, given the values we find F(T) J' J" F, F, -4226 + 5088 + 9314 a = i (5088-4226) = i( + 862) = +431 With these precepts, the required 7nean diffei'ences of interpolation are very I'cadily taken. THE THEORY AND PRACTICE OF INTERPOLATION. 67 38. Bessel's Formula. — We now pass from Stirling's Formula to another, somewhat similar, wherein we employ the odd differences «!, Ci, e,, which fall on the horizontal line between Fo and Fi, and the means of the even differences falling immediately above and below this line. Using the schedule on page (32, let us put Then, since ?>, — h„^c^, and d^ — r/i,:^'",, these equations give ^'o = i-iq , (1, = d-^_e, (107) Let us write the formula (104), for brevity, i.'„ = F^+na+fb,+ Cc+Dd,+ EeJr .... (108) where C = g , i> - 24 ' ^' - 120 ^ ^ Now, by means of (102) and (107), we derive a = ffj — 4-io = fli— K/- — ^f,) = «! — i^' + i^i *o = f> — i ''i c = c, -id, = c,-i(d-ie,) = c,-id + ie, } (110) fZ„ = d —ws^ e = «! — . . Upon substituting these values of a, h„, c, . . . . in (108), we have F„ = F, + 7i{a,-ib + ir,) + f{b-ic,) + C(c,-id + ie,)+I){d-ie,) + E(e,~ . .) + ■ • • Finally, substituting in the last equation the values of (J, D, E, from (109), we obtain „ „ 7i(7j — 1), n(n — l)(n — i) F, = F„+7w,+ ^2 ^ +— i^ ^'■i (>.+ l)n(n-l)(M-2) (n + l) = +30200 Bb = - 3624 C = -0.0040 i\ = +28500 C\ = - 114 Z> = +0.0224 d = +15000 IM = + 336 .-. lO-" = +10000 40. Bacl-ward Interpolation Ixj Bessel's Formula. — To find i^_„ by Bessel's Formula, we conceive the series given on page ()2 to be inverted; the required function is then found by interpo- lating toward from Fi, toward i^_i with the interval n. Hence, the differences to be used in (111) are — We therefore have F_.. = /;- na'+!^ . ?4'-' - "(-^>^-^) .'+.... (111«) the coefficients, as in (111), being taken from Table III Avith the argument n. Example. — Find 10* from the table of §39, by means of (111^/). Taking t = 12, we find ,, = -til' = 0.40 The differences are here the same as in the last example; thus we obtain I\ = +20736 A = +0.40 a' = +18335 -Aa' = - 7334 B = -0.120 y^^' = +.30200 +B.'^^^ = - 3624 C = +0.0040 c' = +28500 -Cc' = - 114 D= +0.0224 —^ = +15000 +7;.'i»l^'= + 336 .-. 10^ = +10000 41. Propertij of Bessel's Coefficients. — If we take from Table III the coefficients for /I", J'", /]", z/^, with the argument n =^ 0.30, and also with ji = 0.70 (=: 1.00 — 0.30), we find the folloAving values: n B C BE 0.30 -.10500 +.00700 +.01934 -.00077 0.70 -.10500 -.00700 +.01934 +.00077 70 THE THEORY AXD PKACTICE OF INTERPOLATION. It will l)c observed that the coefficients are here numerically the same for the arguments // and 1 — n ; having like signs for the even orders, and opposite signs for the odd orders of differences. More generally, let us denote the values of Bessel's coefficients for J", J'", /)'", /T, .... taken with the argument n, by B, C, D, E, . . . ., respectively; and the corresponding values taken with the ai-gument 1 — n by Bi, Ci, Di, E^, . . . . An inspection of Table III then shows that we have B, == +B \ C, = -C ) I), = +I> ) (112) To establish these relations generally, we write (111) in the form F„ = /•; + »»i + i» + C(\ + Dd + i>, + . . . . (113) Now, the value of F^ may also be obtained by interpolating hacl'- loardfy from Fi with the interval 1 — // ; the diffi^rences thus involved will I)e exactly the same as in (113). Hence, after the manner of formula (Ilia), we have F„ = F,- (1-m) «! + BJ> - C\c, + D,d - E^e^+ . . . . (114) But we have, also, ^1 - (i-'0«i = (^i-''i) + "«i = ^0 + ""l Whence, (Hi) becomes F, = 7'; + na, + B,h - C,'\ + l\d - /iV, + . . . . (115) which, subtracted from (113), gives = {B-B{)l,+ {C+C,)c^+{D-D,)d+. . . . (116) The equation (116) is true in all cases to which the formulae of interpolation are applicable; it is therefore true when F(T) is a rational integral function of the second degree. But, in the latter case, the second differences being constant, we have c^ = d = e^ = . . . . = The equation (116) tlien becomes = (B-B,)b THE THEOEY AND PEACTICE OF INTERPOLATION. 71 Hence, since h cannot vanish, we have B^ = +B This result i-ecluces (11 G) to the form = (C+Ci)ci + (Z)-7>i)t/+ (A''+/i\)c, + .... (117) Again, we may suppose J'" constant; tliat is, we may put rf = ?! = ....= The equation (117) then becomes = {C+C\)c, or C^= -C By repeated application of this reasoning, we prove that the rela- tions (112) are true generally. It follows that the numerical process involved in finding F„ by Bessel's Formula is identical whether we interpolate forward from F^^ or backward from F^, except for the terms in F and J'. Hence little or no check is afforded by pei'forming the interpolation by both methods. When such a check is deemed necessary, Bessel's and Stirling's Formulae should both be used. 42. Relative Advayitages of Newton's, Stipo^ing's, and Bessel's Formtdae. — In practice, the only important application of Newton's Formula consists in interpolating functional values near the heginninr/ or end of a given series. The selection of this formula is then a matter of necessity rather than of preference. In all other cases, either of the more rapidly converging formulae of Stirling or Bessel should be employed. Regarding a choice between these two, when Tables II and III are available there would appear to be very little advantage one way or the other. The form given by Bessel is more commonly used, and is perhajis a trifle moi-e accurate in practice than Stirling's form, particularly for values of n in the neighborhood of one-half. When n is quite small, howevei", Stirling's Formula will probably be found more convenient. 72 THE THEOKY ^^'^D PRACTICE OF INTERPOL ATIOX. iSn[)pose we have given a limited table of functions, as follow; F(T) J' J" J'" Jiv F „ F., a" a' "3 '■1 ''0 Assuming that fourth differences must be taken into account, and that fifth differences ai'e to be neglected, the value of i^„ should in this case be comjiuted by Bessel's Formula, which employs the mean of the quantities (/„ and d^ . If, however, the function i^g were not included in this series, then the term fZ, would not be given, and we should proceed b}^ Stirling's Formula, which involves d^ directly. Bessel's Formula is particularly simple and convenient when ?i = ^ , that is, Avhen it is required to find the function which falls midway between F^ and i'^, ; this important case will be fully con- sidered in a later section. 43. Simple Interpolation. — When frequent interpolation is required, as in tables of logarithms, trigonometric functions, etc., the interval of the argument is usually chosen sufficiently small that the effect of second diffei-ences may be neglected. Bessel's Formula gives in this case F„ = F, + na, (118) To interpolate hack wards from i^„, that is, to find i^_„, we obtain from (lllr/), by neglecting second and higher difl'crences. = F,- na' (119) Upon these formulae the process of simple interpolafion is based. The first difference to be used in either case is the value falling between Ff, and the function towaid which the interpolation proceeds. Frequently, where great accuracy is not required, it is sufficient to obtain F„ by simple interpolation even when the second diffi3rences are considerable. In such a case, suj^posing that the third diiferences THE TIIEOKY AND PRACTICE OF INTERPOLATION. 73 are insensible, we observe from Bessei/s Formula that the error of the approximate value of _F„ will be — 8^„ = !i(!^.i" (120) The maximum value of — ^ — ' which obtains for //^|, is — |-; whence we have the following result : ]Vheii second differences are seiisildi/ coustaat, the inajciiuuiit error of //( net ions obtained I)y simple interpolation is i J" . Thus, in Tables I, II, and III, the values of the coefHcients for J" (designated above as B) can nevei' be in error by more than I of 10 units, or 1.2 units in the fifth decimal, when found by simple interpolation. 44. Tnterpolatloa Livolving Second Differences, by Means of a Corrected Fii-st Difference. — When the second differences are con- stant, or nearly so, but too large to neglect, their effect may be included (and hence an accurate value of F„ obtained) by the follow- ing simple method : Since third differences are supposed insensible, Bessel's Formula becomes T1 T, " ('1 — 1) 7 which may be written in the form ^,. = F,+ n 1-n (121) Now, because third differences are negligible, we may write ^o foi- h in (121) ; then, putting V 2 y''» we have \ - / ^ (-^22) F„ = F, + na, The value of F^ is thus obtained almost as readily as in simple interpolation. In forming the quantity -~ (which is simply one-half the complement of n with respect to unity), only an approximate value of 71 is ordinarily required. The value of a, , the corrected first 74 THE THEORY AXD PRACTICE OF INTERPOLATION. (Uffcrence, is tluis found by an easy mental process amounting almost to nirri' inspection. Example. — Find (8.2)^ from the following values of T'^ : T 7-2 J' J" 4 7 10 13 16 49 100 169 + 33 51 + 69 + 18 + 18 Here we have t ^ 1 ?i = 0.4 F^ = 49 a, = 51 /'„ = 18 Hence, by (122), we find a, = 51 - (0.3 X 18) = 45.6 .-. F„ = 49 + (0.4 X 45.6) = 67.24 This result is exact, because the second differences are rigorously constant. 45. BacliiKird Tnterpolation hy Means of a Corrected First Deference. — From (111«), neglecting differences beyond J", we obtain F_., = J^„-/m'+-^ («-l) ft„ + i' F.-na'Jr , n{n-l) or F_.. = F n * Hence, if we put we have «' = a' + l-» (123) (124) Example. — From Hill's Tables of Saturn, the following pertur- bations ai-e taken; find the value corresponding to the argument T = 30G82.38. r F(T) A' J" 28800 29760 30720 31680 32640 12.5751 12.1998 11.8315 11.4700 11.1148 -3753 3683 3615 -3552 + 70 68 + 63 THE THKORY AND PKACTICE OF TNTEKPOLATION. 75 Taking t = 30720, we have /<; = 11.8315 n = '^^'(j^Q^"^'^ = 0.03919 (backward Irom F^) T = 30G82.38 «' = -3G83 o. = 960 /;,, = + 68 Usiiif? 0.04 as a sufficiently accurate value of // in determining a', we find by (124), LZ!! = 1^2! = 0.48 «' = -3683 + (0.48 X 68) = -3650 .-. F_,, = 11.8315 - [0.03919 X (-3650)] = 11.8458 In the present example the algebraic signs of the several quanti- ties of (124) have each been considered. Now it is important to remark that in the majority of cases no attention need be given to these signs; for in this fact lies the chief practical advantage of the method. Thus, in the present example, we are interpolating from the third function toward the second; the value of J' to be corrected is the difference of these two functions, or 3683; the sign we disregard. The correction to be applied to this number is 0.48 X 68, or 33. Again neglecting signs, we simply apply this quantity to 3683 in such a manner as to obtain a result falling somcAvhere between the numbers 3683 and 3615 of the column j'. Hence, we decrease 3683 by 33, thus obtaining 3650 for our corrected first difference, a. Finally, no! = 143, by which amount we increase the function 11.8315 (giving 11.8458), since we observe that the functions are increasing in the direction of the interpolation. A partial exception to this mechanical method of procedure is to be observed when a^ and a have opposite signs ; that is, when //' changes sign in passing the function Fq. In this case the sign of a must be noted; we then have, as in (122) and (124), re THE THEORY AXD PRACTICE OF INTERPOLATION. For exaiiiplo, given the values below : r F{T) J' J" 10 15 20 25 138 538 638 438 + 400 + 100 -200 -300 -300 Suppose it is required to find F, for T = 19. We let t = 20, F^, = 038, and interpolate hacl-wards with n = 0.20. To obtain a, decrease 100 by O.lXIJOO, or 120; whence a' — — 20, and thei-efore lu' = 638 - [0.2 X (-20)] 642 We remark in passing that the value of the corrected first differ- ence, either in forward or backward intei'polation, is always contained between the limits a^ and ■ ■ ■ ■ Systematic Interpolation — Subdivision of Tables. 48. Thus far we have considered interpolation as a process I'or comi^uting the values of functions for occasional oi- .sper/a/ values of the argument, simply. We shall now consider the subject in a broader THE TIIEOKY AND PRACTICE OF INTERPOLATION. 79 sense, and find that interpolation is of great importance as applied in a more extended and systematic manner. When a complicated function is to be computed and tabulated for a lai-o'e number of equidistant v-alues of the argument, oi- when the tabulai" quantities result from a long and laborious calculation, it will be much shorter and easier to make the direct computation foi- a less frequent interval than is finally required, and thence to obtain the intermediate values by systematic interpolation. Foi- example, suppose the function F{T) = 700".4.3 sin2r-l".19 sin4T is to be tabulated for every 10' from 30° to G0° ; we should begin by computing F(^T) for every 4th degree of T. Thus we should obtain the values of F{T) for T = 22°, 26°, 30°, 34°, . . . . 70° ; the calculation being extended somewhat beyond the assigned limits in order to facilitate the interpolation which follows. These quantities having been differenced, and corrected for accidental erroi-s if neces- sary, the middle terms are then found by interpolation to lialves. We thus obtain the series F(^T) corresponding to T = 26°, 28°, 30°, 32°, .... 64° Interpolating again to halves, we have a table of F {^T) for every degree of T. A thii'd interpolation to halves gives the function for every 30'. Finally, interpolating the latter series to thirds, we obtain the required table, giving F(T) for every 10' of the argument T. It is obvious that the labor of computation decreases rapidly Avith each successive interpolation. All of the extended tables in common use, such as tables of loga- rithms, sines, tangents, etc., have been subdivided in this manner, at a saving of labor almost beyond estimation. In fact, interpolation has undoubtedly done more for mathematical science than any other dis- covery, excepting that of logarithms. The following sections will be devoted to the derivation of formulae and precepts which will simplify the process of systematic intei'polation DR. F. McEWEN 80 THE THEOm' AND PRACTICE OF INTERPOLATION. just describL'd. Instead of performing a separate and distinct calcu- lation for each interpolated function, we shall develop a method by which the required values are obtained by .successive (tdditiovs of the computed differences of those values. The most convenient interpolation to perform, either in an isolated case, or as applied to the subdivision of an extended series, is interpo- lation to /i((lres, which gives the function corresponding to the ^iieaii of two consecutive tabular values of the argument. This case will now be considered. 40. Interpolation to Halves. — If, in Bessel's Formula (111), we put y/ = i, the coefficients of j'" and J" vanish, and we get ^ = ^; + i«i-i^ + Tis'^- • • • ■ (125) Since F^ — F^^^a^, we have Also, by (100), we have F, + ^a., = ,^h±k d = d, + d. Hence, (125) may be written in the form .; = 5di _ J (^±^) + ,,.C.±i?. ,_.... (126) which is the foi'mula for interpolation to halves, true to fiflh differences inclusive. The differences are to be taken according to the schedule on page 02. Supposing that fourth differences are so small as to produce no sensible effect, we obtain from (120) the very simple formula true to third diflerences inclusive. Hence, to interpolate a function midtvay between two consecutive tabular values, we have the following THE THEORY AND PRACTICE OF INTERPOLATION. 81 Rule : Fro»i f/te uicdn of fJie fino (/irrji fiDirtioxf^, siihfracf one- eiylifli the mean of the second differences which stand opposite. Tlu' result is true to third differences inclusive. To obtain the value true to fifth differences inclusive, add to the almve residt j|^ of the mean of the corresponding fourth differences. 50. Precepts for Systematic Interpolation to Halves. — The fore- going rule applies either to the interpolation of a single function into the middle, or to that of an entire series of values. For the latter purpose, however, the work may be arranged in a more expeditious manner, as follows: For convenience, we assume for the present that 4th diflerences may be neglected; accordingly, if we put Sn' = t\ W = ^1 - J'i F, - F, 8^ = F,-F, (128) we obtain from (125), So' = i«i - V = ia,- 8,' = A-ttj — 2 h, + L- K + k, (129) The quantities S' defined by (128) are evidently the first differ- ences, oi the interjjolated series ; the alternate terms, S,/^ ^ly §4 ? are computed by (129) from the fii'st and second differences of the given' series of functions ; the values of S/, 83', Sg', .... are not computed. The method and arrangement of the work are shown in the schedule below : T F(T) 8' 8" a (3 J' J" J'" t t + a, « + §<« t + lio F-. Fo F. F, Fz 80' 8/ 83' 84' 85' 80" 81" 8." 8s" 8," 8." -<*•:*■) -<^) -'.(^) a' "1 "2 a.) ''0 82 THE THEORY AND PIt;VCTICE OF USTTERPOLATION. The differences of the given series are placed in the last three columns, under J', //", and J'". The column a is then filled in by writing opposite each of the quantities ji one-half its value. The column ^ is also computed, each term being mhius one-eighth the mean of the tAvo values of J" which stand opposite. The alternate quantities of column 8' are then found, as in (129), by taking the sums of the con-esponding terms in a and fi ; the results are written immediately above the line of the latter terms, so as to fall between F^ and jP, , F^ and F, , etc., i*espectively. Finally, since b}' (128) we have F,^ = F^ + S; , F, = h\ + SJ , F^ = F, + 8,' , .... (130) it is only necessary to add each computed value of 8' to the function immediately preceding, to obtain the required middle functions. Hav- ing thus completed the interpolation, the remaining or alternate values of 8' are filled in by direct diff'erencing. The second differences are then written in the column 8", their regularity proving the accuracy of the work. The given functions, also the computed first differences, etc., are distinguished in the above schedule by heavy type. When it is necessary to take account of 4th and 5th differences, we have only to form an extra column y, to follow jS in the schedule above. Under y we write the terms 3 A^„+'^\ 3 M+f/, 128 V 2 y ' 128 , etc.; the vahies of 8' are then formed by adding the three corresponding terms in a, /3, and y. Example. — Given the values of log sin T for r= 30°, 32°, 34°, .... 42° ; find the value for every degree of T from 32° to 40°, inclusive. In accordance with the method above outlined, we arrange the given functions, with their differences, as follows: THE THEORY AND PRACTICE OF INTERPOLATION. 83 T I^og sin T 8' 8" a (3 J' J" J"' 30 9.69807 31 + 2524 32 33 9.72421 9.73611 + 1190 1143 1103 1063 1024 988 953 + 920 -45 + 1107.5 + 22.4 2335 -189 + 20 34 9.74750 42 169 35 9.75859 40 1083.0 1{\.1 2166 15 3(i 9.70922 39 154 37 9.77946 36 1000.0 ls.:$ 2012 15 38 9.7S934 35 139 39 9.79887 -33 + 930.5 + 10.7 1873 + 10 40 9.S0S07 -129 41 + 1744 42 9.82551 Since -Itli differences may be neglected, only the two columns a and /8 are required for the computation of the differences S'. All the quanti- ties actually used in the process are given in the above table. The computed quantities, together with the giveii values of log sin T, are printed in heavy type, to render this process more evident. 51. To Reduce the Argument Interval of a Given Table from w to 1)10), where „ is a Positive Odd Integer. — As particular cases of this problem, we may take m = .] , i , i, etc. Taking ni = |, we intro- duce two values between every two adjacent functions of the given table; we thus derive the series F F F. F F in which the interval is |w. This process is called interpolation to thirds. To interpolate to fifths, we let ni = l , thus introducing four functions between every two adjacent terms of the original series. "VVe then have the tabular values of F F F. F, F F F, o'S's'J'J'i'i' ■ ■ ■ • the interval being |w. More generally, let us take m = ,, where k is a positive odd integer; we thus introduce /■ — 1 equidistant values of the function between every two adjacent terms of the given series. The resulting series will therefore be F F F F F F ^ (A— Dm) -^ 1; F,^ 84 THE THEORY AXD PRACTICE OF INTERPOLATION. in whic'li the argument interval is mm, or 7^. IN^ow, the two adjacent functions of this interpolated series, which, as a pair, fall itiidica// be- tween Fo »nd F^ , ai-e F and F that is Hence, if we put "m """ "('?) S,' = F — F C13n it follows that S,' is the value of the Jirst difference of the interpo- lated series which falls on the line midwatj between F^ and Fj ; we shall designate this quantity a ntiddle first diff'eretire of the required series. If we now let 1 + m —2- = '* (132) we have 1 — m -2- = ^-'' and (131) becomes 8,' = F,,-F,_,, (133) Hence, to express 8,' in terms of the differences of the given series, we have only to express the values of i'^,, and i^i_„ by Bessel's Formula; thus, abbreviating coefficients, we have, as in (113), F„ = 7'; + ««i + B/j + Cc, + I hi + Ee,+ . . . . (134) Also, by virtue of the property of these coeificients established in §41, we have F,_„ = F„+ (l-)i)a, + Bb - Cc, + D,I - Ee,+ .... (135) The difference of these equations gives 8,' = F„-F,_„ = (271 -1) a, -\- 2 Cc, + 2 Ee,+ .... (136) Now, by (132), we have 1 + m n = — TT — hence, from (111), we find C = ln{n-\){n-~\) = ^ ('«—!) E = ^\^{n + \)n{n-\){n-2){n-\) = ^^{n\■\) (n-2)C = _^ (,„^-l)(,«=_9) THE THEORY AND PRACTICE OF INTKIU'OLATION. 85 Substituting- these values of ii, C, and £J in (13G), we obtain tlu- f'oruuila 85' = wa, + !^ (/'r-l) '\ + j^^ (jn'-l)(„r-d) e, + (1-) by Avliich the iiilddlc first differnuTs may be computed in any ease, |)ix)vided — is a i)ositive odd integer. Let ns now consider the schednle below : T Fcn 8' 8" 8'" J' J" Jill Jiv Jv t—iO F-, 8'-' 8^. 8::; a' b' (-■' d' e' t — III 10 t f + mm i^o' s/ 80'' sr' "1 K '■] '/o ''1 < + 0) — IIIW t + co F,' • • 8/' b: '': The quantities are here arrangx'd in a manner somewhat similai- to the schedule of §50. The given functions, F_,, F„, Fj, . . . . , are separated, snccessively, by h — 1 blank lines or spaces, for the subsequent entry of the interpolated values. The columns 8', 8", and 8"' are also reserved for the dilierences of the interpolated series ; and the diiferences of the given functions are written to the right, in columns J' to J'\ The value of 8,' is now computed by (137) from the ditfercnces rti, ('i, and ("1, which stand opposite. In like manner, 8'^; is computed from the ditferences a, c, and e ; 8/, from a,, Cg, and e.^; and so on. ^ We thus obtain a series of middle first di.ferences, which are tal)ulated under 8' in the schedule above. Now it is clear that if we should interpolate the A- — 1 inter- mediate terms between 8'_, and 8,', between 8,' and 8/, etc., the resultinsf series would constitute the consecutive first diflferences of the inter2wlated series F (T) ; the required functions would then be formed by successive additions of these difierences. The problem of S() THE TIIEOllY AND PKACTTCE OF INTERI'OLATION. interpolating- tlie given sei'ies F {T) is thus vii'tually lediiced to tliat of interpolating the comjnited values of S' in precisely the same )ii<(inicr. Now, let S„" denote the second differenee of the Interpolated series F, which stands opposite Fq] 8,", the second difference opposite F^; etc. It follows that 80" is the middle first difference of the interpolated sei-ies 8', which falls between 8'_^ and 8', ; 8/', that falling between 8/ and 8'; and so on. Hence, we may find 8^", 8/', 82", .... from the computed series 8'_, , 8,', 8,', . . . . , in precisely the manner that the latter quantities are derived from F_i, Fi,, F^, . . . .; that is, l)y application of the general formula (137), mutatis mutandis. For this purpose, we must form the ditferences of the computed series 8'_,, W, 8,', .... Accordingly, let us ]nit, for brevity. and (137) becomes -1) ^^'= UVH.^'""-^)i"'"-^) 8,' = ,iia,+ M'\+ 31' Ci (1.3S) (139) provided differences beyond J' ai-e disregarded. We now form a table of the quantities 8'_,, 8,', 8/, . . . . , and their differences, as follows : Function, = 8' 1st Diff. 2d Diff. 3d 4th 8'., = ma' + 3[r'+ 31' e' S/ = ma, + 3Ic,+ 3I'ei 8',' = ma, + 3Ic^+ 31' e.. mb' + 3fd' mh^ + 31 d^ mb, + 3Id, mc' -1- Me' mc^ + 3Ie, vuV mdo mdi me' me, VI e„ Whence, applying the general formula (130) to the quantities of this table, we obtain 8/' = m (m\ + 3Id,;) + 31 (md^) = m-% + 23Imd^ or, by (138), So" = m\+'^{m."--l)d, (140) by which the quantities 8"_, , 8„", 8,", .... of the former schedule are computed from the differences J" and .1" which stand opposite. Again, we may suppose that the intermediate \alues of 8" have been interi)olated between the computed values 8"_,, So", 8,", . . . . ; this completed series 8" constitutes the consecutive second differences THE THEORY AND PRACTICE OF INTERPOLATIOX. 87 ()(■ the interpolated scries F(T). Finally, wc sliall denote by 8'" the third diiference of the interpolated series F, which stands ojjposite S,' in the given schedule. The (juantity 8'" is therefore the middle iirst difference of the completed series 8", whicli falls between 8„" and 8,"; it bears the same relation to 8o" and 8/', that 8,' bears to 7'^, and Fi . Hence, to find SJ" , let us i)ut 31" = ^(..^-1, and (140) becomes So" /«%„ + iI/"(/„ (141) The diiferences of 8"_i , 8u " , 8i " , .... are therefore as follows: Function, = 8" 1st Diff. 2(1 3d 8", = m%' + M"J' So" = v^\ + ^I"'h 8," = m% -1- MVcl, mh:' + M"e' niH' Whence, applying (as above) the general formula (130), we find 8;" = m{m\ + M"ei)-¥M{w,\) = m^\+ {wU" ^- m-M)e^ 8ubstituting the values of M and M" , we have 81" = '"%+"''("'^-l)e, (142) O In practice, the values of 8'" and 8' are never required, and in many cases the column 8"' is not necessaiy. Supposing, however, that we have computed the (nearly constant) values of S"! , 8, ", 81", . . . by (142), the intermediate terms ai"c then written in by mere inspec- tion. We thus complete the column 8'", — the consecutive third diffei*- ences of the required series F{T). Having also computed the quantities S,,", 8,", 8^", .... and 8'_j, 8,' , 8/ , . . . . , we complete the columns 8" and 8', and hence, also, the interpolated series F(T), by successive additions. We now bring together the formulae for 8/, S^", and 8^", in the order computed in practice, as follows : S;" = m^;+-(m'-l)e, So" = »^\ + '^(^»''-^yio 8/ = ma, + ^ (m"-- 1) c, + j^^ (m"--l)(vi--9) e, (143) S8 THE THEORY AND TKACTICE OF INTEl!lH)LATION. which serve to reduce the tabulai' interval to iii times its original value, ni being the reciprocal of a positive odd integer. It will he observed that the differences re(|uiri'd in computing each of the (piantities 8 are always found on the same line with that (jnantity. 52. fiit('rpolatio)i, to Thirds. — For tliis i)ur])ose, we take vi =: ;| in tlie foi'nuilac (1^:3), and find v = j''o-?i3< y (144) These formulae are more conveniently computed in tlie form s;" = ,',('• «i) (145) S/ = M«i-8i") Example. — Given the value of log tan T for every third degree of T from 27° to 48°, inclusive : find the function for every degree between 33° and 42°. According to the precepts of the last section, we arrange the work as follows : r Log tan T 8' 8" S'" J' J" J'" Jiv o 27 9.70717 +5427 30 9.7«144 + 3.1 5108 -319 + 85 33 34 35 36 9.S12r)2 9.82899 9.84523 9.86120 + 1646.9 1623.8 1603.3 1585.3 1569.6 1556.1 1544 6 -25.9 23.1 20.5 18.0 O.O 2.8 2.6 2.5 2.3 2.2 2.0 4874 234 163 71 -14 12 37 38 39 9.87711 9.89281 9.90837 15.7 13.5 11.5 4711 104 59 6 40 ■ 41 42 9.92382 9.93917 9.9)444 1535.1 + 1527.5 9.5 7.6 - 5.7 1.9 1.9 1.9 + 1.9 4607 4556 - 51 53 + 51 _ 2 45 0.00000 + 4556 48 0.04556 The heavy type shows at a glance the given functions, and like- wise the computed middle differences. We observe that it is here THE THEORY AND I'KACTICE OF INTEKPOLATION. 81) necessary to compute five values of 8'", foiii' values of S", and only three of 8'. These quantities are computed to one more than the number of decimals given in F(T), to avoid accumulation of any appreciable ci'ror in the linal additions, liaving obtained for 8" tlie series + 3.1 2.6 2.2 1.9 +1.9 the intermediate terms are readily inserted, as shown aliove; it is necessary, howevei', to see that the completed series 8" is consistent with the compated values of 8". Thus we must have 2.8 + 2.6 + 2.5 = -(18.0-25.9) = +7.9 2.3+2.2 + 2.0 = -(11.5-18.0) = +6.5 • 2.0 + 1.9 + 1.9 = -( 5.7-11.5) = +5.8 If these relations ai'e not satisfied exactly on first trial, the interpo- lated values of 8"' must be adjusted to fulfill the necessary conditions. The column 8" is now completed by successive additions of the quantities 8". Again, it is necessary to see that the completed sei'ies 8 " agrees with the computed values of 8'. For we must have -(20.5 + 18.0 + 15.7) = 15G9.6 - 1623.8 = -54.2, etc. Since these relations are seldom exact in the beginning, the pro- visional values of 8" will usually require slight alterations. From the final series 8", we obtain 8' by successive additions. As before, an agreement must subsist between the values of 8' and the given set of functions ; that is, between 8' and j'. Thus we should have 2"8' = 1646.9 + 1623.8 + 1603.O = +4874.0 = J', etc. In the latter case, however, a discrepancy not exceeding four or five units in the added decimal may be tolerated. Our final series 8' is therefore satisfactory ; whence we obtain by successive additions the required values of log tan T. 53. Interjwlatioti to Fifths. — Taking ■))i = I in the formulae (143), we obtain V = AC^o-A'^o) } (146) In practice it will suffice to put ^e^ for both rf^e^ and yVs^ij ^^^ formulae (liO) then become, vei-y approximately, Ur DR. GEORGE K iVicEWE^=. 00 THE TIIEOKY AKD PKACTICE OF IXTKRI'OLATION. 8,' = itti-s;" (147) Example. — The following ephemeris gives the moon's K.A. for every ten hours. Obtain the value for every second hour, from Sept. 23'' 20" to Sept. 25'' 12", inclusive. The details of the computation are as follows : Date, 1898 Moon's R.A. 8' 8" 8'" J' J" J'" Jiv il h h 111 8 m s s s ni s B s 8 Sept. 23 IS 24 20.4 + 25 31.1 Sept. 23 10 IS 4!> 57.5 -.034 32 25 11.0 -20.1 -4.2 Sept. 23 20 1» 15 S.5 + 4 59.39 58.39 4 57.36 5 0.31 55.23 54.13 5 3.02 4 51. SO 50.75 49.60 48.44 47.28 4 46.12 44.95 43.79 42 63 -0.076 30 28 26 .024 23 20 1 7 24.3 + 1.2 23 22 24 24 2 19 20 7.9 19 25 6.3 19 30 3.7 1.004 1.0 30 1.054 24 46.7 3.0 24 4 Sept. 24 6 19 35 0.0 10 30 55.2 1.077 1.097 27.3 1.5 24 8 24 10 24 12 19 44 49.3 19 49 42.3 19 54 34.2 1.114 1.128 1.140 J- 1 14 .012 09 07 05 03 -.001 + .002 03 06 24 19.4 1.5 24 14 19 59 25.0 1.149 Sept. 24 1« 24 18 20 4 14.0 20 9 3.0 1.156 1.161 28.8 1.4 24 20 24 22 25 20 13 50.3 20 18 36.4 20 23 21.4 1.164 1.165 1.163 23 50.6 -0.1 Sept. 25 2 20 2S 5.2 1.160 28.9 1.1 25 4 20 32 47.8 41.47 4 40.33 39.19 1-4 38.06 1.154 07 .OOS 09 11 12 25 6 25 8 20 37 29.3 20 42 9.G 1.147 1.139 23 21.7 + 1.0 25 10 Sept. 25 12 20 4G 4S.S 20 51 2().0 1.130 -1.110 27.9 + 0.9 14 + .015 22 53.8 + 1.9 Sept. 25 22 21 11 20.7 + 22 27.8 -26.0 Sept. 26 8 21 36 48.5 THE THEORY AND PRACTICE OF INTERPOLATION. 91 Here we extend the computation of 8 " and S" two j>laces of" deci- mals ; one of which is dropped in computing 8', and the otlier in forming the required functions. The principle and method being the same as in the last example, further explanation is unnecessary. 54. Order of Iiiterpolatioit, to Follow, wJien a Series lieqaires Successive Interpolation to Halves, Thirds, etc — When a table of functions is to be interpolated, successively, one or more times to halves, and also to thirds and Jifths, the easiest method is to proceed in the order named. Thus, if the interval of the original series is w, and that of the final table is w, we may suppose the relation of these quantities to be — u, = 2'.3'.5'".<«' where 1-, I, and )n arc integers. It will then be found most expedient, first, to interpolate to halves, k times ; then to thirds, I times ; and finally to fifths, m times. For example, F being given for every degree, and requii-ed for every minute of arc, we should first interpolate to 30', then to 15', then to 5', and finally to every minute of arc. 55. To Interpolate with a Constant Interval n, an Entire Series of Functions. — Let the given series, with its differences, be as follows : T F(T) J' J" J'" Jiv f. t + (U t + 2^ f + 3 t+ (l + ii)i t + (2 + h)w f + (3 + 7l)a) t + (4 + ?i)<" F,^n F «1 lt.„ «4 A, y^ y!i y4 Now, it was shown in §22 that differences of (inii order may be expressed in terms of the tabular functions. Thus, in pai'ticuhu-, we obtain from the given series F, r., = F,- 3F„_ + 3F, - i^„ = * (t+u,) r, = F, - 3F., + 3F„ - f[ = *(/' + 2u,) (148) where "^1^ (/) denotes, for Ijrevity, the function of / expressed by F„ - 3F^ + 3/; - F_, ; that is, * (t) = F(t + 2,o) - 3F(t + o>) + 3F(l) - F(t-w) Again, in Hke manner, the interpolated series [i'^] gives = F,, 3/'':+,, + 3i^„ F y„ = F.^.„ - 3F„^„ + 3F^^„ - F„ = * (i + «, + Ma,) (140) It follows, then, that the series [/J'"] is simply the scries z/'" inter- polated forwaid with the constant interval n. Moreover, since the above reasoning is perfectly general, this relation holds for an;/ order of differences. Hence, to perform the required interpolation of the series F(T), that is, to obtain the series [i^], we have only to interpolate forward each value of J" with the constant interval >/, thus forming the column [j"]. This process is obviously brief and simple. Then, if we com- pute occasional values of [j'], and also of [-F'], we readily complete the required table by successive additions, as in the preceding problems. Example. — To illustrate the process, we tabulate the "Latitude Reduction" for evei-y fourth degree of latitude (g) from 30° to 82°, and thence derive the series for 45 420 785 032 ,382 ,244 [-)'] + 27.212 13.918 + 0.331 -13.281 26.657 ;j«.525 51.635 62.733 72.650 -Sl.KJS [J"] 13.294 13.5S7 13.«12 13.37« I2.S«S 12.110 11. lis ».S«7 S.4S8 Taking n = 0.25, we compute by Bessel's Formuhi tlK' value.s of v — l' for <7 = 35°, 55°, and 75°, extending the decimal one unit. Similarly, we compute fJiree values of [j'], and all of [J"]; the com- puted quantities being clearly shown by heavier type. Adjusting slightly the series [j"] to conform to the computed values of | yj, we complete the latter column by successive additions. The values of [j'] being found to accord with the computed functions, we complete the entire series as required. Since the computed intermediate values of [.J'] and [i^] serve only as checks, it is obvious that their positions, as also the intervals of their distribution, are entirely arbitrary. These are details to be decided by the computer's judgement in any given case. It may occasionally be practicable to extend the ])rocess to the computation of [J'"]. 94 THE THEOnY AND PRACTICE OF INTERPOLATION. EXAMPLES. 1 . Tabulate the five-place log cosines of 15°, 18°, 21°, 24°, 27°, 80° ; fi-oin these values interpolate log cos T for T =: 17° 43', 23° 8', and 28° 15', respectively. 2. Given the following tal)le : T F(T) T F(T) 10 20 30 17"31 14.68 13.62 40 .50 60 14.16 16.34 20.18 Compute tlie values of F for T = 24.(>, 28.8, 32.3, and 48.5, using either Bessel's or Stirling's Formula. 3. Interpolate the required functions of Example 2 by means of a corrected first ) E F': When it is convenient to proceed hachwards from the argument / with the interval n, we .shall use the expressions F'_„ = F'(i-no>) , F'l„ = F"(t-n^) , F'l'„ = F"'(t-n.o) , (152) THE THEORY AND PKACTICE OF INTERPOLATION. 99 Now, by means of any one of the fundamental formulae of inter- polation, we may expi-ess F^ in the form F,^ = /; + na + Bh + Cr + Dd + Ee + (153) where, in any given case, a, h, c, . . . . are known differences; and where B, C, D, . . . . are definite functions of 71. Let the successive derivatives of B, C, JJ, . . . . , taken with respect to m, be denoted by B' , B" , B'" , . . . . C , C" , C" , . . . . £)' , 1>" , B'" , . . . . E' , E" , E'" , . . . . Then, observing that the coefficient of J"' is always of the degree i in n, we have dB dn B' dC dn C" dl) dn = D' dE dn = E' d^B dri' B" d'C dn^ ~ C" d-D dn-" = D" d^E dn- = Ell d^B dii' ~ d'C dn^ C" d'D dn" = D'" d^E dn" = Eiii d*C dn" " dW dn" = D" d"E da" = E" d^D dn'' = d'E dn'' d^E d7l^ = E^ = (154) Reverting to (151), we have El = IT dF„ dn ~dn ' dT (155) Prom (150) we derive whence dn dT F' = 1 dE\, +CV + //(/+ i;^'e+ . . . .) f;; = 1 (i?"/y+c''v +//'<;+ ^■'"(!+ . . . .) w F'" = ~„ (C"'c + I)"'d + F"'e+ . . . . ) /'i^ = -. (I)"d + E'^e+ . . . . ) i^v = 1 (^v,+ . . . . ) (159) which are the general formulae for computing the derivatives of F{T) in terms of the tabular differences. To derive the formulae for F'^n, F'!_„, F'"„, . . . . , that is, to find the successive derivatives of i^(^ — «w), we have only to alter slightly certain details of the preceding development, as follows : (1) For equation (153) must be substituted the corresponding expression for i^_„, which has the form* . F_„ = F^ - na + Bfi - Cy + D8 - Ft+ .... (160) where a, ^, y are, in general, different fi'om the dift'erences a, h, c, . . . . of (153). (2) In the present case, we have T = t —no> and therefore dn 1 'elf ^ ~ w which must be substituted for equation (15G) above. •Compare (75), (105) and (111") with (7.3), (104) and (111), respectively. THE THEORY AND PRACTICE OF INTERPOLATION. 101 Introducing these changes, and operating as before, we obtain (he required forinnhie, namely, FL„ = - (a-n'/s+Cy-rn+E'^- . . . .) F1„ = I^(n"l3-C"y+/)"S-E"t+ . . . . ) F'l',^ = \{C'"y-Dn + E"U- . . . .) i^'",. = 4 (rPh-E"'f.+ . . . . ) Fl,, = -,{E^c- . . . . ) (lOI) It now remains to apply (159) and (161) specifically to each of the several formulae of interjjolatiou, of which (153) is the general type. It is obvious that a particular set of coefficients, B', B", . . . . , C, C", . . . . , etc., will result in each case. 59. To Comjmte Derivatives of F{T) at or near the Beginning of a Series. — The formulae adapted to this pin-pose are derived from Newton's Formula of interpolation (73), which is — where Fn = ^0 + »'\ + I'h + C^o + ^'''o + Er^^ . . . . ^ (1G2) __ n{ti — \) n- n ^ ^ 2 ~2 _ _ n(n~l)(n — 2) n^ n^ n ^ ii ^ F~ 2^ ■•■ 3 ^ w(w-l)(n-2)(»-3) ^ !^ _ ™' 11 o _ w V ^jg3^ li 24 4 "'"24" 4 m(w-1) (w-4 ) w= «^ 7 3 „ ,. ^ = ^ =120-12 + ^'' -19'^ + n Differentiating these expressions successively with respect to n, as indicated in (154), and substituting the resulting values of B\ B", . . . . , C", C", . . . . , etc., in the general formulae (159), we obtain 102 THE THEORY AND PRACTICE OF INTERPOLATION. F' {t + n^) = - f a„+ (n - ^) i„+ {f-n + i).„ + (f - f «=+ j I n - \) d. + (i'T-/+l«'-^»+iK+ ■ • i^"'(< + «o,) = V'-o+("-3)'^o+(f'-2«+l)''„+ . . . ^ ' ^^^'^^ i'-'v (<+„<„) = l^^(7^+(„_2)e„+ . . . .^ /'^v (, + „„) = Lu + These formulae determine the derivatives of F {T) for any or all values of T between t and t-^-cj, according as we assign different values to n. As in preceding applications, n is always a positive proper fraction. When, as is frequently the case, derivatives are required for some fahular value of the argument, say t, we have only to make « = in (164) ; we thus derive the following simple exj^ressions : ^'"(0 = ^('■o-S'^o+i^o- • • F^-(t) = ^K-2.„+ . . . .) (t) F^(f) = 4 («o- . . . . ) (165) The differences employed in (IW) and (165) must be taken according to the schedule on page 3, as in direct applications of Newton's Formula. The formulae (165) have already been established in §18; for it will be observed that (45) and (165) are identical, since in the foi-mer D,D\D%. . . . are used symbolically to denote o>F' {t), os'F" {t), w'F"'{t),. . . . THE THEORY AND I'UACTICE OF INTEKl'OI.ATION. 103 Owing to the special practical importance ol' the Jirst derivative, the coefficients of F'{t-\-ii(D), namely, B' = n-\ C" = f />' f-f«= + U«-i « + i A'' = fl - i*' + ln--^n + (1(;G) have been tabulated in Tal)le IV for evei-y hundredth of a unit in the argument n. By means of these quantities, we readily compute F'{t-\-nw) from the formula F'{t + n^) = - (a„+£7*„+CV„+Z)V/„+^V„) (167) The formulae (164), (105), and (167) are especially adapted to the computation of derivatives at or near the beginning of a tabular series. We shall now solve a few examples to illustrate their use. Example I.— From the following table of i^(r) =0.3^^—2 ^-+4, compute F"{T) for T = 2.8. T F{,T) J' J" J'" Jiv 2 4 6 8 10 4.0 0.8 48.8 320.8 1104.8 2804.0 - 3.2 + 48.0 272.0 784.0 + 1G99.2 + 51.2 224.0 512.0 + 915.2 + 172.8 288.0 + 403.2 + 115.2 + 115.2 Here we have t = 2 T = 2.8 n = 0.40 a^ = + 48.0 i„ = +224.0 c, = +288.0 cL = +115.2 Hence, using the second equation of (164), we find C" -O.GO D" = f- i« + U = +0.39§ i„ = +224.0 C"c^ = -172.80 Jj'UL = + 45.696 'FJ' = + 96.896 "Whence we obtain F'' = 96.890-^4 + 24.224 104 THE THEORY AND PRACTICE OF INTERPOLATION. Thi.s result is easily verified from the known analytical form of the finiction ; thus, since we derive F{T) = 0.3T''-2T'^ + 4 F'{T) = 1.2rs-4T , F"{_T) = 3.GT^ - i Substituting ^=2.8 in the last equation, we obtain F"(T) = +24.224 as found above. Example II. — From the table of the last example, com^Jute F'(T) for T=zO. Here Ave employ the first of (1G5). Making / = 0, we have a, = -3.2 b^ = +51.2 P^ = +172.8 o, (•„, .... from the given table, we find, in accordance with (167), a„ = +0.023858 B' = -0.35 f>^ = +8891 B\ = — 3111.9 C" = +0.19458 r, = - 638 C'-o = — 124.1 /;' = -0.12881 (/„ = - 160 D'd^ = + 20.6 E' = +0.09358 "o = + 15 E'e, = + 1.4 log (wF>„) = 8.314- •94 ■■■ "i^'„ = + 0.020644 log ft) = 8.843937 logii"„ = 9.470857 •■• -f''„ = +0.295704 This result is easily verified by observing that F'(T) = ^(siuJT) = siii2r which, for T ^ 8° 36', becomes F'(T) = sin 17° 12' = 0.295708 The former value is thus seen to be very nearly exact. If the variation in F(T) corresponding to an increase of one degree in T were required in the present example, the i-esult would be, simply, F'(T) = 0.020644-^-4 = +0.005161 60. To Comjyute Derivatives of F(^T) at or near the End of a Series. — In this case the requisite formulae are derived from ]S[ewton's Formula for backward interpolation (75), namely. F^ - na_^ + 7?6_2 - Cc^ + Dd^^ - Ee_, + (168) where B, C, D, . . . . have the values given by (16.3), as before ; and where the differences a_i, h_.,, c_3, .... are taken according to the schedule below : T F(r) J' J" Jill Jiv Jv t — 5(0 t — 4ft) t — 3o) If -2ft) fZ F-, «-5 C-4 d_ e-7 e-5 t O) F-, «-2 b^. C-8 t Fo a-1 106 THE THEOKY AND PliACTICE OF DJTERPOLATION. Comparing (1(58) with the general formula (160), we have Therefore, substituting the pi-eviously determined values of B', B", . . . . , C, C", . . . . , etc., in the general formulae (161), we obtain + (?,-'/+ !«'-!«+ i)«-6- ••• F"'(t-no>) = ^, (c_,-(n-?,) d_, + (f-2n+ |) e_,- . . . .) ) (1«'') F'^ (t-7iM>) = -Jd_^-{n~2)e_,+ CO V F" {t-7lw) = -, ((■_5- . . Making n^O in (169), we have F' (t) = - ("_i+ */'_,+ i '■-3+ i'U+iC-5+ . . . .) F" (t) = ki/'-2+''-s+n'U+le_, +■ • • •) ^""(0 = \(''-3+Sf?-4+l«-5+ . . . .) ^ (170) to ^'" (0 = h ('^-.+ 2^'-5+ . . . . ) As above, we emphasize the relative importance of the Jirst deriv- ative in practice : thus, for brevity, we wi'ite the first of equations (169) in the form F'{t-7iw) = ~(a_,-B'l>_,+ C'c_,-D'd_,+ E'e_,- . . . .) (171) the coefficients B', C", D', E' being taken from Table IV with the argument n. Formulae (169), (170), and (171) are particularly useful in the computation of derivatives at or near the end of a series of functions. THE TIIKORY AND PRACTICE OF INTERPOLATION. 107 Moreover, when tlie interval m approaches unity, formulae (109) and (171) are convenient for computing derivatives corresponding to the argument f -\- nco, since they enable us to proceed backwards from the argument t -\- w with the interval 1 — n. We shall now solve several examples to illustrate these ai)phcations. Example I. — From the following ephemeris of the moon's right- ascension (a), compute the hourly change in a at the instant Feb. 3'' 20" 24". Date 189S Moon's R.A. a J' J" J'" Jiv Jv d h Feb. 1 1 12 2 2 12 3 3 12 4 h m 8 4 49 39.68 5 16 0.86 5 42 26.85 6 8 51.58 6 35 9.06 7 1 13.92 7 27 1.71 m s + 26 21.18 26 25.99 26 24.73 26 17.48 26 4.86 + 25 47.79 B + 4.81 - 1.26 7.25 12.62 -17.07 8 -6.07 5.99 5.37 -4.45 8 + 0.08 0.62 + 0.92 s + 0.54 + 0.30 Since the assigned unit of time is 1 hour, we have w = 12 ; hence, letting t = Feb. 4'' O'', we find 4<1 Qh Qm _ 3d 20^ 24™ = 0.30 which is the interval reckoned hackwards fi'om t = Feb. 4'' 0''. De- noting the quantity sought by Ja, we then have JU = F'(f — 7l_. = - 3.414 C = +0.07833 c_3 = - 4.45 + C"-'_„ = - 0.349 u = _ 0.03800 d_^ = + 0.92 _J/,+ 7ic +(f -,V) {t + n.) = -l(c+nrf„+(f-J)e+ . . . .) ^ ^^^^^ F" {t + nm) = - ('«?„+ we + . iTv (^ + „,) = ^^,{e+ . . . .) 110 THE THEORY AND PRACTICE OF INTERPOLATION. Making n = in (174), tlie latter become F' (0 = ^(.,-^c^^,e- . . . .) ^'"(0 = -. (''o-A.''o+ . . . .) (175) ^-(0 = -.K-. • ■ •) ) = i^(^«-,«/„+(f-i)e- . . . .) F" (f — no>) = -, (d^ — ne + . . . (176) F- (?-«a,) = -^[e- The coefficients for the computation of F'(f±uco), namely B' = n , I)' = f-f^ C - "' - 1 E' = #i - s' + J. ^ ' are given in Table V with the argument n. The quantity F'{T) is thus readily computed (for any value of T) by either one or both of the formulae F'{t+nm) = ^{a + 7ih^+C'r + D'd^ + F'e) (178) F'(t-no,) = '^(a-nb^+C'c-B'cl^ + F'e) (179) in which the odd differences are algebraic means of the tabular differ- ences, taken as indicated below : THE TIIEOKY AND rEACTICE OF INTERPOLATION. Ill T F(T) J' J" J"' Jlv Jv t — (1) ^-i «' ?/ c' rf' e< t Fo {a) ^'o p. r/„ t + F'{t)-]--^F"(l)Jr-—-F"'{t)^ 11 li (180) Arranging Stirling's Formula (104) according to ascending powers of n, we find F(t + nuy) = /;+„,(«_! c+^V - • • ■)+ j^(''o-iV'^o+ • • •) + (181) Whence, by equating coeflScients of like powers of n in the equivalent expressions (180) and (181), we obtain ^Fi (0 = a-lc + ^\e „^F'"(t) = c -ie+ . . (ISlrt) which agree vdth the formulae (175). Again, by Taylor's Theorem, we have F' (t + nw) = F' (t) + na,F" (t) + ^ F'" (t) + F"{t + WO)) = F" (f) + nu,F"'(t) + ^ F" (t) + 112 THE THEOKY AND PRACTICE OF INTERPOLATION. which may be written in the form 1 / n- F"(t+ nu,) = -U^F"(t)+ nm'F"'(f)+ — m*F''- (f) + Substituting in these equations the expressions for wi^' (/), ^F"{t), . , . . , as given by (181«), we get F' {t-\- n"{t.^n^) = -3 F^ (t+na,) = -, ) + + !L(e_ . .) + . . \3_ (c-ie+ . .) + «K-- O + ^-Ce-- •) + (d,- . .) + 7i(e- . .)+ • • (e- . .)+ . . (182) These expressions, upon being arranged according to the succes- sive orders of differences, will be found identical with the formulae (174). For some purposes, however, the present foi-ni is more con- venient. It is quite common, particularly in an astronomical ejihemeris, to tabulate the values of F' {T) corresponding to the tabular values of F{T). Such a table woidd run as follows :* T F(T) F'{T) t-2^ I'-. F'(t-2o,) t — CO ^'-1 F'(t-w) t Fo F\t) t + w l'\ F'(t + a,-) t + 2io 1'. F'(t+2io) * It is evident tliat F'(<+H(j) can be derived from tlie colunm F' (T) by direct interpolation : moreover, wlien tlie tal)ular values of F' (T) are tlms available, this method of coiiumting F' {l+iiu) is more expeditious than the use of formula (178). TIIK TIIEOKY AND PRACTICE OF INTEKPOLATION. IV.l The first of llio fonniilac (17/)) is almost invariably used for this purpose, because of its simplicity and rapid convergence; this formula is, in fact, the most important and useful of those which pertain to the computation of derivatives. For this I'cason we fornuilate tiie following Rui.E for computing the lirst derivative of a tabular function corresponding to one of the given functional values : From the inean of tJui two first differeiices which immediately pixcede and follow the function in question, subtract one-sixth (J) the mean of the correspond- ing third differences, and divide the result hij the tabular interval. This rule neglects only 5th and higher diffei'cnces. To include 5th and 6th differences, add to the above terms (Ixfore dividing by co) one- thirtieth (g'o) the mean of the corresjwnding ffth differences, and, divide by 0) as hefore. It will evidently suffice, in most cases, to apply only the first part of the above rule. Several examples will now be solved as an exercise in the nse of the preceding formulae. Example I. — Given the following ephemeris of the sun's decli- nation (8) : compute the houi'ly differ&ice in 8 for the dates Jan. 7, 10, 13, and 16. Date Sun's Decl. J' J" J'" 1 ,■ Diff. for 1S9S S 6'' 1 hour O / // / // / // // II II // Jan. 1 4 7 10 i;: -22 59 2.4 22 41 38.5 22 20 12.4 21 54 49.4 21 25 35.9 + 17 23.9 21 26.1 25 23.0 29 13.5 32 56.9 36 32.2 + .39 58.2 + 4 2.2 3 56.9 3 50.5 3 43.4 -5.3 6.4 7.1 8.1 -9.3 + 1404.55 1638.25 1865.20 + 0.98 1.12 1.27 + 19.52 22.77 25.92 16 19 22 20 52 39.0 20 16 6.8 -19 36 8.6 3 35.3 + 3 26.0 + 2084.55 + 1.45 + 28.97 The term 3^^ e in the first of (175) is hei-e insensible ; hence, for each of the given dates we have only to compute the quantity Accordingly, in column a we write the required mean first differences, expressed in seconds of ai-c. The next column contains minus one- lU THE THEORY AND PRACTICE OF INTERPOLATION. sixth of the corresponding mean third differences. Finally, since w = 72 hours, we Avrite in the last column ^\^ of the quantities formed by suniming the corresponding terms of the two preceding columns. We thus obtain the hourly diffei'cnces required. Example II. — Compute, from the ephemeris of Ihe last example, the (Idili/ motion in declination Ibi- the date Jan. G'' 13'' 30"'. AVe proceed hacJctimrds from Jan. 7, using the formula (179), and taking the coefficients from Table Y with the aigunient 7' 0' 0"'_6"13"30"' 3" 10'>.5 72'' = 0.14583 find II a = +23'24!55 11 = 0.14583 \ = +230.9 -7ih^ = - 34.55 C" = -0.1560 c = - 5.85 -1 C'f = + 0.91 D' = -0.012 d, = - 1.1 -D'd^ = - 0.01 .-. ,oF'_„ = +22 50.90 Whence, for the daily motion in S, Jan. (5'' 13'' 30'", we obtain F'_„ = 22' 50".90-:-3 = +7' 30".97 Example III. — The following table gives F(T) = e% where e denotes the base of natural logarithms: compute F'(T) for ^=0.30. T F{T) = cT J' J" J/" Jiv Jv 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.000000 1.105171 1.221403 1.349S59 1.491825 1.048721 1.822119 2.0i;^753 + 105171 110232 128450 141966 156890 173398 + 191634 + 11061 12224 13510 14930 16502 + 18236 + 1163 1286 1420 1572 + 1734 + 123 134 152 + 162 + 11 18 + 10 Using the first of (175), we lind i<''(0.30) = .^(135211-^ + ^) = 1.34986 It will be observed that our I'csult is substantially equal to the value o{ F{T) for the same argument, T ^ 0.30 : this is required by the relation F{T) = F>{T) = F"{T) =....= e'' />^ = +0.014930 c = +149G 7,r. = + 927.5 "(/^ = + ic.c e = + 14 ii:"e = - 1.6 THE TIIKOIJY AND PKACTICK OF INTEKPOLATION. 115 Example IV.— From the tabic of Example III, compute F" (T) for T = 0.462. Taking t = 0.4 and // = 0.(52, we obtain, by means of tin- second of (174), 71. = 0.(52 1)" = f- -jij = +0.1089 Jit, ^ n''_ n ^ _0.115 ' .-. a,^FJ' = +0.015872.5 .-. /<;" = +1.58725 The trne mathematical value is — F".(T) = F(T) = e'' = eo-''- = 1.587245 . . . 62. Derivatives from Bessel's Formula. — Other useful formulae, convenient for the computation of tabular derivatives, are those dei'ived from Bessel's Formula of interpolation (111). The latter may be written in the form F,^ = F^ + ,w^ + m, + CV, + Ihl + Ec,^-\- . . . . (183) where the dijfferences are taken as in the schedule on page 62, h and d being the mean differences defined by (106) ; and where B, C, . . . . have the following values : n(7i — 1) n- n ~ 2 ~ 2^ ~ 2 7i{ii — \)(ii — \) n^ 71^ 11 ^ 6 ^ TT ~ 4 ■•■ 12 _ (7i + l)n{n-l)(n-2) _ w^ »' n^ n ~ 24 24~"l2~2i"''l2 _ (n + l)n(n—l)(n—2}(n—i) _ n^ m* ?r _?j_ ~ 120 120~48"'"4S~]20 (184) Deriving from (184) the values of B', B", . . . . , C", G", . . . . , etc., according to (154), and substituting the.se in the general foi'm- ulae (159), we obtain 116 THE TI1EOI5Y AND I'HACTICE OF INTKKPOLATION. + (s4 — T 2 + -J 4 "■ 1 2 cj) ''l + • • F'"{t + nu>) = l('.j+(M_i-)fZ+(f -;])e,+ . . . Putting n := in (185), we get tt) P" (i) = ^ (^'-i 'i - A '^ + 5^4 «i + • • • • ) 0)" jP"'(A = 1 (r,-i, /+()* + . . . . ) (0 (1) P'- (0 = 4 (^' - . . . . ) (isr.) (186) Again, putting n = I in (185), we obtain the following simple foi'mulae: F' (f+io,) = - („,_,l^,^+.,3_e__ . . . . ) F"(t+U) = \(l>-^^d+ . . . .) /""«+.U) = "3(q-ic,+ . . . .) /.^'^(/+^o.) = 1 (r/- . . . . ) 7''" (Z+Jo,) = i(c,- . . . .) (187) wliicli (Ictenniiic llic (liM'ivativcs of F {T) at points tiiidira;/ between the labular values of the; funetion. It is important to observe that, •The coefficient of e^ vanishes. THE TllEOUY AND PUACTJCE OF INTEKPOLATION. 117 unless third (liU'e'rciicrs ;ii'C considci-iiblf, a close ') , d = i(d^ + d') (18S) and (111^/) becomes F_,^ = 7-; - //-'' + BJj - C-i + Dd - A'c'*+ .... (189) Comparing this expression with the general formula (KJO), we find that a, ^, y, 8, e, . . . . , in the latter, are replaced by ', B", . . . . , C, C", . . . . , etc., in the formulae (161), we obtain F' (t-no>) = i ('«'-(«-i)^+(r-i;+ ,'.)';'-(?;'-f-ij+ .5)'^ 4. ( II* n'^ I n 1 \ ^1 T I-.' 4 15+5 4 T50 J '^ ■ F" (t-nu,) = ^Ji-{>i-i)c' + (yr-v,-^,)d-(f-f+^^)e'i- . w- \ F"'(t-uo.) = ^(.■•-(,i-i)d + (f-l)e'- . . . A ) (1^") F'" (t-nm) = -^Jd-{n-i)e' + F- (t-n^) = i The values of B', C, D', and F', as computed from the expres sions B' = n - i , D C" = 1' - 1 + ^5 , F ,,/ n" n' nil \ ~^~^"'^^'' } (lyi) / n n in 1 I ^ ^ — -24 1 S T 24 ~ T55 ) 118 THE THEORY AND PRACTICE OF INTERPOLATION. are given in Tabic YI witli tlu' arnuuK'nt «. By means of tliese co- efficients, values of F\T) are ivadily computed from either one of the formulae 1 F' (C + Ho,) = - ((/,+ //'/,+ C V, + />',/ + £'V,) Hi 1 (192) F' (I- 7(0)) = - («' - B'h + C V' - D'd + E 'el] (193) in Avhicli the enni differences arc vteaHs, taken as indicated l)elow : T F(r^ J' J" J'" Jlv Jv f — (0 i^-i // ^' ^/.' ih) (■' (r/.) e! t i^o ''. ^^0 rtl {!>) c\ (^/) Ci t + (0 ^1 K rf, Several examples Avill now be solved. Example I. — Given the following- table of natural sines T i^(T)EsinT J' J" A'l' Jiv O 40 42 44 46 48 50 0.6427876 0.6691306 0.6946584 0.7193398 0.7431448 0.7660444 + 263430 255278 246814 238050 + 228996 -8152 8464 8764 -9054 -312 300 -290 + 12 + 10 Let it be required to find F'{T) for r=45°. Taking t = 14°, we have = 2° = ^ = 0.0349066 « = t Hence, using the first of (187), we find r, = -300; a, = +0.0246814 12.5 1 f = + .-. mF\ = +0.0246826.5 .-. F' = +0.707106 The true value of thi.s ([uaiitity is — F'{T) = cos T = cos45° = 0.707107 THE TUEOKY AND TKACTICE ()¥ INTEKPOI.ATIOX. lli» Example II. — Fi-om tlie preceding table, conipute the value of F"{r) for T=U°4S'. We take / = 44° ; hence )i = 0.40. Accordingly, from the second of (185), we obtain b = _o.0008614 C" = n-i = -0.10 c, = -300 (7% = + 30 Z>" = ^'-n_ I = -0.203 -F'J = -0.0008586 F'J = -0.70465 The actual value is — F"(T) = -sin T -sin 44° 48' = -0.7046.'^ Example III. — The table below gives the "Washington mean time of moon's ui)per transit at the meridian of Washington : Washington Moon Culminations. Date 1898 Mean Time of Transit J' J" J'" Jiv Mar. 22 23 24 25 26 27 28 li m 15.57 1 1.00 1 47.29 2 34.88 3 23.83 4 13.84 5 4.24 III + 45.43 46.29 47.59 48.95 50.01 + 50.40 in + 0.86 1.30 1.36 1.06 + 0.39 in + 0.44 + 0.06 -0.30 -0.67 111 -0.38 0.36 -0.37 Before proposing an example fi-oni this ephemeris, it is proper to remark that the tabular function is the time of the moon's arrival at a succession of meridians (in reality one fixed meridian) whose com- mon difference of longitude is 24 hoiu-s. The argument of the series is therefore the terrestrial lomjitudc traversed by the moon, counted west from the Washington mei'idian : the iutercal of this argument is 24 hours of longitude. Now, let IJ denote the differeiuT in. time of transit for 1 /lonr of longitude. This quantity is simply the first derivative of the tabular function: computed for the instant of transit at a meridian I hours west of Washington, the quantity D expresses the amount by which the local time of transit at the meridian /-|-1 hours would exceed the local time of transit at the meridian / hours, supposing the rate of 120 THE THEORY AND PKACTICE OF IXTKUPOLATION. retardation to remain constant between the two transits, and eqnal to what it is at the moment of the (ii'st. Thns, il' />,, is tlie value of I) for thi' instant of ti-ansit at AV^ashington on Mar. 24, tlie loeal time of moon's transit at a station 20 minutes west of Washington is given with sufficient precision by the formula T = Mar. '.'4'' 1" 47"'.l-'9 + ], 1\ Now, by the first of equations (18(5), we lind ibr llu' value of />„, D^ = F'(t) = ,i^-(47.o!)-if+f -^') = 1"'.054 Hence the preceding equation gives T = Mar. 24" 1" 47"'.94 In this manner the local time of transit is simply and accurately determined for any number of stations within half an hour of the AVashington mei'idian. To find the local time of moon's transit over a meridian 3 hours west of Washington, on the 2-lth day of March, we have only to in- terpolate the Washington time of transit between the tabular values for Mar. 24 and Mar. 25, as given above, the interval fi-om the former being Cjll n = 24.. = 0-125 Finally, if it were required to compute the local time of transit for several stations whose longitudes range from 2| to 3^ hours west of Washington, we should find the time for the 3 hour meridian by direct intei-polation, as explained above. AYe should also compute D = F'{T) for the same meridian ; that is, for ?< = 0.125. Then the local time of transit at any adjacent meridian, whose longitude from Washington is 3'" -|- A.""", is given b}' the simple formida ^ = ^■+60^ wliere t^ is the time of transit at the 3 hour meridian. Example IV — From the preceding ephemeris, compute the differ- ence in time of transit for 1 hour of longitude (D) at the instant of DR. GEORGE F. McEWEM TTIK TIIKOKY AND PltAC'TTrK OF TNTKIM'OLATTON. 121 moon's transit over tlic iiicridian of San Francisco, Mar. 25, 1898; the longitude from AVasliington being taken as 3'' 1"' 30* = 3''. 025. Here we use the fbnnula (I!*-!) : thus, taking the coefficients (roni Table VI (with the argument n = 8.025 -h 24 = 0.12(;04), and tiic differ- ences from the given ephemei'is, we obtain />" = -0.3740 C" = -f- 0.0282 D' = +0.0692 l> = +1.21 c, = _0..30 d = -0.365 «, = +48.95 III/, = _ O.!;-).-; CVj = - 0.008 D'd = - 0.025 .-. oyF'„ = +48.464 .-. D = /'''„ = 4S'".464+-24 = +2'".019 Example V. — Use the above table of Moon Culminations to find the variation in D for 24 hours of longitude, at the instant of moon's lower transit over the meridian of Washington, Mar. 24, 1898. The lower transit at Washington is evidently the upper transit over the meridian 12 hours west. Hence, denoting the required vari- ation by V, and regarding 1 hour of longitude as the unit, we find by the second of (187), for t = Mar. 24, = Jj (1-33 + 2\ X 0.37) = +0'".059 ■) ()3. Interpolation of Function,,^ hij 3Ieans of their Tabular First Derivatives. — As already observed, it frequently happens that a table giving F(T) also contains the values of F\T) which correspond to the tabular functions. The object in thus tabulating the derivative is to facilitate the interpolation of intermediate values of F(T). To derive the formula upon which this method is based, we consider the schedule below, where the differences are those of the series F'(T) : T F(T) F'{T) IstDiff. 2cl 3d t -2w t — w ^-1 fi' f t + (O Fo' 7'V «-i A. (5 + 2(0 F, i'V 122 THE TUKOKY AND I'ltACTICE OF INTERPOLATION. We sliall iissiime Uiat tlic diirfreiUTS of 1^ (T) beyond .I"' may be disregarded; hence tlie differences of jP'(J') beyond y may be neg- lected in the above schednle. N^ow, by Taylor's Theorem, we have n^(i>' „ ,, wV ,, ,,, "■'to* „. ,^„., ^ " [3_ li Again, since dF' J-F' d^Fi " ~ dt ' " ~ df' ' " ~ dt' ' ' we ol)tain, by means of llie fornnilae (175), in wliich we have put, for brevity, a = H"' + "i) , V =- Hy' + y.) (I '•»<■') Substituting these expressions for i^„", i^^,,'", and 1^^''' in (H>1), the latter becomes F„ = 7'; + ^coT-; + — {a-}., y) + f3^+ y [2_ li li which may be Avritten /; = /; + ««. f a;'+ !; a + f p^ + -, (f -i) y) (197) By means of tliis foi-mnla Ave compute F„ in tei'ins of the differences of F'(T), instead of tlie differences of F (T) direct, as in the usual formulae of inter])ohition. Substituting — u foi- // in (1-*T), we have F_„ = /;_ MO, (^z-;'- !;« + f /?„-;;, (f-l)y) (198) The values of B = f , r=-rl,(f-l) (199) THE THEORY AND I'llACTICE OF JNTERPOLATION. 123 are given in T;il)le VIII with llu; argiunent n. By means of these coefficients we readily compute (200) (201) The coeHicients in Tal)le VIII are not extended beyond m^O.60, since by this method it is invariably more convenient to jjrocced from the nearest function F^ . Example. — From the American Epherneris for 1898 we take tlie heliocentric longitude of Mercury, together with tlie dallij motiou, in longitude, for a portion of the month of October. The differences of the daily motion are then taken, as shown below : Dale Helioc Long, of Daily Motion a |8 r 6 1898 MiTciir;/ O 1 II o / // / // / ff II fl Oct. 11 176 51 7.8 4 2 34.3 -14 0.0 12 17.5 10 40.4 9 9.4 - 7 45.8 13 15 184 41 59.2 192 6 33.3 3 48 34.3 3 36 16.8 + 1 42.5 1 37.1 —5.4 6.1 -7.4 -0.7 17 199 8 10.6 3 25 36.4 1 31.0 -1.3 19 205 49 59.6 3 16 27.0 -1-1 23.6 21 212 14 54.7 3 8 41.2 Let it be required to find the heliocentric longitude of Mercury for the date Oct. 15'' U" 21'" .0. Here we have T = Oct. 15-' 14" 24"\0 = Oct. 15''.60 MO) = T -t = 0''.60 n = 0.30 Hence, using Table YIII, in coiniection with (200), we obtain F^ = 192 6' 33.3 ™ = +0.15 B = +0.0150 r = -0.02,"9 Whence a = -11 28.95 |8„ = + 1 37.1 -y = _ 5.75 /;' = +3 36 16.8 |« = - 1 43.34 B/3„ = + 1.46 Ty = + 0.14 Sum, 1) = +3 34 35.06 F.. = F„+ ,im . D = 194° 15' 18".3 124 THE TllKOKV AND ri;Af'TlCE OF INTEKi'OLATJON. Differencing the given series of longitudes ami applying Bessel's Formula of iutei-polation, we find /-; = 194° 15' 1S".2 ()4. ^ljq>Jlcatl<)n of the Prevcdltiy JSLihod of Inicrpohdion irhen the Second Diferences of the Series F(T) are JSfearhj Constant. — When the 3d and -tth differences of F {T) arc small enough to be neglected, we may omit the terms containing ^^ and y in the formulae (197) and (198) : we therefore obtain F_., = /;_«,.( a;' -^«) (203) It will be interesting to determine the error of these apiJroximate formulae as applied Avhen the 3d diffei'cnces of F(T) are apjjreciable. For this purpose •\ve write (197) in the form K = ^ +"-(/•;'+ ^0 + f -/8o+(5i-T"^)"'y Hence, if we disregard 4th differences of F{T), and thus neglect y, it follows that the error in question is — c = ±f «,/?„ (204) Now, from (175), ^ve have also, from (195), Whence and (204) becomes ^■'"(0 = ;-. = ^' F'"(t) = % «/J„ = c = z/'" (20o) = ± » J'" (200) Since in practice the maxiuuun value of n is 0.50, it follows that the maximum cri-or resulting from an application of the formulae (202) and (203), when 3d differences of F{2') arc sensible, is ^V--^'"- Hence, even when third differences are considerable, these formulae are suf- ficiently accurate for many pui'|)oses. THE THKOIIY AND I'K'AOTIOIO OF INTKIiPOLATION. 125 Thai tlic foniiiiliie (202) and (2().'5) are rujorou^^J ;/ true wlicn the 8d ditfereiKHis of Fl^T) are zero may be eleai'ly shown from g'eo- metrical eonsiderations, as follows : The 2d ditterences of F {T) being supposed constant, it follows fi-om Theorem VI that the function is necessai'ily of the form F{T) = a^T- + a,T-{- a.-. (207) N^ow, if in the accompanying figure we draw the rectangular co- ordinate axes OT and OY, and plot the curve defined analytically by (207) (regarding // = F (T) as the ordinate coi'responding to tlie abscissa T), it is evident that we ol)tain a parahola whose axis is parallel to OY. Let us now take OM = t OS = f + 0, ON = t + noj Whence MN = 7iu) MP = F{t) = /; NQ = F{t + ni^) = F„ Draw the tangents PA, QL ; also, draw PD \\ QL and PB \\ MN. ilF Then, denoting ^m l^y ^ni we have FJ = tmAPB FJ = tan DP D Hence we find NA = MP + PB tan APB = F,+ iio,FJ NI) = MP + PB tan DPB = /l + nu,?\' It is therefore evident that to find NQ =^ F,,, which lies between NA and ND, we must employ a value of F' somewhere between the values i^„' and F^. N^ow, let KE be the ordinate erected at the mid- dle point of MN, and EH the tangent at E. Then, by an elementary 12(5 THE TIIEOliY AND PK.VCTICE OF INTERPOLATION. theorem of the parabola, llie cliord l'(^ is jxinillcl to EII, and we have, therefore, NO = MP + rr. tan QT'li = F„ + no>F! (208) which agrees with the formula (202). We have shown above that tlie maximum error i)roduced by appl}'- ing this formuhi when tlie second diiferences of F (^T) ai'c not constant, is 4V ^"'- Hence, unless the 2d differences of F\T) are considerable, we may compute F„ by the following Rule : Fhid hij simple inter jwlation the value of the talmlar derivative tohich belongs midioay hetiveen the required function and the nearest tabular function (F^) ; multiplij this quantity (F!/) hi/ the units contained, in the entire interval [T — /), and ajiply the product to F^. Example I. — Given the following eiihemeris of the moon's decli- nation (8) : compute the value for the date July 9' .'3" 18™.0. Date 1898 Moon's Decl. S Diff. for 1 Minute a ^ July 9 1 9 4 9 7 July 9 10 + 6 2 14.1 6 43 39.0 7 24 37.4 + 8 5 8.0 + 13.876 13.732 13.582 + 13.422 It -0.144 0.150 -0.160 It — .006 -.010 Here w = 3" = 180'" ; hence, taking t = July 9'' 4", we find 78° n = = 0.433 = 0.217 180'" • 2 Accordingly, the value of F' interpolated for half the interval, or 39 minutes, is — F'„ = 7'V + la = 13".732 - 0.217 X 0".147 = 13".700 Whence we obtain S = 6° 43' 39".0 + 78 X 13".700 = 7° 1' 27".0 Since the value of n is nearly one-half, we may interpolate bach- wards from July 9'' 7'' with equal facility : thus we find n = 0.507 S = 0.283 .-. FLn = 13".582 + 0.283 X 0".155 = 13".626 THE TIIKOUV AND IMtACTICE OF INTERPOLATION. 121 Wliciu'c; 8 = 7° 24' ;57".4 - 102 x 13".(;2G = 7° 1' 27".55 which substantially agrees with Ihe above result. Example U. — From the Ibllowing table of the moon's horizoiilal parallMx (77). inlerpolale the value for July 10" IG" 24'".0. Date 1898 July 10.0 10.5 11.0 11.5 Moon's Ilor. Parallax Diff. for 1 Hour a 56' 2g!i 56 2.5 55 40.7 55 21.1 -2.04 1.89 1.73 -1.55 + 0.15 O.k; + 0.18 Here we have T = July 10'' 1G".40 o) = 12 hours „ = ^ = 0.367 t = July 10'' 12".00 » = 0.183 We therefore obtain Fi = -1 ".89 +0.183 X0".1G = -1".8G .-. -K = 5G' 2".5 - 4.4 X l"-86 = 55' 54 ".3 Interpolating bachvards from July 11'' 0'', we find ,r = 55' 40".7 + 7.G X 1".78 = 55' 54".2 65, Choice of Fori)iulae in a Given Case. — When derivatives are required to lie computed at or near either the heginning or the enxl of a tabular series, the formulae derived from Newton's Formula of interpolation must necessarily be employed. In all oilier eases, the choice lies between Stirling's and Bessel's forms, and should be decided by the value of n. When n = 0, the formidae (175) are un- questionably the best. When n. = .1, the group (187) is especially convenient. As a general rule, subject to change in certain cases, it may be stated that when n lies between the limits 0.25 and 0.75, the formulae derived from Bessel's Formula of interpolation will be found most convenient : for other values of n, those derived from Stirling's Formula should be employed. 128 THE TIIEOKY AND PllACTICE OF INTERPOLATION. EXAMPLES. 1. Given the follovvino: table of " Latitude Reduction 9 9"-?' ? V — f' O 5 10 o' o!oo 1 59.53 3 55.47 o 15 20 25 5 44.32 7 22.80 8 47.93 Compute the variation of f^r — ) = F^ + nJ^' + BJJ' + CJ,'" + DJ^''- + .... where B, C, D, . . . . denote the binomial coefficients of the ?«th order. Multiplying by dn, and integrating, we obtain CF(t + )iu,)d,i = C{F^ + 7iJ^' + BJ^"+CJ^'"+ . . . .)dn or fF(t + tia>)dn = nF„+ f J^' + J," CBdn + J^'" Ccdn+ . . . . + 31 (212) where 31 is the constant of integration. If, for brevity, we put 13 = Chdn , y = Ccdn , 8 = Chd7i , .... (213) then, from the preceding equation, we derive £F(t + n^)dn = F^ + iJ^' + f3J," + yJ,'" + U;^ + .... (214) Whence we obtain, in succession, fF{t + no,)dn == CF(f + w + 7tM>)dn =^ Fi + i/l,' + fiJ," + yJ,"'+ .... CF(t + v^)d7i = CF(t + 2u, + 7iw)dn= F., + i/1„' + (3.'V' + y.V''+ .... (215) ("F(t + jio>)d7i = CF(t + i-U + 7>u,)dn = /',_i + i4'_i + ;8J;i, + yz/,-'i+ • • Summing the integrals expressed in (214) and (215), we find r=i— 1 r -i— 1 r =i— l r = i— 1 j"F(t + 7iu,)d7i = 2 F,+ i 2 J,' + I3 ^ J/'+y 2 ^l'"+ .... (216) r=0 r=0 r=0 r=0 The numerical values of ft, y, 8, ... . (sometimes called the coefficient!^ of quadrature) must now be determined. These may be THE THEORY AND PKACTICE OF INTERPOLATION. I'Mi found directly by integrating' the expressions for B, C, D, . . . . , as expanded in (163), and then taking the Hniits of n according to (213). But the following' indirect method seems preferable, since it adds a significance to the result. Let us i)ut Q =r(l + _y)"f/« =!"(! +»// + /?/+ C// + /y+ . . . .)du (217) where j/ is supposed constant. Then, if we also put Q' = fli+!/ydn we shall have Q' = 1 + i y + p,/ + y,/ + h/ + a/ + t/ + . . . . (218) the coefficients being those defined in (213). Again, put that is and we find or (1 + //)" = z (219) ?ilog(l+//) = logs dz log(l + y) . dii = — log(l+y) We therefore obtain /C C dz z (1+v)" (1 + M)"c??i = I zdn = I , — TT-i-— • = , — ^, . ^ + const. = / ,.•' . + const. ^ •" J Jlog(i + (/) log(l + y)^ log(l+y)^ Whence (1+y)" Q' = S^vr In log(H-y)J 11 = log(l+//) \ 2"^3 4 "^5 Expanding the last expression by the Binomial Theorem, or by direct division, we obtain Q' = i + i.'/-TVy'+5Vi/''-7¥o//' + Tia/-ag^l,T/+ • • • • (221) Whence, comparing (218) and (221), we find (222) li = T5 € = + tIs y = + 5^ t = £6 3 s 134 THE TIIEOEY AND TKAOTICE OF LNTEKTOLATION. which are the numerical values of the coefficients of formula (21(5). It therefore ajipeai-s that the fundamental coefficients of quadrature are those in the expansion of [log (l-|-y)]~'. Let us now regard the functions F^, F-^, Fo_, . . . . Fi as first differences of an auxiliary functional series which Ave shall designate 'F. A schedule containing the new series may be conveniently arranged as follows : T 'F ¥(!•) J' J" J'" 'P^ t K IF, ^J t + (O J", ^o" 'F„ •J/ J'" t + 2u> 'l'\ F., J/' 'F , ■ ^'.- , J"' t + (l-l)w 'Fi ^',-1 ^',- 1 4'i. t + 1(0 'F Fi The value of 'F^, is entirely arbitrary. Having assigned a con- venient value to this quantity, the remaining tei-ms in the series are readily formed by successive additions, thus : '/'■' = '7^ + F 'F — 'F + F IF ., = 'F- + F- "VVe shall now put the second member of (21(3) under a form more convenient for computation. By Theorem I, we have ^K = K + F> + F, + . . . . + i'Vi = 'F, - 'F^ 2 ^J = ^o' +^i' + 4' + • • • • +'^'-1 - F,- /; r=0 r = ( - 1 2 J,'l = JJI + J,'l + zlJ' + . . . . +JIU = JJ - z/„' r=0 r =i— 1 ^/i;"= 4"'+Ji"'+j„"'+ .... +j;!!i = J/i-J,ii and hence (216) becomes £F(t + no.)dn = (iF,-iF^) + i{F,-F;) + fi(Ji-J^i) + y{J,'i-J^ii) + S(z//"- j;") + e(jr-4') + ■ (223) (224) THE THEORY AND rRACTICE OF INTEKPOLATION. 135 This formula possesses the disadvantage of involving differences ^.') ^/') ^i", ■ ■ ■ ■ which are not furnished by the foregoing schedule. To o])viate this difficulty, wc proceed as follov^s : Put q = 'F^ + h^F, + 13 1,' + yj," + a /,'" + £. ir + ii^ + . . . . and (224) may then be written £Fit+n,.)dn = q - {'F^ + iF^ + l3J,' + yJ," + UJ"+ . . . .) (225) (226) Upon giving to n, in formula (75), the values -\-l, 0, — 1, — 2, — 3, — 4, . . . . , successively, we obtain '^.- ='i^m-^. F, =F, 4' = ^'i- -1 + J'U + AIL, + ^;l, + JU+ ■ ■ ■ ■ 4" = J'^„ + 2J!!J, + 3z)ti + ^^U+'- ■ ■ ■ 4'" = Jill, + SzT'I, + QJU + . . . . Jr = AU + ^JU + . . . . (227) If these expressions be substituted in (225), we shall have q in terms of the known tabular differences, and hence obtain the requii'ed integral from (22()). To avoid the labor of numerical reduction inci- dent to this substitution, we derive the result in the following indirect manner : Put log (1 + x) ^ r- 1 Also, take u * - 1-M and we have a;-' = lL-^{y-u) = ?i-' - M° x" = M» X = u (1-m)-i= U^l,}+ m'+ M^+ «'^+ . . . . x^ = u\\-^^)-''■ = m2+ 2w''+3j/.^+4«^+ . . . . x' = M=(l-i/,)-'= m3+3?<*+6h=+ . . . . X* = !«*(!-«)-'= M*+4?4='+ . . . . (228) (229) (230) 13G THE THEORY AND PllACTICE OF INTERPOLATION. If now, we substitute these expressions for «"', .-e", x, of, ... . in the second member of (228), we obtain 6 in terms of ir\ u", u, ti'\ .... But it will be observed that this operation is identical in algebraic form with the substitution above proposed with respect to (227) and (225) ; for the operation involves the quantities $■ X-', x", X, x% x\ . . . . ; »-', u", 11, u\ 11*, . . . . ; while the q oi)ei'ation involves, in precisely the same algebraic relations, the quantities „ . IW W A I /I II /I III -IF F /ti //" //'" Hence the i-esult for q will immediately follow when the result for 6 has been derived. But we may obtain as a function of u, in the form required, more simply than by direct substitution of the expres- sions (230) in (228). For, by (229), we have 1 + .T = ,r^ 1 — M whence log(l+.r) = _log(l-«) (231) Therefore, by (228), we find = ] = -^ ^ = u-^-irc^+pii-y,r+8a''-m^ + iu^- .... (232) log(l+.x) log(l — (>) Accordingly, writing q for 0, 'i^i+, for «"', Fi for ti", j'i_^ for m, etc., as justified by the pi-eceding reasoning, we obtain q = 'F,^, -iF, + fSJ\_, - yJ'.L, + 8J,^'3 - ^^t, + K-^U - . . . . (233) Substituting this value of q in (22G), and grouping hke terms, we get J> (t + nu,) dn = ('7^,+, - 'F;) - k (F, + i^„) + /? ( J',_, - J'„) -y(J,(i,+ j;') + 8(^r3— 'o"')-'(^i-4 + ^o'')+ • • • • (234) Whence, restoring the values of /8, 7, S, • . • . , as given in (222), and applying (211), we have F{T)dT = to 1 F{t + tiu,) dn = o,\i'F,^,-'F,) -iiF,+ F,) - ,V (^'.-1-^') - .-4 i-JiU + ^o") - 7^5 {^'A-^n - xtiT (4'l4 + ^^o") - « SMii i^U-J.') - .... I (235) THE TUEOKY AND PltACTIUE OF INTEKPOLATION. 137 AVhen the tabulation of the function extends beyond the value Fi, it is sometimes more convenient to employ the following formula, easily obtained from (224) : = ^\{'F,-'F,)+i, {F,-F,) - ,\, (.//-.//) + ,', (^/'-z/„") -7V5(4"'-^„"')+TiTi(4'^— ''V')-iig|ia4^-^o^)+ . . . .\ (236) "We here emphasize the fact that the value of 'F^ is wholly arbi- trary. G8. As an example in the use of formula (285), let it be required to find* J ■•44° COS TdT JO" using six places of decimals. The first consideration concerns the tabular interval to be em- ployed. It is desirable to tabulate as few values of the function as ai-e consistent with a convenient schedule of differences. In all cases the differences should sensibly vanish beyond the thii'd or fourth order. Adopting w ^ 4° as a suitable interval in the j^resent instance, we obtain the following table of F(T) = cos T : T 'F F{T) = cosT J' J" J'" Jiv 20 24 28 32 36 40 44 0.000000 0.939()93 1.853238 2.736186 3.584234 4.393251 5.159295 5.878635 0.939693 0.913545 0.882948 0.848048 0.809017 0.766044 0.719340 -26148 30597 34900 39031 42973 -46704 -4449 4303 4131 3942 -3731 + 146 172 189 + 211 + 26 17 +22 Taking t = 20°, and assuming the arbitrary quantity 'Fg= 0, we complete the column 'F by successive additions. Whence, by (235), we find * In selecting examples of numerical integration for the present chapter, we have in most cases chosen for F{T) some simple, inteijrable function, whose tabular values are readily taken or formed from various tables in common use. By such selection we gain in simplicity, while losing little or nothing of generality ; and, moreover, from thus knowing a priori the true value of the integral sought, we are at once informed as to the final accuracy of each application. 138 THE THEORY AND PRACTICE OF INTERPOLATION. = 6) F^ +F^ = +I.(i5<)033 A' -A' = - 20556 J," +J," = - 8180 A"'-A"' = + 65 Jjiv + J^iv _ + 48 'F-, -'7'; = +5.878635 - i^ {F^ + /; ) = -0.829516.5 - ^ (JJ -J^') = + 1713.0 - 3.^ (A" + A" ) = + 340.8 - t'/o W- A"') = - 1-' - Tile (4'^ + ^^n = - 0^ log^ = 0.703392 logo, = 8.843937 log A' = 9.547329 sum, Z = + 5.051170 CD = * = IE A' = 0.352638 Since Ccos TdT^ sin T, we find for the true value of the defi- nite integrral. A" = sin 44° -sin 20° = 0.694658 - 0.342020 0.352638 If it be required to compute X = coi COS TdT 20° from the foregoing table, formula (236) at once serves the purpose. Thus we obtain (i = 2) 'F„ -'F^ = +1.853238 F„ - F^ = -56745 + H^2 - ^0 ) = - 28372.5 J^' - J^' = - 8762 - A (A' -A') = + 729.3 ///' _j;' = + 318 + ^VW-^o") = + 13.3 J^n, _ j^ni = + 43 -tV\(^2"'- A'") = - 1.1 Z = +1.825607 .-. X = 0.127451 Here the true value evidently is — X = sin 28° - sin 20° = 0.127451 69. Precepts for Computing the Definite Integnil ivhen One or Botli Limits Fail to Coincide with some Tabular Yalue of the Argu- ment T. — Thus far we have considered the limits of the integral X J'KT" F{T) dT to be of the form where t" and i" are integers, and hence T' and T" are two jiarticular THE THEORY AND PRACTICE OF rN^TERPOLATION. 139 values of T for which F (T) has been tabuhitcd. We sliall now con- sider the more general 2'i"oblem of finding X when the limits have the form T' = t + n'u> , T" = t + n"a, where ii and n" are non-integers — that is, either proper fractions or mixed numbers. To illustrate the significance of the problem in question, suppose it were required to find by mechanical quadrature the value of /»42'' 46' 64" = I COS TdT i/Sl" 13' 37" Obviously, it would be impracticable to tabulate the function for a series of equidistant values of T, of which T' = 21° 13' 37" and T" =^ 42° 46' 54" are two particular terms. We may, however, employ the same table as was used in the preceding examples, con- structed for T = 20°, 24°, 28°, .... 44°, and obtain the required result by interpolation. Thus, in the. examples jiist mentioned, we have computed the values of X from the lower limit T' = 20° to the up])er limits T" = 44° and 28°, respectively. In like manner, keeping the lower limit always ^ 20°, we may find the integral cor- responding to each of the following values of the upper limit, viz.: T" = 20°, 24°, 28°, .... 44°, respectively; that is, for each of the tabular values of T. Then, having differenced the resulting values of the integral, we may i-eadily find by inter- polation the values which correspond to the upper limits 21° 13' 37" and 42° 46' 54". Denoting these interpolated values by X' and X" respectively, we have A''= j cos TdT , X" = \ cos TdT and therefore /»42» 46' 64" X = I COS TdT t/Sl" 13' 87' A'"- XI We leave the detailed solution of this example to the student as a valuable exercise, exhibiting the spirit of the method employed in pi'oblems of this type. The process actually used differs somewhat in 140 THE TUKOKY AND I'KACTICE OF INTERPOLATION. form from tlio method here explained ; but the principle remains the same. We proceed to develop the general formulae. 70. Let US put T. = j F{t + n(j>)dn (237) and *(/) = 'i<: + ^ F, + /3.7/ + yj." + 84'" + Uf + . . . . (238) where * denotes an integer. Then (221) becomes I. = *(0-*(0) (239) Let US now suppose that (239) has been computed for i = 0, 1, 2, 3, 4, .... , in succession. Then, from the sei-ies of values /„ = *(0)-*(0) I, = *(l)-*(0) /„ = * (2) - * (0) (240) thus detei'inined, it is evident that any intei-mediate value, say /„, can be found by interpolation. To derive a general formula foi- this pui'- pose, we must express the differences of the series (240). Now, by (238), we have * (0) = 'F^ + iF„ + pJJ + yJ," + 8. /,,'" + £J,r + * (1) = '/; + i i-'i + /?,_// + y. ;," + 8. 1,'" + eJ," + *(2) = '/; + ^F„+ 13 J J + y.JJ' + 8.JJ" + cJ^^ + (241) whence, observing the general relation we derive the following schedule of differences Function 1st Differences 2d Differences 3(1 Differences 7o = *(0)-*(0) /i = *(l)-*(0) /„ = *(2)-*(0) 4 = *(3)-*(0) /;+^j;+/?Jo"+yj„"'+. . . F, + iJ,' + fiJ," + yJ,'" + . . . F„ + \/h' + p/l„" + yJ„"> + . . . . In' + i.]„" + 13. !„'" + . . . .// + *J," + /3. /,'" + . . . .IJ + IM' + I3.J.,"' + . . . ./,"+IJl"'+. . . Therefore, applying Newton's Formula of interpolation, we have /„ = /„ + ,i(lst ])iff.) + B(2d Diff.) + C7(3d Diff.) + . = *(0) -*(()) +«(7:,+i/V + M" + 7-V"+ • • •) ) + D(JJ>'-h . .) + THE THEORY AND TRACTICE OF INTERPOLATION. 141 By transposing the term —^(0) to the first member, and substi- tuting for ■^(0) in the second member the expression given by (241), we find /„+*(0) = ('F„ + iF, + l3J,' + y.J," + BJ,'"+. . . .) + n(F, + iJ,'+pj„" + y.l,"'+ . . . .) + 2?(Jo' + *^„" + /3.-V"+ . .) + C(.J," + kJ,"'-t. .)+^W'+- •)+• • Upon ai-ranging the last expression according to the coefficients 1, I, |8, y, 8, . . . . , it becomes /„ + *(0) = ('7^„+ nF„ + BJ„'+CJ„" + I)J„'"+ . . . .) + i(F,+ Hzl„' + BJ,"+CJo"'+ . . . .) +P(JJ+nJ„"+BJo'"+ , + 8(Jo"'+ . + . Now, it will be observed that the first polynomial in the second member of this equation is simply the expression for 'F„, — the quantity derived from the series 'F^, 'F^, 'F.;,, .... by interpolation. Simi- larly, the remaining parentheses contain the expressions for i^„, j'„, //„", . . . . , likewise derived by interpolation fi'om their respective series. AVe therefore have J„ + *(0) = 'F„ + ^F, + li.J'„ + y.lj' + h.lj"+ . . . . = *(m) (242) Whence CF(t + nu>)dn = J„ = *(,i)_*(0) (243) 71. In like manner, if we put (r(0 = 'F,^,-^F, + (iJ',_,-y.JlU + U',1,- . . . . (244) then, by (234), we have /, =iiF{t+nu>)dn = ,y(j)_*(0) Therefore, by interpolation (reasoning precisely as above), we obtain r^X^ + Mw) dn = (f (n) — * (0) (245) 142 THE THEOET AND PRACTICE OF LNTEKPOLATION. Again, writing n for the upper limit n in (243), and v!' for n in (245), we get CF{t+ni^)dn = *(«')-*(0) , (F{t + nu>)dn = g(7i")-*(0) the difference of which gives CF(t + nw)dn = g (»")-* (re') (246) Upon substituting in equations (243) and (245) the expressions for ^ and '/' as given by (238) and (244), and restoring the numeri- cal values of yS, y, 8, . . . . from (222), we obtain i"F{T)dT = o>CF(t + n,D)dn = <„K'i';-'i';) + H^;-^o)-i\(-J'„-^'«)+ .'4 i^J'-^o") - t'A (4/"-- '„'") + T§TJ K^-40-^Sn5 (^-//j; + . . . . I (247) Jf-h'tO) /*n F( T)dT = ,a\F{t + nm) dn = „K'i^„^.-'V„)-i(J^„+^„)-,->, (J'„_,_J'„)_^i, (4,','_2+^o") -7V.G^;:^3-^„"')-TtTr(^r-4+jr)-,gi3n(^i-o-^j)- • • • -^ (248) In like manner, we derive from (246), X!+n"(0 /' n" i'X 2') rfT = 0)1 /''((; + « F, j;' 'F, J', J'/' t + 2^ ■ F., -';' t + (i' — l)o) F. , 4''i 'F, , J',. , 4"\ t + iia F, z/;' t^- (/+!)«) '^:+! ^'Vi '^',+« ^',+1 ^;;i ^;;j j;;'. t + (i+2)«) >:,. ^;« Reverting now to (104), an inspection of this equation shows clearly the law of fornaation of the successive coefficients in the second member : hence, adding the term in ./", we have "2 ') + 2" ^° + 6 (^ 2 ) + 2r~ ^^ + 120 [~^ J+ 720 ^» • • • • ^^^ Multiplying by dn and integrating, we obtain fF{t + wo.) tin = uF^ + f (ju + j[) + s" Ji' + 5^5 (I*- »i=) ( j::; + z^:") (252) 3/ being the constant of integration. If this integral is now taken between the limits m = — | and ?z = + i , the coefficients of J', J'", J^, . . . . evidently vanish, and we find, therefore, JF{t+ no,) dn ^ F,^ ^, J„" - sU^ z/J7 + ^ifi^^^ JJ' (253) (254) 148 THE THEORY AND PRACTICE OF INTERPOLATION. In like manner, we derive jF[t + nu>)d>i = F,+ ^\ 4" - jijj Jr + iniVVs^ ^V- • • • • Whence, by summation, we obtain J7(/ + «a,)=0 (256) in formula (255), the latter becomes fFlt+ ««,) dn = ('F,^^-'F_{) + jij (j;+j - JLj) - ^u^ (j;;'i - j::.;) Finally, therefore, we obtain F{T)dT = w 1 i''(i! + wo))(7ra = o,|('i.V,_'i<'_j)+ ^V (z^',.+,-zi'_j)-5Uij(-^.';'i-^^)+ .AV«n(/^JVi-^Ii)- ■■■■\ (258) When several values of an integral are to be computed from a given series, each having the lower limit f — iw, it will be more convenient and expeditious to determine the arbitrary quantity '-F_^ such that the sum of the terms with subscript — I is equal to zero. The foi-mula (258) may therefore be written as below : /K" 1 //' J. 1 7 //'" 3B7 x/v 1 \ -'-.J — — 51^^-!+ 57e(J ^-! SJT57BB5 '^-S+ • • • J F(T)dT = 0, iF{f + no,)dn (259) 1—10) «^-l I = «»('^.+J+5'*^'.«-5H0^m+^5\VBis4''+,- • • •) / THE THEORY AND PRACTICE OF INTEKPOLATION. 1^9 As an application of (258), let it be reqnired to find .r = Csec-TdT Taking w = 3°, t = 31° 30', and 'F^i = 0, vvc tabnlate F(T) = sec- T as follows : T 'F F(T) = sec^ T J' J" J'" Jiv 25° 30' 28 30 31 30 34 30 37 30 40 30 43 30 46 30 49 30 0.00000 1.37552 2.84788 4.43667 6.16612 S.06005 1.22751 1.29480 1.37552 1.47236 1.58879 1.72945 1.90053 2.11045 2.37089 + 6729 8072 9684 11643 14066 17108 20992 + 26044 + 1343 1612 1959 2423 3042 3884 + 5052 + 2(59 347 404 619 842 + 110S + 78 117 155 223 + 326 Owing to the rapid convergence of the coefficients in (258), the effect of fifth differences is here insensible : hence, nsing but three terms of this formula, we obtain (i = 4) z/',j-J'_, = +12920 J'^'-J'li = + 899 -'F_, = +8.06665 -J'_^) = + 538.3 -z?-) = - 2.7 log V = 0.906982 log o> = 8.718999 log X = 9.625981 V = +8.07201 a, = 3° = ,r + 60 .: X = 0.422650 Verification : Since /■ sec-Tdl'' = tsLXiT we have X = tan 45° tan 30° iV3 0.422650 To illustrate the application of formida (259) when several values are assigned in succession to the integer ^, we solve below an example which proceeds according to the evident relation = '.-j;:(s.)- DR. GEr 150 THE TIIEOKY AND PRACTICE OF INTEKPOLATION. where / denotes the value of any coordinate at the instant T, and /„ its value at the epoch T^. In particular, let us put I — the lieliooeuti-ie longitude of Mars for any date T; — = the daily motion in lonuitnde ; (IT ■' 2'^ = 1898 June 13, Greenwich mean noon ; ? = 1° 47' 14".3 = the heliocentric longitude for the date T^; and let it be required to compute the longitude (I) for (Jreenwich mean noon of the dates (1898) June 21, June 29, July 7, July 15, and July 23; the values of the daily motion being taken from the American Ei^hemeris for 1898. The complete solution is conveniently arranged in tabular form as follows : Date 1898 .(r)=s(fy J' A" T 'u + '^ +£ I June 1 9 17 25 o / n 5 1 36.8 4 59 51.3 4 57 45.4 4 55 21.0 -105.6 125.9 144.4 160.8 ff -20.4 18.5 16.4 June 13 21 29 / // 1 47 19.5 6 45 4.9 11 40 25.9 n -5.2 6.0 6.7 1 47 14 6 44 59 11 40 19 July 3 4 52 40.2 175.0 14.2 July 7 16 33 6.1 7.3 16 32 59 11 4 49 45.2 187.5 12.0 15 21 22 51.3 7.8 21 22 44 19 4 46 37.7 198.1 10.6 23 26 9 29.0 -8.3 26 9 21 27 4 43 19.6 — 206.3 — 8.2 Aug. 4 4 39 53.3 The function tabulated in column F{T) is eight times the daily motion in I : it is so multiplied, because the unit of the derivative being one day, we have w = 8 ; and thus the final multiplication by this factor is avoided. Upon taking t = June 17, the formula (259) is at once applicable. AVe have, therefore, since differences beyond J" are negligible. and l-L = ^ ilT — I F J- 1 //' THE TIIKORY AND PRACTICE OF INTERPOLATION. 151 the factor w having been previously ai)plied. Whence the expression for / becomes Thus, the value of / for ant/ date T being found by adding the constant /„ to the integral taken from T^ to T, it is clear that we have merely to increase the above value of 'i^_, by the quantity 4 = 1° 47' 14". 3 in order to avoid the subsequent addition of this constant to each computed value of the integral. Accordingly, under the heading /(,-|-'i^, on the line t — ,] w (=: Jnne 13), Ave write the quantity 1° 47' 19 ".5 ; the remaining numbers of this column are then formed in the usual manner by successive additions of the functions F. Each term of the series thus formed is evidently greater by Z^ than if the latter constant had been excluded from the initial term. Under -|- ^\J' are written the values derived from the corres- ponding terms in J'. The sum ^o~h -^4'2V^' '^ then tabulated in the final column, I, which therefore gives the heliocentric longitude of Mars for the dates indicated in column T. 73. AjypUcations in which the Limits Fall Otherwise than Midway Between Tabular Values of the Argument and Function. — If we put ea+v) = 'F,,, + ^\ j',„ - jH\cF„-'F_o + ^,(^'n-^'-i)-z\hO^::'-^^)+,ifj^^(J:-Jii)- ■ ■ -l (264) J'»(-(-n"(i) /• n" F(T)dT = ^ \F{f + nw)dn = «>K'^:-'-'^.0+5L(z/V'-z/'„0-jHTj(-<"- T" = 0.48 = 0.20 + 0.28 = < + 2.8 «, Hence, at the upper limit, the value of n is — » = 2.8 = 2.5 + 0.30 Accordingly, we find 'F,,, .J'„, and ./;;' by interpolating forward from the quantities 'i^„„ J'„5, and ./.;'^ with the interval 0.30. From the table above, we take 'F„,^ = +4.058247 //'„.; = +0.156896 /I';;, = +0.001572 Hence, making the required interpolations by means of Bessel's Formula, and jjroceeding according to (266), we find 'F,^ = +4.535670.3 z/',. = +0.161674 + „\ //'„ = + 6736.4 j'j' ^ + 1019 -^H^^n' = - 4.8 i: = +4.542402 .-. X = +0.4542402 The true mathematical value of X is easily found : thus, since we have X = e"-'^ - c°'= = 0.454240159 74. Quadrature as Based upon Bessel's Formula of Interpo- lation. — Another set of formulae for mechanical quadrature, similar to those ah-eady developed, may be derived in the same manner from Bessel's expression for F{t-\-no}). However, since these formulae may be obtained more conveniently by a direct transformation of those developed in the preceding section, we choose tlie latter course. Putting n' = i, and /// = 0, in formula (263), we have f>(i + K«)) d)i =6 (i) - e (0) (267) We also have, by (260), e(i) = 'F^+^,J'.-s\lo-Jl"+^^^x>^^- ■ • ■ • (2G8) 154 THE TIIEOKY AND rilACTICE OF INTERPOLATION. Referring now to tlie general schednle on page 147, it will be observed that the quantities are not explicitly given, but must be found by interpolating fo halves between 'i^,_i and 'F^+i, j'^-i and J'i+i, etc., respectively. For this purpose, let us denote the algebraic iiieans of the latter pairs of quantities by ('F,), (/',), (i't"), . . . . ; that is, let us put i'F,) = i{'F,_,+ 'F,^,) (269) Applying formula (126), we have, therefore. 'F, = i'F,) - i(^'<) + TS^(^r)-T^,(^r)*+ • • ■ ■ ^'i = (J',) - I (j;") + -ris C'^O - . . • . J[" = (JD - i (Jj) + • • • • ^"i = {.■11) - . . . . Upon substituting these values of 'Fi, z/'j, .J-", .... and reducing, we get e (i) = i'F,) - ^y, (J',)+ -,y„ (J-") - ^hih (^0+ • • • • Putting i = 0, this becomes 6(0) = ('i^o)-TV(-^'o) + TVirK")-irMkK)+ • • ■ • Whence, from (267), we derive j"F{t+nco)dn = e(!)-e(0) + AV [(^A"') - «')] - TT jUtt [K) - (^:)] + • (270) in (268), (271) (272) (273) *It is evident from (111) that the coefficient for the sixth difference in Bessel's Formula is- (m+2)(M+l) n (n— l)(ft— 2)(n— 3) l!l which, for /( = i, yields the value given in the text. THE THEORY AND PKACTICE OF INTEKPOLATION. 155 Again, putting n ^ i in (2(32), we liave C'F{t + nu,)dn = 6l(;)-6l(-i) = C^) - Vs (./'.) + 7V„ (./;") - AlU (^0 + . . . . -'F_, - ^^J'^,+ ,\U ^-i - sbWso ^Lj+ . . . . (274) In like manner, making n" ^ i-^\, and y/ =: 0, in (203), wc obtain CF{f + iiiS) dn = (9 (; + i) - (9 (0) - C^o) + tV (^'o) - tVV K") + AiU W - . . . . (275) Finally, substituting u" = « and u =z 0, in (2Go), the latter becomes n''(i! + Mo))(Z?t = e(^n)-0(O) — 'P 4. 1 ,/' 17 ,/"'j_ 3 7 ,/v _ - C^o) + tV ( J'o) - tVtT (^-^o") + IfiliTT W - . . . . (276) The equations (273), (27-4), (275) and (27(5) give, respectively, the following formulae of quadrature : ("F(T)dT = o> CF(t + ii[(^d-Uon+ • • • . I (277) J"I+)d7i = - K'^.) - iV (^',) + tVs (4'") - ^JSiij (-'O + • • • • -'F_, - ji, J'_, + sHs ^-; - uirW^TJ ^Ij + . . . . ^ (278) F{T)dT = wj /'(;'+ "o,)fZw - C^'o) + iV (J'o) - ^ViT (-^D + aiU (^J) - .... ^ (279) r/'''(7')(ZT = u,CF{t + noi)dii = (0 1 '/^;+ .)j j'„ - ^fj^ <' + ^^Wbtj ^;; - . . . . - C^o) + A (^'o) - t'sV C^o") + K-Jlit* (^oO - .... I (280) 156 THE THEORY AJSTD PKACTICE OF INTERPOLATION. ill which / denotes an integer and n a non-integer ; where 'F-i is wholly arbitrary ; and where ('Fi), {-J'i), • • • • 'i"d {'F^), (z/'o), • • • • are vitans of corresponding tabular (juantities, as defined by (269). If, in the formulae (277), (279), and (280), we take C^) - iVC''o)-rVo(X') + ann(^o)- • • • • then the sum of the terms with subscript zero will vaiiisli. But, since the 231'eceding condition is evidently satisfied if we take 'i^_5 = -i^o+TV(-J'o)-T'5VK")+tri^io W- • • • • (281) The formi;lae (277), (278), (279) and (280) may therefore be computed as follows : 'F_, = - i-Po + TV (^'o) - tVo i^o') + ,llh (-Jo) - • • • • \ j"F'{T)dT ^ u>CF(t + no,)dn (282) = -s('^:)-TV(-j'.) + 7va-';")-ffJik(^o+ • • • • I / CF'('T)dT = u,i'F(t+iiu,)dn (283) = <-|('^.)-TV(-^',) + Tyo(4"')-ITilk(^^I)+ • • • • w '^ = -iJ'\+T\ u'o) - tVc c^o") + auij (-'o)- .... \ F(T)dT = = 20° = F{T) = wTsm T, as follows t :=■ 10° = :j^ , and tabulate T 'F F(T)=u>r&\uT J' A" J"' J" J' o - 50 30 - 10 + 10 30 50 70 90 110 130 + 150 0.00000 0.01058 0.10197 0.33532 0.73607 1.28438 + 0.23335 0.09139 0.01058 0.01058 0.09139 0.23335 0.40075 0.54831 0.62974 0.60671 + 0.45693 -14196 - 8081 + 8081 14196 16740 14756 + 8143 - 2303 -14978 + 6115 8081 8081 6115 + 2544 - 1984 6613 10446 -12675 + 1966 -1966 3571 4528 4629 3833 — 2229 -1906 1966 1 605 957 - 101 + 796 + 1004 + 361 648 856 897 + 808 The value of X is now readily found hj (278). Taking the arbi- trary quantity 'F-t^ =: 0, we complete the column F as above : we then have 'F_, = J'_, = J'l[ = JU = Whence, proceeding by (278), we find (i = 4) ('^4) = = i('F,, + 'F,,) = +1.01022.5 (J'O = i(J'3j + ^'«) = +0.11449.5 - iV (^'4) = - 954.1 {J':')= i(J^ + J^) = - 4231 +tVVK') = - 64.6 (JI) = ii^l,+JW = + 852 -AUA^d^ - 2.7 .-. X = +1.00001 Verification : Since C T sin TdT = sin T - TcosT we have X = .sin T - Tcos T ExAJViTLE II. — Compute the value of 158 THE THEORY AND TRACTICE OF INTERPOLATION. Here we take a> = 0.1, ^ = 0.9, and tabulate i^(T)=(l-)-0,lT')-f, as below : T 'F FCr)E (1+0.1 7'2)-3 J' J" J'" 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 -0.44672 + 0.44302 1.30980 2.15234 + 2.96960 0.93076 0.91115 0.88974 0.86678 0.84254 0.81726 0.79119 0.76455 -1961 2141 2296 2424 2528 2607 -26G4 -180 155 128 104 79 - 57 +25 27 24 25 + 22 Proceeding by means of (282), we compute 'F-^ as follows F^ = +0.88974 (J'„) = - 2218 «') = + 26 - ^ F^ = -0.44487 + t'j (^'o) = - 184-8 -tVuK") = - 0.4 .-. 'F_, = -0.44672 Whence, having completed the column 'F, we conclude the com- putation by (282), with the following results : (i = 3) HF^) = +2.56097 (jy = -0.02567.5 - ^^ (jy = + 214.0 (z/^'O = + 23.5 +7VVK") = + 0-4 Z = +2.56311 A' = +0.256311 Since /< dT (1+0.1 T'')S (1+0.1 T^ we find for the true value of X, X = 1.121936 - 0.865625 = 0.256311 Example III. — Let it be required to find X tan ' 5 = fsec^ TdT Expressing the assigned limits in degrees of arc, they become = 45° tan-'i] = 56°18'35".77 = 56°.30994 THE THEORY AND PKAOTICE OF ISTTEKPOLATION. 159 We now take w ^ 2° ^ g"^ , t =^ 45°, and tabulate the follow- ing values of F (T) = cosec* T : r 'F F{T) = M sec* r J' J" j"i Jlv J" 41 43 45 47 49 51 53 55 57 59 61 -0.00819 + 0.07144 0.23279 0.42121 0.64376 0.90987 1.2.3238 + 1.62909 0.10759 0.12201 0.13963 0.1(;i35 0.18842 0.22255 0.26611 0.32251 0.39071 0.49608 0.63186 + 1442 1762 2172 2707 3413 4356 5640 7420 9937 + 13578 + 320 410 635 706 943 1284 1780 2517 + 3641 + 90 125 171 237 341 496 737 + 1124 + 35 46 66 104 155 241 + 387 + 11 20 38 61 86 + 146 Here we employ formula (285) ; in which, for the upper limit, we have n = (56°.30994-45°) -^2° = .5.65497 = 5.5 + 0.15497 For the value of 'F-^ , we find i.; = +0.13963 (J'^-) = + 1967 (4',")= + 108 - i F^ = -0.06981.5 + iV (^'o) = + l«'^-9 -7VVK")= - 1-6 .-. IF , = -0.06819 Whence, completing the series 'F, and observing that the values of 'F„, J'„, and J,'," are obtained from their respective series by interpo- lation with the interval 0.15497, we find 'F„ = +1.28846.8 + A ^'» = + 323.1 J\ = +0.07754 z/;'" = + 787 _ 17 /I'" — _ 2.3 .-. X == +1.29168 Verification : The expression for the indefinite integral is — Csev.'TdT = taur + J tailor Therefore with which the above result substantially agrees. 160 the theory and practice op interpolation. Double Integration by Quadratures. 75. Having derived various foi-mulae for the mechanical (juadra- ture of single integrals, the corresjionding formulae for douhle integra- tion are now readily deduced. These will serve to compute integrals of the form F(T)dT"- (286) independently of the analytical nature of the function F{T), provided T' and T" are numerically assigned. To define the quantity Y more explicitly, let us put ('f{T) (IT = /(T) + M (286a) where M is the constant of integration. We then have Y = CflT)aT +M(T"-T') (287) •J T' It is therefore evident that unless the constant M has a definite value in any given case, the value of Y will be indeterminate. In practical applications, however, the quantity 31 is generally known from the fact that the fird integral has an assigned value (usually zero) corresponding to the lower limit of integration. If we now i)ut T = t + n<^ , T' = f + n'w , T" = f + )i"w we have (IT- = m-dn- (288) and hence (28G) becomes Y = C CF(T)dT' = ,^-' ffF(t + >io,)dn^ (289) upon which relation the subsequent formulae are based. 76. Dotihlt Integration (ts Jiased upon Newton's Forrnida of Interpolation. — If we substitute, successively, n and n' for n in (2-13), and take the difference of the two results, we obtain CF{t-\-nio)dn = *(„'/)_*(„') (290) tiik; theory and pkactice of interpolation. 101 From the foni) of (2!)0) it follows that the expression for the indefinite integral is — CF{t-^nu>)d7i = *(«) or, by (238), fF(f + 7iw) (In = i^F„ (In = 'F„+ \ F„ + liJ'„ + yj;,' + 8. /;," + . . . (291) the constant of integration being contained in 'F„ , which depends upon the arbitrary quantity 'F^ . Multiplying this equation by dii, and integrating, we get C ('F{t + nu>)(hi- = C'F„dn-\-\i F„dn-\r pLt'„d>i.^yCr,;(hi-^oC.i;;'dn-\r . . . (292) Let us now consider a new series, namely — lip II w lip lip up the term "F^ being arbiti'ary, and the subsequent terms so determined that the qnantities 'F 'F 'F 'F ^0' ^1^ ■'^2) • • • • -f,-+I are the successive first differences of the proposed series. The man- ner of arranging the series "F, 'F, and F, together with the differences of F, is shown in the schedule below : T iiF 'F F{T) J' J" J'" Jlv "F^ 'K f "F, 'F, K ^\ t + O) "Fn K ■^0 " 'F., ^'1 <' /' + 2w "/'; - ^2 •^'2 ^r //;" J'v t + 3o) "F, F, ^2' z/r t + (/-2)a) ■ "^,-1 '^'',-1 F,-. ^'-2 ^j;:. /^;-3 '^;i4 t + (j-l)o) "F, ^,-1 ^,-2 '/', ^',-1 t + /u) lip lip ip F, 162 THE THEORY AND PRACTICE OF INTERPOLATION. Now, since tlie differences ^"^' may be regarded as a series of functions whose 1st, 2d, .... differences are //»'+", J'-'+-'> . . . . , it is clear tliat formula (291) may be applied successively to each of the integrals in the second member of (292). Accordingly, we have dn (In I3fj'„dn bC/i::' dn £ Cj'J dn dn = (293) Summing these expressions, we find, in accordance with (292), f CF{t + nu>) dn'' = "F„+ 'F„ + (1 + 2/3) F,^ + (/?+2y) //'„ + {p-" + y + 28) Jl' + (2^y + 8 + 2£) J',l ' + . (294) Upon substituting the numerical values of ;8, y, S, . . . . from (222), formula (291) becomes ^ ^F{t+n>^) div = "F^ + '/'„ + tV i^„ - ^iT ^;,' + -k ^:." - (294a) the coefficient of j'„ reducing to zero. We proceed to determine the expansion to Avhieh the coefficients of this formula belong. For brevity, let us write (291) in the form rp-' ('! + «'-) ''"•' = "^n + 'F^ + aF„ + hJ\ + cA'i + dJ':: + Now, from (228), we have Also, let us put w = X-- + .(,-• + ax" + h.r + ,'.,■■ + d.r^ + . . . . (295) (296) (297) THE THEORY AND PRACTICE OF INTERPOI.ATION. Ki:} in which the coefficients are taken as in (29.3). Whence, since the second member of (295) is tiie combined sum ol" tlie second members in (293), it is evident tliat (297) may be resolved, conversely, as follows : IV = a;-2 + I a;-' + fix" -\- yx + hx- ■\- . + i(a-'+ix''+/8«+y.«'+ . . . .) + y(x+ ia:-4- . + . which may be written w = a;-i (a;-i + k x" + fix + yx- + &x^+ . . . . ) + ix''(x-^+ix'' + I3x+ yx-+8x''i. . . . .) + fix (X-' +lx''+fix+ yx' + 8x'+ . . . . ) + yx' (X-' + ixO + fix+ yx- + 8.r^+ . . . . ) + 8.--^ (a;-i + i x" + fix + y.<--^ + Sx'+ . . . . ) + = (,*-> +ix°+ fir + y.'- + . . . . ){x-' + *.'•"+ fix + yx' + = (,;-i+^,,.o+/J,,. +y,r-^+8,,3+ . . . .y Therefore, by (296), we have = ^log(l + .r) _ r~- 4- 1 — 1 -I 1 r° 1 ;■- -U i r^ '-SI r* J- 19 v^ — ^ -f- u, -r 15 ■< 240 • r 5^40 .*- ir(T4 KO > ITOT^ •'' .»■• X' a;' .?• = '•^•-2 + -3-4+5 (298) Comparing (297) and (298), it follows that the coefficients of the former, and hence, also, those of (295), are the coefficients in the expansion of [log (1 -(-.*■) ]~^, as developed in (298). "Whence, in- troducing these valnes of a, b, c, d, . . . . in (295), Ave obtain ffF(t + no,)dn:'="F„+ 'F„+ ^^^ F,-^x^ j;'+ ^i, ^'-^ho '^u + T;iU ^l---- (299) as was found directly — in part — in (294^/). Let us now j^ut X(»,) = "F„ + IF,, + aF„ + hJ'„ + cJ',; + d.J',l' + eJ';; + . . . . — ■'^n-r ^ n'T 15 -'^ n -r '-'-' H — 54(1 ^n + Jiff ~' ,i — SC4S5 ^" r and (299) becomes C CF{t + n,^)dn- = A («) (.300) (301) l(Ji THE THEORY AND PRACTICE OF INTERPOLATION. Whence, if the intcg-ral be taken between the two fractional limits, n and n", we shall have f pV + »<")''«' = ^("") - >•(«•') ('"^02) And if we make the upper limit an integer, say n" = i, we have r C'f (f + ?! «,) dn' = X(!) -\ (w ') (303) The last formula involves the disadvantage of employing differ- ences z/,', Ji", Ji'", .... which are not given when the tabulation of F {T) ends with the quantity Fi- To remedy this defect, we i^ro- ceed as follows : Put V = A(t) = "Fi+ 'F. + nF, + h.l', + C.I'' + dJ'." + eJY + .... (304) and sul)stitute for "i^, , 'Fi, F^, z//, j/', .... the expressions 'IF, ="7^.+„ - 2'i^,.+i + F, 'F -- 'F — F F, = F- j.' = j;_j + ,/;i, + j'^ + z7;i, + . J'' = j;i, + 2j;:;3 + 3J!i, + . j;" = • J,::; + 3j;i, + . JY = J;i4 + . (305) AVlience the integral (303) may at once be expressed in terms of the available differences, //'{_, , /I'i^ ? ^'iU > substitution, let us put, as in (229), . . However, to avoid direct (.30G) and we shall have x-^ = ic- (1 — ?(.)- = U-- - 2 (/-I + ?(» .T-' = H-'(l-!0 -- «-i - »° x" = n" X = «(!-?<)-' = », + „■' + !!» + «^ + . . . . ,r2 ^ „2(l_„)-2 = ir+2u'' + Zu*+ . . . . a« = u^(l-ii)-^ = m»+-3j!^+ . . . . .r^ = u\l-v)-^ ^ (^^-1- . . . . (307) TIIK TIIKOIIY AND rUACTICE OF INTERPOLATION. Hyi) Again, from (297), we liave w = X-- + .?-' + a.,-" + bx + c.r- + J.r^ + e.i-*+ .... (308) Now, it is evidcMit tlial if the expressions (307) he substituted in the second member of (308), the algebraic process will he identi- cal in form with that of substituting the expressions (305) in (;}04). The w operation involves the quantities w ; x~% x~\ x", X, X-, ,/•", .... ; tr^ u~\ u", a, u"; u^, . . . . ; while the v operation involves, in exactly the same manner, the (juan- tities V • "F 'F F A' A" A'" • "F 'F,^ - '7'%, + aF, - h.J',_, + cj;i^ - dJlll, + eJ^ - . . . . Denoting this expression by ■n-(')j '^^^^^ restoring the numerical values of a, b, c, . . . . from (300), we have V = ■n-{i) = "F,^„ - 'F,^, + aF, - hJ\_, + cj;i„ - rfj;i'3 + eJ'I^ _ . . . . = "i^+, - 'i^+i +i,F,- ^_U ^'U - ^.U ^;-3 - ,T*Ho 41. - ... . (310) IGG TTIK TIIEOKY AND ntACTTrE OF INTERPOLATION. Whence, by (304) and (310), X (i) = V = IT (i) and tlu' fonmda (303) becomes, therefore, C ^'F(t + nm)dn'' = Trii) -\{n') ■ (311) In the fonmda just proved tlie quantity / denotes an intcg-er. Now, by the general method of interpolation employed in §70, it is easily shown tliat (311) is true for non-integral valnes of i. Thus, writing n" for /, this formula becomes r rF{t + ««,) (hi" = TT (n") - X (■»') (312) We now bring together equations (300), (310), (302), (312) and (289), in the order named ; observing that in the first two of these we may write "i^„_^, for "jP„-(-'i^„ and for "F,,.^^ — '-^«+i > respec- tively. Thus we obtain the following group : \(n) = "F„^,+ j\,F„-,uj': +^i„j;"- ,Hif.^r +ai?^: - . . . ^(n) = "F„^, + ^^ f„-^i^j:u-^u^::i,- ,m^ j^,-^iuj:_,- . . . C ("F{t + «'" F = | \F(T)dT-' = (0^ i F(t + iio,)d>r From this group are immediately derived all of the formulae given in the following section. 77. We have already remarked that in tlie jirocess of single in- tegration the value of the definite integral is wholly independent of the absolute value of 'F^, which may therefore be assigned arbitrarily. Similarly, in double integration, the quantity "F^ may be taken at pleasure, the integral being independent of its absolute value. Per contra, the douhlt integral will evidently vary with the value assigned to 'F^. Hence, unless 'F^ is fixed by some special considei-ation, the value of the double integral is indeterminate — a conclusion already derived from (287). (313) THE TIIKORY AND PRACTICE OF INTERPOLATION. IC) N^ow, as was ])iwiously remarked, the value of the tirst integral corresponding to the lower limit is usually known in practical appli- cations. We shall tlu'refore denote by 11^ the value of f F {T) )dn = «,('i'\-i7^o+^.y'„+yj;'+8j;"+£jr+ . . . .) or, upon restoring the numerical values of /3, y, S, . . . . from (222), and transposing, J^i — h n- -f'o + TJ-' o~ 5-1 -^0 + T5(r -^0 Tito ^^0 +c;o4B(J'^o .... K''^*) (0 which determines 'F^, and hence, also, the double integral Y, provided H^ is known. In practice the value of //(, is frequently zero. Using (314) in conjunction with the relations (313), we obtain the several groups of quadrature formulae given below : // -t^l — r T-'^o I T5 -^ — 54 -^0 T 755 -"o Till) ^0 T 1I04 sTJ ^0 .... (1) rr^x'T)rf2'- = u>^C j'F{t + uw)
- ('(>(<;+ ».a,)(7)r = or\c'F„,-"F„^^+^,(F,-F,:)-^i^(j['-j::)+,u(/ii"-^n - ^§fk (4^ -^!n + iritis (^r -^i) - • ■ • -I (317) -H'„ ^'1 = h t -'^0+ TI -^ 24 -^0+ -7'2<5 '^0 ro iJ ^0 + ff 4S0 ^0 ( )i^(2')(Zr- = coH /^ (/! + ««,) rf»r (318) = «>^K"^-."+l-"^..'+l)+ TV(i^,..-^',.) -^U(^n"-^'^')+ j4ttK"-^;.'') 168 THE THEORY AND PRACTICE OF INTERPOLATION. The foregoing formulae arc applicable when the upper limit falls near the heginnhxi of the tabular serit's. When the u|)per iinut.s falls at or neai- the (nd of the given series, the ibllowing formulae — like- wise derived from (313) — may be employed : w C("F"(T')dT- = o^-'CJ'F(t+nw),bi' \ (319) jjF{T)JT- = wj I /•'(/+)( a,) -f (F(t + »M>)d>r- \ (322) In applications of all the preceding formulae, the value of "1^\ (or of "Fg when employed) is wholly ai-biti-ary, and therefore may be assigned at pleasure in every case. Hut when (315), (316), (319) and (320) are applicable, it is frequently convenient to determine "-F, such that /(/,' I A' J_ 1 ,/" 1 /)'" _l. 22 1 //iv 19 //»4_ =0 The formulae in question then take the form as follows : THE THEORY AND PUACTIOE OF FNTEUI'OI.ATIOX. 1()9 jr -'I — ^ 1 2 ' T^ 1 i '^ 2 4 -^O T 7 50 -^0 TOO ^0 + (iiJ480 • '" • • ■ ■ II V — I A' X 1 /" - I 7'" J- '."-'1 /I" 1!) /'J- 1 1 •-' ^ (I T^ 54" ^0 -.Mu "o 'r (;n4gs -^o (S04S ■'-'u T • • • • /> /'r+.(u /• /■' /''(7')'/r- = or /'(/ + //<„)(/«■- If — ?4. 1 // 4_ 1 /' 1 /"X 'D /I'" ••! /'*■ X s«3 /' -' I ~ ^ T 7 ' OT 1 3" -^ 24 -^0 T 7 25 ^'u ~ 1 B (f -^0 + iS (i 4 S3 • ^u ~" • • • • -■ 1 T-.' ' 1^ 54 • 'o 24 ■ 'o + (SSjiSTJ ^0 — 604 5 "^0+ • • • • 0*f4-'itu /» /•« F{T)dT- = (0=1 j /''(<+««) c/«'^ = <"'("^,+i+A^-45^;.'+5iuX'-BsUo^;:+ais^^:;- • • • •) -• 1 — T ^ ■< oT^ 15^0 24 '-'" T^ 720 -'o 1 ilO -^0 T j;04«(T -^'a — .... (O '//F" I P X 1 //" 1 //'"X 2 '2 I //iv 19 ,-/v 1 -' 1 — 12^0 I 54(7 '^(i 24 (T "^0 Tiirf4sTJ^u ITOJB'^i)"'" .... I li^(T)f/r'- = i-3 C0450^ii— 1 Btr4S"'n-5 — • • • ) (.-^23) (324) (325) (326) The diffeixniees which appear in tlie Ibregoing fornmhie, together with the auxiliary functions 'F and "F, are to be taken according to the schedule on page IGl. The symbol i denotes a positive integer, while 'II designates a fractional or mixed number : so that all functions and differences whose subscripts involve n must be derived from their respective series by inter j)olation. Finally, the quantity //„ de- notes — as previously defined — the value of j F (T)dT when t is substituted for T : so that we have JL = fF^T)dl^l (327) ' Jt=i It may happen occasionally that the value of JT^ is luiknown, while the value of {F{T)dT corresponding to T = /-(-mw is known for a particular value of n. Denoting this quantity by H^, 170 THE THEORY AND PKACTICE OF INTKIM'OLATION. we may, by any one of the foregoing methods, compute the definite F{T)dT = integral //.. - IL and hence ihid (327o) H^ = 7/, - A" with which value we proceed a« before. Several examples will now be solved as an exercise to illustrate the formulae given above. Example I. — Let it be required to find Y = C i'cosTdT- ow the supposition that |'cosTfZJ'=2 when T = 0. AVe tabulate and difference the following values of i^(T) = cos T : T I'F 'F F(T)=cosT J' J" J'" Jiv O 10 20 30 40 50 60 70 80 90 0.00000 11.95916 24.90313 38.78679 53.53648 69.05221 85.21073 101.86925 118.86979 136.04398 11.95916 12.94397 13.88366 14.74969 15.51573 16.15852 16.65852 17.00054 17.17419 1.00000 0.98481 0.93969 0.80603 0.76604 0.64279 0.50000 0.34202 0.17365 0.00000 - 1519 4512 7366 9999 12325 14279 15798 16837 -17365 -2993 2854 2633 2326 1954 1519 1039 - 528 + 139 221 307 372 435 480 + 511 + 82 86 65 63 45 + 31 Accordingly, we have t = 0° = 10° = 18 7/„ i = 9 Proceeding by ('^10), the computation of 'F^ is as follows Ih^'- = + 11.45915.6 /; = +1.00000 + i ^; = + 0.50000.0 --^'o= - 1519 + ,V ^^'o = — 120.6 <= - 2993 - -h -k; = + 124.7 <'= + 139 + 7¥ff -^o" = + 3.7 J!,-= + 82 .3 ,i\V — 1.5 .-. 'F, = + 11.95916 THE THEOltY AND PRACTICE OP INTERPOLATION. 171 Tilt! column F is now completed by 8ucces^sive adtlitioiis ; Jicnce, also, the colunm "F, having first assumed "Fi = 0. AVheuce, by (319), the remainder of the computation is as follows : "h\^ = +136.04398 "t\ = o.(h;)OOo ("F,„-"F,) = +136.04398 F^ = 0.00000 ^0 = + 1.00000 + ^5 (-^9-^0) = - 0.08333.3 J'^' = - 528 4' = - 2993 - .-1^ (^i'-^;') = - 10.3 4'."= + 511 <' = + 139 - jItt (^;" + /^;')= - 2.7 jr= + .31 ^i^ = + 82 -iT§liTr(4^-4'') = + 0.2 log^ = 2.1334129 ^ = +135.96052 log ui' = 8.4837548 log Y = 0.6171677 .-. Y = 4.141595 To verify this result, we have = C fees Ti .' Jo COS TdT = sin T+ C IT-' = -con I'+C'T = l + iC\ where C is the constant of tlie first integration. To determine (J, the first of these relations grives If. = sin T+ C = C whence and therefore C = 2 r = 1 + ,r = 4.141593 Example II. — Compute the value of Y =J£t^^^IT^ which corresponds to 11^ = 0. Here we tabulate and difference F(T) T~^ as below T "F IF F{T) = T-"' J' J" J'" 2.0 2.1 2.2 2.3 2.4 2.5 -0.02082 + 0.10210 0.45178 1.00807 1.75340 + 2.67234 + 0.12292 0.34968 0.55629 0.74533 + 0.91894 0.25000 0.22676 0.20661 0.18904 0.17361 0.16000 -2324 2015 1757 1543 -1361 + 309 258 214 + 182 -51 44 -32 172 THE THEORY AND PRACTICE OF INTERPOLATION. AVc have, tluMefore, t = 2.0 U.l IL = whence, proceeding by (o26), the comi)utation of i^, and "F^ is as follows : + I i?; = +0.12500 + ^V J\ = - 19.3.7 - ,'., /; = -0.02083.3 -/.j;,' = - 12.9 +.U< = + 1-3 0.2 .-. 'F^ = +0.12292 .-. "F^ = —0.02082 From the completed table we now iind n = (2.468 -2.0) H- 0.1 = 4.68 = o - 0.32 F„ = +0.16418 //" = + 191 ^::-.= 36 "/;+! = +2.36025.6 + J. F„ = + 1368.2 — 5^5 ^»-2 = — 0.8 -^U^'Z, = + 0.1 2 = +2.37393 .-. Y = +0.0237393 This result is easily verified, foi- we have Cl'-'dT = - ,+ C Y = - losr,. r+ CT = - los, 1.2.34 + 0.468 C also = //„ = -j,+ C c = ^ = -i+c Hence Y = -log,1.234 + 0.234 = -0.2102609+0.234 = +0.0237391 with which the above result substantially ayrees. Example III. — From the table of the preceding example, find the value of Y = C Cr-'dT'' THE TIIKOKY AND TltACTICE OF INTERPOLATION. 173 Here we employ formnlu (324), in which we take 2.15 - 2.0 0.1 We therefore obtain («+l= 2+i) F„ = +0.21633 ./j;; = + 235 .;;;' = - 38 = 1.50 = 1 + "/^„^j = +0.24992.0 + ^ij F, = + 1802.8 _4„j;' = - 1.0 + ^u^:' = - 0-2 V = +0.26794 .-. Y = +0.0026794 The true mathematical value of Y is — Y = 0.075 - log, 1.075 = +0.0026793 78. Double Integration as Based upon Stirling's and Bessel's Formulae of Interpolation. — Let the schedule of functions (including 'F and "F) and differences to be used in the subsequent formulae of quadrature be as follows : T I'F 'F F(T) J' J" J'" t -2^ P-2 J' , ^-'2 -^"3 t — u> "F , F , ■^"1 + c)J'':;+. . . . Whence, restoring the values of a, h, c, . . . . from (329), and reducing, we obtain JJ^(i + «a.)fZ«^ = "i^„+ .'.i?',-,'^ //;.'+ ^^\V^ Jj.'- . . . . (332) If, as in (327), we denote by //„ the value of fF(T) (IT which obtains for T = t, then, by (328), we have //„ '^ = \' CF(T}dT = J CF{t + nM,)dn and hence, by (272), 0.0 (0) (333) THE TIIKOIIY AND ritAOTICE OF INTERPOLATION. 175 Upon substituting- i = in the first of equations (269), we get i'F,) = i{'F_,+ 'F,) = 'F,-iF^ whicli, together with (333), gives 'F, = — ° + i K + iV (-J'o) - tV-o (4',") + r^Uh i^l) - . . . . (334) where the differences enclosed within parentheses are means of the corresponding- tabular quantities, as defined by (269). By employing simultaneously the relations (332) and (334), and assigning various limits to the integral, we obtain the following group of formulae : 'F, = —'+i F^+ tV (J'o) - 7'5'u (4;") + a^ h i^l) - . . . . \ (O I C ("F{T)dT- = 0,-C CF{f + no,) dn- } ^^^^^ = a>^K"^-"^)+i\(^-^)-^in(4'-^:')+^AViy(4"-^ir)- . . -I / 'F — X 4 A' J- 1 ( 1' \ — 11 {.i"'\±. 1 fn { /P\ — \ CCF{T)dT- = ,j'ffi\l + Hu.)dn' ) (^^^) = co'i("i^,-"i^o)4-iV(^'^..-^';)--io(4';-4;')+irnV^5(4:-40- • • -i / 'F, = ^+iF„+ rV (^'0) - tVo (^o") + ^h'ih (^0) - . . . . \ <" ' f r rF("r) rf r^ = 1("^,-"^'„)+TV(^<-^,.)-.iTT(4'-4'.')+5^VHw(4'-^L')- • • -I / 'F^ = ^ + i^';+TV(-''o)-TViT(4") + ai^n(4o- • • • • \ f r^( T) ,Z r= = .0= f CF(t + nu) d7i^ I ^^^^^ = a,^K"^."-"^,')+i'5(^;"-^.)-5i5(^;.''-^:o+ir^Vo(-^i^'-4;)-. . i / In the jjreceding group the value of '' F^ is wholly arbitrary. We may, however, determine the quantity "F^ such that the sum of the terms in (335) and (336) having the subscript zero will vanish : these formulae may therefore be written — 176 THE THEORY AND PRACTICE OF INTERPOLATION. - ' ta F{T)dT- = «,-j I F(i; + wco)(/w--^ (U "P — _ 1 P J- 1 7" _ ■■' I y/'* -I- (339) (340) Let us now denote the second member of (332) by y {11) ; that is, lot us put 7(m) = "^,.+ 1^5^, -sinX+rrriVss'^;:- ■ • ■ • (341) Making- n:=l-\-l, this becomes yii+i) = "F,+i + i\ ^+s - 5^ ■^;;j + TTiJVss ^^i - . . . . (342) It will be observed from the foregoing schedule that "i'Vi? ^'+*' j",.,.^, .... are not explicitly given, but must be derived from their respective series by interpolation to haJres. For tliis purpose, let us put, in analogy with (269), (i^,+5) = i{F^+F.+.) (4-;*) = i(^,''+4';o (343) then, after the manner of (270), we shall have "i^,.+, = {"F.^0- H-p.+i) + tIsC^'Is) - TT/^^i (4;«) + ^+» = (^+j - h (•^;;s) + tI^ (-^i+j) - (j;+j) - i (^;;o + 4;* (344) Upon sul)stituting these expressions in the second member of (.342), and reducing, we find y (i + i) = i"F,^0 - ^\ (F,+i) + tHit (^Hi) - T/AVtf (4+j) + (345) THE THEORY AND PRACTICE OF INTERPOLATrON. 177 Again, by means of (.332) and (341), we derive C CF(t + nw)(/n- = y(n") ~y(7i') (346) Finally, denoting by //_+ the value of fF(T)dT when T=t—},cj, we shall have, by (328^0' jr_ ) (In „,(IP X I ,7' '7 /I'" J. .■) G 7 //v \ which gives 'P — < 1/1' J_ 17 A'"— ■■"5 7 y^V 4_ (347) By assigning various values to the limits n and n' in (346), and employing either (341) or (345) as required in each particular case ; and finally, by using either (334) or (347) to determine the series 'F, according as the assigned lower limit is not or is equal to we derive the group of formulae given below : 1 2 > II V 1 jf s. 1 /I" 3 1 A" ±. /» /•H-(*+i)(u /• /•i+J j JF(T}dT'' = «.M ji^(;! + woj)(ZM^ '^! = ^"+i^„+TV(^'o)-T'5V(X')+aik(4)-- • • • "F^ = ««y convenient value ; arbitrarily assigned. ( ji^(r)(Zr2 = wM ( /''(!;+ WO)) <^ra^ = -^K"^+5)-5'4(^+j + tU« (-/:;«) -T^'5^(^i;o+ • • • II V 1 P 4. I //" 3 1 //iv . ITf — zi I //' -4- 17 A'" 3 (i 7 //'J. rr^x"r)(/T- = (!!+wco)r;«-^ . — ,.?i"F X- 1 T*" 1 //" X 3 1 //iv \ — <" (, -' ! + IJ -'^i 540 ^i + STJ4STr ^i — ■ • • • j (348) (349) (350) 178 THE TllEOKY AND rKACTICE OP INTEEPOLATION. H IW -i 1 ,/' I 17 /*'" 3(17 /f _1_ w (351) 'F_ H -i (D 54 '^ -1 + 57ffir ^— J ■■! (; 7 J — TJtTTCSS ^1>+ "ifl = any convenient value; arhitrarily assirjtied. The last formula may also bo written in the following form (352) /f_. 3 G 7 A'' X- F{T)dT- = u)M j F(C + »; = +0.029974 Upon completing the table as shown above, and continuing the computation by (353), we obtain (i = 4) ("iJ^j) = +0.240524.0 (F,,) = -0.002332 -j',(^40 = + 97.2 (j;;) = - 106 +^i|^(z/:0 = - 0-9 .-. Y = +0.240620 We easily verify this result analytically as follows : -M cos TclT M J sin= T sin 2' = .1/iog, tan i r + cr + C // sin^ T sin 2' ■ 3/cosraT- ""su?r = log,„taiii2'+ CT+ C : Y logi„tani T + CT r=BO«=T^'gT But .-. C = .-. F= logi„tanr^j -logiotanT— j THE TIIEOKY AND PKACTICK OF INTERPOLATION. 183 Now we find log tan 25° = 9.068072.5 - 10 los tan 15° = 9.428052.5 - 10 .-. Y = 0.240620 which agrees exactly with the former result. Example IV. — From the table and data of Examjile III, compute the integral _ /Vj/cosra: // sm'T Here we employ (351), taking i =i 32° as before ; we then have for the value of n at the uppei- limit, n = (45° -32°)-:- 4° == 3.25 = 3 + 0.25 We therefore obtain "F^ = +0.189420.3 7*; = -0.002993 +,V-f» = - 249.4 A':^ = - 163 -gj-ff^'.' = + 0-7 .-. r = +0.189172 Verifying this result as in the last example, we find Y = Iogi„tan22°30'-logiotanl5° = +0.189172 Example V. — As a final exercise, combining both single and double integration, and illustrating, moreover, the use of formula (339) when several values are assigned in succession to the integer ?', we shall conclude these examples with a complete and detailed solution of the following problem : A particle P of unit mass is impelled along a straight line AB by a varying force whose expression is 20000 T ~^ ; where T is the time in seconds after a definite epoch, and the implied unit of length is one foot. It is required to find by quadratures the velocity, v^ and the distance, AP = a?, for the times T = 102, 104, 106, 108 and 110 seconds, respectively; assuming that v^ = O.G feet per second and x^ =: 8 feet when T^ = 100 seconds. 184 THE THEORY AND PRACTICE OE INTERPOLATION. Since the mass of P is unity, we have, simply, d-x _ 20000 whence by a single integration fZ.-r /20000(Zr + "o («) and by double integration rfiomodT" w We shall iii'st compute the required values of x as given by equa- tion (^), effecting the double integration by means of (339). The details of the computation are shown in the following table : Table (A). r 96 98 100 102 104 106 108 110 112 114 40000 T-" 0.04521 .04250 .04000 .03769 .03556 .03358 .03175 .03005 .02847 0.02700 -271 250 231 213 198 183 170 158 -147 + 21 19 IS 15 15 13 12 + 11 + 0.53730 .57980 .61980 .65749 .69305 .72663 .75838 .78843 + 0.81690 ''F + ix,= a + 3.99667 4.61647 5.27396 5.96701 6.69364 + 7.46202 + I'a F= h + 0.00333 314 296 280 265 + 0.00250 x = a-\-h 4.00000 4.61961 5.27692 5.96981 6.69629 7.45452 8.0000 9.2392 10.5538 11.9396 13.3926 14.9090 Since we shall afterwards use this same table in finding v by single integration, it is here convenient to tabulate &> times the given function : thus avoiding the final multiplication by w in computing v, and reducing the corresponding factor in the case of x from w^ to w. Accordingly, we tabulate under F {T) the function F(T) E 200000)2'-^ = 40000 r-= Assume t = 100, aud proceed by (339). To determine 'i^^, it must be observed that since F{T), j', j", .... already contain H the factor w, it is here necessary to multiply the independent term — ^ THE THEORY AND PRACTICE OF USTTERPOLATION. 185 TT by the same factor: so that, writinj? i\{= IT^ for " in the first equation of (339), and omitting insensible terms, we have % = V, + kF,^^\{J\) (y) Hence, substituting r„ = 0.(), F^ = 0.04000, (j'„) = .] ( r ,-|- j;) = —0.00240, we find 'F), = +0.G1980, and thus complete the series 'F as given above. The second equation of (339) gives simply, "i^„ =^ — J.j i^„, the term in ./' being insensible. But since, by equation (/3), we should afterwards have to add the constant ,r„ to each computed value of the double integral taken from T^ to T, it is expedient to tabulate in place of "F^^ the quantity "F.-\-- = "F^+h-'-o = --hF,+ ^.^ = 4.0 -0.00333 = +3.99G67 and thence complete the series as given under "F -\- \ x^ rE (t. The reason for this procedure is easily made aj^parent : for the final equa- tion of (339) gives (since w^ must now be replaced by w) // 20000(^T2 3 = „,("i^,+ -,Vi^,) and substituting this expression in equation (yS), we obtain y. X = 0, ("F, + -r'i F,) + .r„ = „, ("F, + -° + -rV F,) (8) O) Therefore, upon forming the column -j-iV-^^= ^j ^^ given above, we have from (S) ix = "F, + i.r„+^y^F, = a + h whence the required values of x are derived and tabulated in the final column of Table (A). 186 THE THEORY AND PRACTICE OF INTERPOLATION. For the computation of tlie velocity v we employ formula (282), the first equation of which gives or, by adding i^o to both members. But we shall avoid subsequent additions of the constant V(„ required by equation (a), if we increase this value of 'F'i by the term v^ =■ 0-^ j that is, if we take which is the same as the expression (y), iised for determining the series 'F in Table (A). The latter series is therefore to be employed in finding v, the computation of which is as follows ; Table (B). T CF) (J') -A.(-J') i'={'F)-^^(J<) 96 + 0.51470 + 24 + 0.51494 98 .55855 -260 22 .55877 100 .59980 240 20 .00000 102 .63805 222 18 .03883 104 .67527 205 17 .07544 100 .70984 190 10 .71000 108 .74251 170 15 .74200 110 .77341 104 14 .77355 112 .80207 -152 13 .80280 114 + 0.83040 • ■ + 12 + 0.83052 Recalling the fact that functions and difterences in parentheses are means taken according to (269), the method of forming the second, third and foui-th columns of this table from the quantities of Table (A) is obvious. Now, since the fiictor w has been previously introduced, the second equation of (282) gives V = ('7^,)--,V(^'.) from which expression the required values of v are computed and tabu- lated in the final column of Table (B). THE THEORY AND PKACTICE OE INTERPOLATION. 187 This completes tlie solution of the problem. An interesting check is derived, however, by observing- that equation (a) gives dT+,- W whence x may be obtained from the series v by single integration. For this purpose we make f{T) = oiV ^=. 2v, and thus form the table below : Tai!le (C). T f(T) = 2v 8' 8" '/+■<■» (:f)+Xo=c (8') -TV(8')='i x = c + d 96 98 100 102 104 106 108 110 112 114 1.0299 1.1175 1.2000 1.2777 1.3509 1.4200 1.4853 1.5471 1.6056 1.6610 + 876 825 111 732 691 (;53 618 585 + 554 -51 48 45 41 38 35 33 -31 + 7.4067 8.6067 9.8844 11.2353 12.6553 14.1406 + 15.6877 8.0067 9.2455 10.5598 11.9453 13.3979 14.9141 + 801 754 711 672 636 + 602 -67 63 59 66 53 -50 8.0000 9.2392 10.5539 11.9397 13.3926 14.9091 Here again we take t =^ 100, and employ (282), which gives '/_, = -i/o+iVCS'o) = -0.6000 + 0.0067 = -0.5933 Increasing this value by Xo = 8.0, to pi-ovide for the constant .r,, in equation (e), we get -|-7.4067, which number is written under [/'-}-. ''o, on the line t — i w. Completing this column by successive additions of the functions /, we next form the series of mean values tabulated under ('/')-(- cCq = c. The columns (8') and — tV (^') =^ ^^''^ then computed, and finally the column .c = c -(- d. These values of x agree substantially with those given in Table (A). From the given analytical expression for the force, together with the initial conditions of the problem, we easily find V = 1.6-lOOOOr-- , .r = 1.6r+ 10000 r-i - 252 whence, making T = 110, we obtain V = 0.77355 and x = 14.9091 which further verify the results derived by quadratures. 188 THE THEORY AND TRACTICE OF INTERPOLATION. 79. It is Avorth while to inquire what change takes jilace in the vahie of the double integral ■^o' = 0/^^) dT' when, in a particular problem, the quantity II is changed from an assigned value W to a new value II". This is easily answered. For, if we change //' to //", the value of the first integral — corresponding to any particular value of T — is thereby increased by the quantity //" — //'; or, what amounts to the same thing, the constant of the first integration, M in (286a), is thus increased by //" — //'. There- fore, by (287), it is evident that 1" is increased by the quantity {ir—H) (T"—T'). THE THEORY AND PRACTICE OF INTERPOLATION. 189 EXAMPLES. 1. Given the semi-major axis of an ellipse, a = 1, and the semi-minor axis, b = 0.8, to find the length of the elliptic quadrant. A7IS. l.J:1808. [Note : — Take tlie eccentric angle E as independent variable, and hence find TT s = I Vl - e^ cos^^ dE where n is the eccentricity, and s the required length.] 2. Given the equation of a cardioid, ?- = 1 -|- cos d : to find, by mechanical quadrature, the length of that part of the curve comprised between the initial line and a line through the pole at right-angles to the initial line. Ans. 2.82843. 3. The equation of a curve being y z= x^ V2 — sin x , find the area included between the curve, the axis of x, and the two oi-dinates, a? = f and .?;=fTr. J.H.y. 0.180518. 4. Compute the value of IT -//;. Vl - 0.82 sin^r assuming that the first integral vanishes at the loAver limit. Ans. 0.139727. 5. Given a curve in a vertical plane whose points satisfy the relation dhj _ 4x^-3 DR. GEORbh I-. ivicLWLN 190 THE TllEOKY AND PRACTICE OF INTERPOLATION. — the axis of y being vertical. Find the difference of level between two points whose abscissae arc 1.000 and 1.473, respectively ; assum- ing the direction of the curve to be horizontal at the first point. Am. 0.044228. 6. By what amount would the preceding result be changed by supposing the tangent to the curve at the first point to be inclined 45° to the horizontal ? [Note : — This question should be answered mentally.] CHAPTER V. MISCELLANEOUS PROBLEMS AND APPLICATIONS. 80. The present short chapter will be devoted to the solution of a number of problems and examples involving certain principles and precepts hitherto established. 81. Problem I.— To _/zWZ ,S' = 1''+ 2''-)-3''+ . . . . + »■", where Tc and r are integers. The method of solution is best illustrated by assigning a particu- lar value to k. Thus, let it be required to find S = V-\-2' + o'+ . . . . + r* We tabulate below and difference the values of T* which corre- spond to T ^ 1, 2, 3, 4, 5 and 6. Thus we find : T 'F F(T) E r< J' J" J"' Jiv Jv 'K 1 'F, 1 15 o >F„ IG 65 50 60 3 81 175 110 84 24 4 256 369 194 108 24 5 625 671 302 6 1296 r-1 'Fr.y (,-1)^ r IF, ,.4 Now, by Theorem V, the 4th diflTerences of F{T) are constant, and hence the 5th and higher difierences all vanish. Whence, if we 192 THE THEORY AND PRACTICE OF INTERPOLATION. considei" the auxiliary series 'F — defined as in Chapter IV — we shall have, by the fundamental formula (73), ^F, = '/;+,.+ !:(^(i5)+'±^)i!::z2)^^^^ + ,.(,_l)(,._2)(,-3) (60) + r{r-l). (r-4) 11 (24) = '^0 + 3(5 ('• + 1)(2/- + 1) (3.^+3.-1) Therefore, by Theorem I, we have r 30 S = 'F^- 'F„ = — (r + l)(2r+l)(3?-^ + 3r-l) (355) which is the required expression for the sum of the fourth powers of the first r integers. 82. Problem II. — Given a series of functions, F_3, F_o, F_i, F^, F^, F2, . . . . , and an assigned intermediate value, F„ : To find the corresponding interval n. First Solution : The simjjlest method is to determine by inspec- tion an approximate value of n, and then find by direct interpolation the values of the function corresponding to three or four closely equi- distant values of n that shall embrace the required interval. The latter is then readily found by a simple interpolation. Example. — From the following ephemeris find the time when the logarithm of 3Iercurifs distance from the Earth = 9.79G8280 : that is, given F,, = 9.7968280, to find n. The tabular quantities are here given for eveiy second Greenwich mean noon. Date 1898 Log. Dist. of 5 from ® J' J" J"' Jlv Jv May 8 10 12 14 16 18 20 9.7500700 9.7652375 9.7768883 9.7905482 9.8057806 9.8221946 9.8394585 + 91009 110508 130599 152:524 104140 + 172639 + 24839 20091 15725 11816 + 8499 -4748 43(;6 3909 -3317 + 382 457 + 592 + 75 + 135 We observe that the given logarithm falls somewhere between the tabular values for May 14 and 16, and soon find that the interval THE THEORY AND PRACTICE OF INTERPOLATION. 193 (from the former date) is somewhat greater than 0.4. Hence we take F^ = 9.7905-182, and interpolate — hy Bessel's Formnla — tlie functions corresponding to n = 0.38, 0.41, and 0.44. Thus, computing and dif- ferencing these vahies, we find n F,. J' J" 0.38 0.41 0.44 9.7961736 9.7966267 9.7970810 + 4531 + 4543 + 12 Whence, if we denote by u,' the interval at which the required function lies beyond the middle function in this new series, we shall have, by neglecting the small second difference, n' = 2013-^4543 = 0.44, nearly. But if great accuracy is required, we may easily take account of the second difference by the method of the corrected first difference (§44). Thus, in the last table, we find that the corrected first difference which corresponds to n = 0.44 is 4540 ; hence we have »' = 2013 -f 4540 = 0.4434 .-. n = 0.41 + 0.4434 X 0.03 = 0.423302 The required time is, therefore, T = May 14" + 0.423302 X 48" = May 14'> 20" 19'" 6«.6 83. Second Solution of Problem II. — Given F^, to find the value of n. Let ni denote an approximate value of n, true to the nearest tenth of a unit, and put n = m + z (356) Then Ave have F, = i-'^^, = F[f+{m+z)ui] = F[_(t + m,^) + z,^-] = F{t-]rm^) + zu>F' {t+m)-[- .... Since we have supposed z not to exceed 0.05, it is permissible to neglect z^, z'^, . . . . in the last expression, which becomes, there- fore, F,^ = 7^,„+.ta.i':,+ i.~Vi?';; (357) 19-1: THE THEORY AND PRACTICE OF INTERPOLATION. To find z from this equation, we first neglect the small term in z^, and thus obtain an aj)proximate value which we shall call .*'. In this manner we find -f „ - ^» X = " ,„ " ' (358) This approximate value of z will now suffice for substitution in the last term of (357). Accordingly, we obtain -("^^^ ■ (359) (360) s : -X whence, piitti ng y = \ we have z — x — y and equation (35G) becomes n = VI + X — y (301) Finally, to express F,„, uFm, and at^J^m iii terms of the differences of the given series F, it will be expedient to emjiloy Stirling's Formula of interpolation, together Avith the expressions for F^ and F^l as developed in §61. The above solution may then be expressed as follows : Determine m = an approximate value of n, true to tlie nearest tenth of a unit. Thence find F„== F^ + ma + B\ + Co + DJ^ + . . . . Z>i = u>Fl, = a+mh^+ C'r+ />'J„= - 2.6 Z)'t/„ = - 10.4 A'e = + lA. E'e ^ + 1.5 .-. F^ = 9.7964755 .-. D^ = +151101 .-. IK = +14070 i?;. = 9.7968280 .-. F,-fI, = + 3525 Whence K = A+-A = +14070 -+151101 = +0.0931 X = (i.;_i';,)-i- z>, = +3525 -H 151101 = +0.023329 and we finally obtain m = 0.400000 X = +0.023329 (Table VII) - y = - 26 .-. n = 0.423303 which agrees within one unit with the foi'mer result. 84. Problem III. — To solve any numerical equation ivJiatever involving hut one unhnown quantitij. The given equation, whether simple or complex, algebraic or trans- cendental, may be written in the form F{T) = The problem therefore reduces to the question of finding n when F,^ is known and equal to zero — which is the same as Problem II. 196 THE THEORY AND PRACTICE OF INTERPOLATION. Example. — Solve the transcendental equation T - 20° sin T = 45° Avhere T is expressed in degrees of arc. This equation may be written F(T) = T - 20° sin r - 45° = which by trial we find to be satisfied by a value of T not far from 63° ; hence we tabulate F (T) for T = 62°, 63°, and 64°, as follows : T F{T) J' J" O 62 63 64 -o!g590 + 0.1799 + 1.0241 + 8389 + 8442 + 53 Here we have given F„ ^= 0, to find n. Whence, employing the corrected Ji7-st difference (§45), we find 1799 r = 63° - —^ X 1° = 62°.7861 8410 , 8.5. Problem IV. — Gtiven a series of numerical functions em- bracing a maximum or minimum value : To find the value of the argument ivhich correspotids to the maximum or minimum function. Find by inspection the tabular function which falls nearest the required maximiuu or minimum value. Call this tabular function F^. Then, from the schedule T F(T) J' J" J'" Jiv t — t t + O) F, a' b' d' we have, by the first of equations (182), = F'{t + n,^) {a-l c +. . . .)+ «(i„_,i., d„ +. . . .) + iM=(c- ....)+ i ""K- ••••) + •■■• F'{T) = F'{t + nu>) 1 THE THEORY AKD PRACTICE OF INTERPOLATION. 197 Therefore, since the condition of niaximnni or minimum requires that F\T) =0, we have, by neglecting 5th differences, („-i c) + (/.„_ j>, d^)n + ^en^ + ,1 tiy = (363) whicli detei-mines the value of n, and hence, also, the value of T, at the point of maximum or minimum of F(T). This equation may be readily solved by successive approximations, by first neglecting the terms containing n^ and n', and afterwards substituting therein the approximate value of n thus found, and so on ; or, we may consider the solution of (363) from the standpoint of Problem III, — which may be regarded as the more direct of the two methods. Example. — The following ephemeris gives the log radius vector of 3I((rs with respect to the Sun (log r) . Find the time of iDerihelion passage of the planet. Date 1898 Log »• J' J" J'" Jiv April 6 14 22 30 May 8 16 24 0.1416628 0.1409303 0.1404822 0.1403232 0.1404553 0.1408772 0.1415840 -7325 4481 -1590 + 1321 4219 + 7068 + 2844 2891 2911 2898 + 2849 + 47 + 20 -13 -49 -27 33 -36 Here we are required to find the instant when log r is a mini- mum. Since it is evident that this condition occurs only a few hours from April 30, we take F„ = 0.1403232. Whence, from the above table, we find a = - 134.5 a — (\ c = - 135 ho = +2911 *o-i'A = +2914 (■ = + 3.5 ^« = + 2 do = - 33 if'o = - 6 and therefore, by (363), -135 + 291 4?i + 2)1^ - 6«« = or 291471 = 135 - 2n- + 6n' 198 TILE TIIEOKY AND PliACTICE OF INTERPOLATION. Neglecting the last two terms of this equation, we have, for an approximate value of n, n = 135 -^ 2914 = 0.040, nearly ; and since for this value of n the small terms sensibly vanish, we obtain as our final value 71 = 135 -^ 2914 = 0.04633 The date of perihelion passag'e is, therefore, T = April 30'! + 0.04633 X S X 24" = April 30" 8".895 86. PPtOHLEM V. — Given a series of numerical values {F_s, F_2, F'_i, Fq, Fi, F2, . . . .) of any function F(T) which is analytically unknown: To find an apjn'oximate algebraic exj^ression for F (T) in terms of the variable argument. Let us put T = T -t (364) and Tayloe's Theorem gives F{T) = F(t + T) = F{t) + rF' {t)+~ F" (t) + '^ F'" (t) + . . . . (365) Upon substituting in (365) the expressions for F'{t), F"{f), F"'(t), . . . . , as given by (I'J^'j), we obtain F(T) = F(f) + 1 (a - i c + jV « ) ^ + -V (^0- .'. ''0 + ■■■•) ^' 2^ + r^(«-i« + - • ■ ■)r»+_^(f7o-. • ■ .)r*+-j^(e-. . . .)r^+. . . . (366) which expresses F (T) as a rational integral function of r, with known numerical coefficients ; t being the value of the variable argument counted from the fixed epoch t, as defined ])y (364). Example. — From Newcomb's Astronomical Constants we take the following table of the mean obliquity of the ecliptic (e) for evei-y fifth century : THE THEORY AJJD PRACTICE OF INTERPOLATION. 199 Year Obliquity J' J" J'" 500 1000 1500 2000 2500 1 u 23 41 43.78 37 57.97 34 8.07 30 15.43 26 21.41 23 22 27.37 ' II -3 45.81 3 49.90 3 52.64 3 54.02 -3 54.04 II -4.09 2.74 1.38 -0.02 + 1.35 1.36 + 1.36 Let it be required to express e in terms of t, the latter being counted from the year 1000 in terms of a centuiy as the unit. Since we adopt one century as the unit of time, it is necessary to express w in the same unit ; therefore we have oj = 5 t = lOOOy F(t) = 23° 34' 8".07 a = -3' 51".27 = -231".27 a-lc = -231 ".496 Whence, by (366), we obtain \ = -2".74 c = +1".355 ■)^li = 50 a,'!! = 750 Coefficient of t = -231.496 +- 5 = -46.299 " « t' = - 2.74 -i- 50 = - 0.0548 " " T« = + 1.355 +- 750 = + 0.00181 Accordingly, the required expression for the obliquity is — £ = 23° 34' S".07 -46".299r- 0".0548T= + 0".00181r^ Verification : Putting r =^ 10 in this formula, we should get the obliquity for 2000. Xoav we find (Per 2000) £ = 23° 34' 8".07 - 462".99 - 5".48 + 1".81 = 23° 26' 21".41 which agrees exactly with the tabular ^'alue above. It will be observed that the solution given by (.366) restricts the epoch, or origin from which t is counted, to some tabular value of the argument, as t. Should the assigned epoch be some intermediate value of T, say T, , it Avill only be necessary to write and we have T, = T. F{T) = F{T, + r,) = F{T,)+r,F<{T,) + ^^F"{2\) + DR. GEO; ;cEWEN 200 THE THEOKT AND PKACTICE OF INTERPOLATION. Therefore, if we put T^ = t + mtsi ~\ we shall have j,^^^ _ f,„ + T,F:„Jr 4 F,;, +^ f:: + .... ( ^'^'"^"^ whei-c Ti (= T — T,) is the value of the variable argument counted from the assigned epoch T^ . Accordingly, if we compute l)y the usual methods the values of F,n, jF"^, F'„^, F',',^, . . . . , and sul)sti- tute these in (36G('<), Ave shall obtain the expression required. As an example, let us express the obliquity (e) as a function of the time (r,) counted from the epoch 1600.0 in terms of a century as the unit. Reverting to the above table, we take t = ISOQy T^ = 16005- //, = 0.20 Whence we find F„ = 23° 29' 28".69 F',,, = _46".761 F;1 = _0".0443 F'^' = +0".01088 Substituting these values in the formula (3G6a), we obtain the required expression, namely, £ = 23° 29' 28".69 -46".7G1t, -0".0222Tr +0".00181 tJ 87. CtEOMetiucal Problem. — A circular well four feet in diameter is centrally intersected by a hoi-izontal cylindrical shaft whose diameter is one foot. Find the volume of the portion of the shaft within the well. Solution : Consitler a vertical section or lamina of the shaft parallel to its axis, at a horizontal distance x from the latter, and having the differential thickness dx. Then, if we denote the radii of well and shaft by li and r, respectively, we shall have for the length of this rectangular section and for its breadth, or height, h = 2V?-^— ai« THE THKORY AND PRACTICE OF INTERPOLATION. Therefore, the volume of the differential section is — dV = Ihdx = ^-J{B^-x^{r^-x^)dx V = ^£'J{B-~x^)(r''-x^dx 201 whence Upon siib.stitntin<;- tlie g-iven values of R and /■ in this formula, it becomes r = 8j„V(4-.r-^)a-x-^)rf.r . This expression belongs to the class of functions known as ellip- tic integrals, and therefore cannot be integrated directly. Accordingly, we 2)roceed to evaluate 1' by mechanical quadrature. For this purpose it will be convenient to put whence X = \ sin dx = if cos 6d6 and the preceding expression for V becomes ■'Ode (367) We now tabulate F (6) = cocos'^e >^16—sur0 (where w = 10° 77 -i- 18) as follows : 6 'F F{e) zJ' J" A"t Jiv O - 15 - 5 + 5 15 25 36 45 55 65 75 85 95 + 105 0.0000 0.6927 ].3427 1.9129 2.3765 2.7201 2.9449 3.0063 3.1117 3.1168 0.6500 0.6927 0.6927 0.6500 0.5702 0.4636 0.3436 0.2248 0.1214 0.0454 0.0051 0.0051 0.0454 + 427 - 427 798 1066 1200 1188 1034 760 - 403 + 403 -371 427 427 371 268 -134 + 12 154 274 357 403 403 + 357 - 56 + 56 103 134 146 142 120 83 + 46 - 46 + 56 56 47 31 + 12 - 4 22 37 37 46 -46 202 THE TIIEOKY AND PKACTICE OF INTERPOLATION. Accordingly, we take < = 5° t = 8 t + iw = 85° and proceed by formula (259) : thus, observing that Ji, , J'_[i, . ... . and J'._^.^, z/;|'j , .... are aU zero, and remenibcring that the factor w has already been introduced, we find 'F_, = and V = '7''.+,j = 3.11G8 cubic feet 88. Various other problems and applications of a similai- nature might be added ; indeed. Astronomy itself presents a large variety of such. But the leading principles of our subject have ah'eady been developed, explained, and exemplified. We therefore feel confident in leaving the student who has thoroughly mastered these principles, believing him fully capable of solving any further questions or prob- lems that may arise in his practice. THE THEORY AND PRACTICE OF nSTTERPOLATION. 203 EXAMPLES. 1. Derive the exjjression for the sum of the cubes of the iii-st r integers. Ans. \r\r-\-l)-. 2. Find from the following- ephemeris the instant when Autumn commences ; that is, the instant when the Sun's right-ascension (a) equals twelve hours. Date 1898 Sun's R.A. a Date 1898 Sun's R.A. a Sept. 13 16 19 22 h in 8 11 25 47.56 11 36 33.99 11 47 20.29 11 58 6.94 Sept. 25 28 Oct. 1 4 h m 8 12 8 54.44 12 19 43.35 12 30 34.30 12 41 27.92 Ans. Sept. 22^' 12" 34'".8. 3. From the ephemeris of the moon's latitude given below, determine the instant of a-reatest latitude north. Date 1898 July 9.0 9.5 10.0 Moon's Latitude o / n + 5 7 9.3 5 14 28.1 + 5 17 38.3 Date 1898 Moon's Latitude July 10.5 11.0 11.5 + 5 16 48.7 6 12 9.7 + 5 3 52.8 Ans. July 10" 3" 27™.4. 4. Given the equation sin (s-43°) = 0.92 sin^s to determine the root which foils in the second quadrant. Ans. 101° 17' 43" 5. Given the following table of the longitude of Mercury's as- cending node (^) : 204 THE TIIKOItY AND rUAOTICE OF INTERPOLATION. Year d 1700 44 46 34.42 1800 45 57 39.28 1900 47 8 45.40 2000 48 19 52.78 2100 49 31 1.42 Express 6 as a function of r ; where t is the ehipsed time from 1900, reckoned in terms of one century as the unit. Ans. e = 47° 8' 4,5".40+4266".7.3t+0".(5307^ APPENDIX. ON THE SYMBOLIC METHOD OF DEVELOPMENT. 89. While many of the fonnuhie and results in the foregoing text have been derived by somewhat indirect methods, yet the pro- cesses emjiloyed in every ease have involved nothing but purely alge- braic operations and principles. For the benefit of such students as may be interested, we shall now devote a brief space to the more direct and potent foi-m of develojMuent known as the sipnhoUc method. In this our only purpose is to exhibit the simple manner in which the fundamental formulae of the text may be deduced ; leaving the student to enter for himself upon the broader field thus opened by suggestion. 90. Let us define the symbol of opfratloii A by the relation AF{T) = F{T+^)-F{T) (368) from which we formulate the following Definition: The operation of A upon any futictiou of T j>i-o- duces the increment in the function ivhich corresponds to the finite increment w in the variable T. The relation (368) may be more briefly expressed in the form ^Fn = ^,.+1 - F,^ = K (369) where n can have any value. Thus, taking n = 0, and referring to the schedule on page 15, we have /\F^ = F,-F^ = j; (370) Similai'ly Ai^i = F„_ - F^ ^ /l[ \ AF„ = F, — F„ = j:, ( \ F — F — F — /I' 1 206 APPENDIX. Thus it is evident that the effect of operating with A upon any tabu- lar function is simply to form the fir.^t differejice of that function and tlie succccdini>- tabular value. Whence it is evident that we have 'o AAFi = A(J;) = Jy AAF. = A (.7.') = z/:' (372) It follows that the operation of AA upon any tabular function produces the second difierence beai'ing the same subscript. But this double operation of A may be conveniently characterized by A- ; hence we write A^i'; = J; , ^'F, = ^[' , , ^"-F, = J'J (373) In like manner, i denoting any integer, we have A'F„ = A(A'->F„) - A(J<;-") = 4'' A'/; = ACA'-^i-;) = A(.'Ji'-^') = Ji" (374) A'/: = A (A'-'/;) = A(J<'-") = z?i" and, moi'e generally, n being a non-integer, A'/',, = (AAA ... . (■ times) i^„ = 4" (375) 91. Let us now consider the operation of differentiating F'(T) with respect to T and multiplying the derivative by w. Denoting the operator in this process by D, we then have dF, also D^„ = 0,^ = mF,: (376) D'F„ = DD/; = '- — {'-Fl) = u;'F:' (377) D'F„ = (DDD .... / tinie,s) /''„ x= L -^V/'',, = i^F^'^ (378) 92. The fundamental laws or i)rinciples governing the combination of symbols of quant'dij in algebraic operations are the following : APPENDIX. 207 I. The Distributive Law, by vii'tue of which a (y' + ositive integral indices are concerned, the symbol A obeys the Index Law. 93. Retaining the limitations and the notation nsed above, similar results are easily obtained for the operator D, as follows : 208 APPENDIX. (1) D (/-„+/„+ ....) = -^(^'.+/.+ . . . .) = "7^ + '*';rr+ • • • • = Di^„+D/„+ .... (2) D.,F„ = (-/^,)<'^„ = «-^' = «D^„ These relations prove that — Avithiii the limitations imposed — the sym- bol D obeys the fundamental laws of algebraic combination. 94. To a limited extent it is necessary to consider negative powers of A and D. Now the meaning and nsc of A~", A~-, . . . . , and of D"', D~", .... are easily understood : thus, from the foregoing- definitions, we have A('i^„) = F„ where 'F„ is defined as in the schedule on page 134. Then, in analogy with the usual mode of expressing inverse functions, Ave may write 'F„ = A-'F„ Whence we have A A-'/; = AC/',,) = F„ (379) which shows (1) that the operation of AA~'(= A") leaves the sub- ject function iinaltei'ed, and (2) that negative jwivers of A also oheij the Index Law. The relation £^-'F„ = 'F„ (380) may be taken as the definition of the operator A~'. Similarly, Ave have A-=i^„ = "F„ , A-^i-; = "'F„ , (381) Again, consider the relation D7^„ = J^ = V (382) Avhich, fi'om the ])oint of view above taken, may be written /; = D-i« (383) APPENDIX. 209 Then Ave liaAe DD-'y = DF„ = V (384) whence we see that negative powers oC D likcAvisc follow tlie Index Law. Moreover, from equation (382), we obtain dF„ = 0,-hHlT and therefore F„ = o,-'CrdT which, with (383), gives D-i(. = m-'l'vdT (.S85) It follows that the operation of D^' is equivalent to an integration. More specifically : Operathig upon ant/ function with D~^ integrates that fmidion with resjject to T and divides the residtiiig integral liij w. In like manner we have \r"-F„ = (0-= rfF^cZT^ (386) and so on. 95. Having thus defined and explained the use of the symbols of operation. A"-, A"', A", A, A% . . . . , and 0"=, D"', D", D, D% . . . . ; and having shown that these symbols may in general be combined algebraically as if they were merely symbols of quantity, we noAV pro- ceed to dei'ive the fundamental relations of the text, as originally proposed. 96. The theorem of the change in sign of the odd orders of differences caused by inverting a given series of functions is easily proved. To this end, let us suppose that .J;/', of the direct or given series, becomes [j;'] when that series has been inverted. Then, since A/^, = F,^,-F, = j; we have -A7^, = F,-F,^, = [j;] Whence, regarding —A as operator, it follows that (-A)'^^, = [./;■], (-A)^7<^ = [j;-], . . . ., (-A)'/-^ = [j;"] and therefore [JS^'] = (-A)'7^,. = (-l)'A"^. = (-l)-Jr' (387) which establishes Theorem III. 210 APPENDIX 97. By definition, we have hence A/''„ = /'„+, - F„ (1 + A)F„ = F„-\-AF„ = F,^^, = F{t + n^> + u,) = f,^.f:^If: + '^f:'+. . . . D- D« = F„ + DF„ + ^^ F,. + ^ F,,+ . . . . = {}+^+^+^+ ■ ■ ■ ■) ^'. = «''^» Avhere e is the base of the natural s^'stem of logarithms. AVe have, therefore, 1 + A = e'> (388) Avhicli is the fundamental relation l)etween A and D. 98. From (388), we get 0= D' D^ A=f"-l = D+^+l^ + ^+.... (389) and hence, by involution. A- = D- + D''+V,D'-i-TD'^+ • ■ • • A"* = D" +ft D^+ i D^ + :^ 0-^+ . . . . A' = D' + i D'+'+ J^ (3i+ 1) D'+-+ . . (390) These expressions are equivalent to the formulae (21). Again, from the last of (390), we derive A'F, = (D' + '+-Fl'+"-^ + . . . . (391) where for brevity we have written ~}j = >N D (394) and since, by (388), we have 1 + A = e° we must have also 1 + 9 = ed = e"" (395) AVhence we find 1 + 9 = (1 + A)"' = 1+ »/ A+— ^^2^ i A- + — ^^ ^ '- AH .... and therefore 7)1 (m — l) „ vi(m — l)(m — 2) 9 = m A + -^^ '- A'+ — ^ ^ ' A' + d'^ = »rA'^+ »r(wi — 1) A^ \ (^qq) 3" = m^A^+ which are equivalent to the relations expressed in (64). 101. The equation AF„ = F,-F^ may be written in the form (l + A)/'; = I'\ (397) 212 APPENDIX. Hence the binomial 1 + A may be defined as an operator whose effect is to raise by unity the sul)script of the subject function. Whence we have (i+A//; = (i + A)/-; = F, ^ ^"■'^> and generally (l + A)"7^„ = F,, (.199) We therefore obtain .. L X r. /i . «(» — !) „ n(Ti~l)(n~2) , F„ = (1 + A)"i^„ = fl+»A+ ^^ ^ A-+-^ ^ -^A' + /': or „ ^ ., w(n-l) ^„ n(n-l)(n-2) „„ F„ = i^„ + «J;,+ -^-j2-^ ^i + -^ ^ ^" + . . . . (400) which is the fundamental formula of interpolation due to Xewton". 102. We now find it convenient to introduce a new symbol of oi^eration, which, from its similarity and relation to A, we shall desig- nate V: this operator is defined by the equation V/-, = F, - F,_, = j;_, (401) From this relation we at once derive v=7': = vj,'_i = //;i„ v*Fi = vj;:'3 = Jii, V3^ = vj;:. = ^- , ^^^^^ whence it appears that the operation of V upon any tabular function produces the difference of order /■ which falls upon the upward in- clined diagonal through that function ; whereas the successive opera- tions of A produce, as already shown, those differences falling upon the doivnward diagonal line. Moreover, from the complete similarity of character of these two operators, it is obvious that V likewise follows the fundamental laws of algebraic combination. The relation between V and A is easily found : thus, from (401), we obtain (1-V)/.; = F,_, (403) also, from (307), we have (l + A)/',_i = F. (404) APPENDIX. 213 Whence we find (l + AXl-V)F, = (l + A)/v, = /'■ and therefore 1- V = (l + A)-i (405) which gives log(l-V) = -Iog(H-A) (406) Again, combining (388) and (405), we obtain 1 - V = e-° (407) 103. As an immediate apphcation of the preceding relations, let lis derive the formula (75). By means of (388), equation (399) becomes /•„ = (1 + A)"/; = e-/; whence, changing the sign of n, we find /'-,. = «-"°^; = i^-yj'^ = (i-v)"/'; nfn — 1) ^^o n(n — l)(n — 2) __, , ,, = (1-"V+ ^^ W ^-^ ^ ^ VH . . . .)/'\ Therefore F_„ = F,- nJL. + '^^ J-1, - ^^lumUlI^:) j,^+ (408) which is Newton's Formula for backward interpolation, as given by (75). 104. Formula (60) of the text is easily deduced by means of the identity A = (1 + A)-1 Thus we find A'F, = i(l + A)-l\-F, . = |(i+A)' -/(i+Ar'+i^^(i+A)-^- .... J/-; whence, by (399), we obtain J(.) = F, - IF^_, + '^ /■_, - '^IzM^ F,_, + . . . . (409) which is the same as equation (66). 214 APPENDIX. 105. AVe now paf^s to the derivation of the fundamental forni- uhie of mechanical quadrature. Since D = log (1 -(- A), avc have f A- A^ A* \"' D-'/'„ = |log(l + A)|-'/'„=(^A-^ + ^-^-+ . . . .j F„ = (A-'+i-iVA + 5'jA--,'^-,A'+ ,il,;A^-,;i!;S„AH • • • ■ ) F„ Whence, interpreting the first member according to (385), and the term A^'F,, as in (o80), we find o,-'fF„dT= 'F„+iF„-^,j: + ^,j:: -j^^j::'+^s^j^-^^i^,j:+ . . . (4io) This is the fundamental i-elation of qnadi-ature, from which the formnla (a) of (250) is at once derived. To obtain {h) of (250) in- volving the ditferences z/,'_, , .:/,','_2, ^L'is, . . . . , we have only to employ the relation (40(j), and the above development becomes D-i/; = !iog(i+A)i-'^':, = s-iog(]-v)r'^''„ — V V ^ 1 J V i 4 V 7 J II V 1 1; V II II 4 s V ■ • • • ^ -^ 11 the interpretation of which gives agreeing with fornnda (/>) of (250). 106. Similarly, we obtain for the second integration / A- A^ A'' \~= D-=i^„ = siog(i+A)r=^'; = A- :;+':-—+ .... F„ = (A-=+A-'+ ,',-,], A-+.]nA^-^iini. AH, ;/,^.A^- . . . )/;. Now the first pair of terms in the right-hand member may be written (A-= + A-')/'; = A-=(l + A)/'; = A-=F„+, = "/:.+! and therefore the pi'eceding expression becomes ,.,-- C C F rlT'^ — " F A. Ji F — 1 ,1" J- I //'"_ 22 1 /livi V9 //v _ M1'>"1 <" I I^,."-' — ■'^i.+l + T^f-'^n 54lJ^'i + 5?TJ -^.1 TrSJSO'^" + fflJJS '^'i • • • \*^-) from which (324) immediately follows. APPENDIX. 215 Again, we find D--/''„ = Jlog(l + A )<-'/•', = >-loS(l-Vj5-=7-',. ( V' V'* V" V" \"'t, = (v+f+f +^-+|-+ . . . .) /'; = (V"— v-'+ A-auV— .].,v'-«n5ioV^-,;j^sv'- • • )^:, (4i;^) Transforming- the first two terms of the last expression, wc olj- tain ( V-- - V-') F„ = v-= (1 - V) /'',. = V -= ( 1 + A )-' F„ Now, because the operation of 1 -)- A raises by unity the subscript of the subject function ({^101), it follows that the operation of (l-)-A)~' diminishes that subscrii)t by one unit. Accordingly, we have (v-^-v-^/'; = v-ni + A)-'/:, = v-=i;;_i = "/;+. and hence the relation (413) gives „,-2 ( i V ,lTi '//,' J_ 1 F 1 ,/" — 1 ./'" 22' /-/'^ 19 /P |'11J.\ which is equivalent to the formula (.326). These expressions complete the fundamental relations of mechanical quadrature. TABLES. 218 Table I. — Xewtox's Interpolating Coefficients. 3 k o O < c 5 > -1 la Q CO OC CO t?^ (M ^ ^ — I- -* 3 + cc ac o oi >.o 1 00 o CO «re cyD TH cq c-i cq oi C C-1 -f CD CO CO OO CO 1 I s i O CV CO -* O C-1 C^l 1-0 CJ C-1 »0 1- OC OC OO C5 C5 o S (N r- i-H S5 o :o «D lO »iO ^ ■<* CO 1 O CO rt CD ^ CO Cq Cq rH T-( GO CO n » CO CD O CO Cq ffl O LO o t- C-l lO 00 05 o CO CO 00 o oi »o Oi C5 05 o o o O O O iH tH T-l lO O LO O LO O X> lO C-l t^ CO X O' 01 CO' O O -TH TH tH © lO © >o © CI LO 00 © © LO © I- X C ^ rt T- TH C-l f-~^ -^^ T-^ ^r^ T^ lO © LO © lO © X 10 Ol t- © tH 01 CO CO 01 Ol 01 01 01 7~< T^ T^ T^ -^H © LO © LO © Ol LO X © © ^ -Y -f ~f i'. 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CO CO CO CO "* Tt* -S* tP ■^ tJ* ^^ ^ + lO O lO c >o o t- CI 10 CO C5 o CO tH 00 LO' CI o C5 O CO 00 00 CO o o o o o o LO o >o o >o C5 CO lO CI t- CO CO o t-- CO t- t^ t- CO CO O C C C' o O no O lO o CI lO CO Ci o O CO CI CO lO CO lO LO ^ -* O O O C' C: lO O' lO O 10 C5 CO lO CI 1- C CO CI 00 CO 'a- CO 00 CI CI o o o c o O 1-0 O lO o CI lO 00 C5 C: C3 -sjt C5 -^ C' 1— i 1— I O O w O CO o o c Interval lO CO l- t^ I- I- o O lO O 05 00 lO CI l- LO -H -* -* ^ CO CI CI CI CI CI CI O LO O I'O o CI lO CO' C2 o CO CI 1-1 O 'O CI CI CI CI CI LO O LO C LO C5 CO LO CI l^ 00 t- CO LO CO O >0 O LO o CI lO 00 03 o CI O 00 CO o iH 1-1 o o o LO O l-O O lO CI 00 LO CI t^ CI O CO CO 00 C; O Ci Ci C3 1-1 tH C' o c Interval O— IC1CO-*L0 COt-OOOiO tHCICO'^iO cot-cocao ■^CICO'^LO lOlOLOLOlClO LOLOlOLOCO OCOCOCOCO COCOCOCOl- t—t-t-t— t- d d 220 Table II. — Stirling's Interpolating Coefficients. o O (M ^ ^ c; o t- t^ (^ tT -f CO C-l r-l rt — + + 05 T* Ci CO t^ O to 05 t-i "* ^ o: I- t- 00 OC 00 oo O O O O O O o o o c; o o 0 CO O t- CI -Tf 1.0 O O l^ occc cocoo + + + + + rt< CO CI ■rt rH ^ <* O 00 O CI Tit C) CI CI CO CO CO o o o o o o o o o o o o CI CO •* » X' :o cc O CI -T CO CO -* -* 'j* O O' C' o o o o o o o o c^ o t~ c t- C5 T-J CO CO 1< ^ l-O 1.0 lO c o o o o c o o o o CO lO CC O CO OO CO LO t- LO ?r CO tt to C C' C' O CT' o c c o «=• 1-0 I- C5 S 1-1 c. 1-1 ?t CO 00 ^0 t~ I- I- t^ O O O C' o O C' o o c O t~ (M t- 00 r- r- o CO O CI T-H t- O O -f I- O CI o 03 C ^^ CO' -i* LO 00 tC ^ -c -* -* o o o o o o O t- tH CI C5 I- CO C <-^ -n iO I- CrS C' rH -* -^ ^ O i.O o o o o o C) CI C! tH O CI CI 1-1 tH O' CI CO ^ LO CO O lO IQ 'O »0 o o o o o ».0' lO CI ^ tH OO CD 'd* 1-1 00 CO i^ X) c; C5 LO O »0 lO o o o o o o ^ CO t- CO o ^ O lO C lO O -rt rt CI CI CO CO CO CO CO o o o o o lO O O lO Ift ITS CC t— 00 O CN (N f3 iC ^ ^ tM CO O »C3 to >0 CO r^ CO OS ■^ "^ "^ ^^ + lO O lO o o o CI <» -t< CI o o 1-1 CO CO c; CI "O CO CO CO CO '^ -t< o o o o o o lO C LO C i-O O CI -f CO CI CO -^ -+ l^ 1-1 -f lO l.O l- CO o c o c o O LO O lO o CO •* CI o c -* CO c> -o o CO CO i— t- X) o o o o o LO O «I O lO C CI -C CO C] -f CO CI CO iH (Z) OO C-. c: 'O O O O C 1-1 O lO o »o © CO -*■ CI o o '-0' O KO O LO C 1-1 IH C) CJ + + + + + + Interval lO CO [~ CO C: O CI CI M CI CI CO i-( CI CO '^ 1-0 CO CO CO CO CO CO t^ 00 03 o CO CO CO CO ^ T-l C5 CO 1* l-O *^3^ ^^ ^^ ^^ ^^ O t- 00 c^ o ^ ^ ^ ^ iO + + O CO t- O CO CO GC CI 1-0 I- O OCOCOOCOCO C3COCOC5C1 O O O rt -H 1-1 -H C) CI CI CO oooooo ooooo ooooco ooooo tH CO -)< lO CO CO CO CO 1-0 Til CDOCllOt/3 1-lTfl-OCO COCO-*-*-t< lOlOlOCOCD ooooo ooooo ooooo ooooo CI OJ t~ CO C5 CO 00 ■rt ^ CO CO CO t^ I- I- ooooo ooooo + + + + + + o* O C^ (M CO CO in OCDt-GO CI OiOi-HC^ C^ CO -^ -t^ »f3 CD CD 1— t— 00 O O CI ^ I- o O O O C C rt O O O O O' o oooooo oooooo ^^^ , ---- " " " T 10 O CO' CO 1-^ 1-H CI CI ot. -f ooooo OOOOO ooooo O O C-. C CI i-O 10 O X 3-. ooooo ooooo ooooo ^ [- 1-^ 1-0 o O' ■rH CO' rt' CO 1— ' T— < T-< 1— 1 1— * ooooo ooooo CO CI c: O '* [^ C5 O CI -* ■^ 1-1 CI CI CI ooooo ooooo t-CDI:-CDlO 1(5 ira-*CO(M »-l 00300CD -* COC^OSOO O c0C^O5CO OCDCOCDCO CD CDCDCDCO CD COiOlOlO lO i.TtO'^-* rl^ iJ"i*COCO O t- CO O CO tH O CO CO C CO CO C 1-1 CO' 1-0' CO cc oooooo OOOOOO CO r-l l-O X O O CO CI 'X lO o 1—1 CO -f CO O tH 1— ( 1— * tH OOOOO' iH iH O X -1< iH I- CO X -* X o 1-1 C) -f 1-1 1-1 CI CI CI Ooooo X 1-1 CO CI o .C3 1-0 O lO o 10 I- O O CI CI CI CI CO CO OOOOO CO 05 1-1 O CO •* CO CO I- O CO '^ CO t- 05 CO 00 CO CO CO ooooo CD I- a; c: iO »^ »A o «D t- 00 Cs + O rH M CO "* + "=l« O 10 O lO O l-O O O CI -* X CI O O O O O iH OOOOOO OOOOOO O 1-0 O lO o OO -t C) o o -^ CI CO -t 10 ooooo ooooo lO O lO O lO O CI T)< 00 CI CO t-- X C5 1-1 O O O O 1-1 ooooo O lO O lO o 00 -^ C) O o CI -^ CO 00 o iH tH iH iH CI ooooo lo © ira o lo © CI »* 00 CI CI ■* CO 00 iH CI CI CI CI CO ooooo + + + + + + Interval O iH CI CO ■* lO ooooo© o O t- X o © © © O O 1-1 tH CI CO -f 10 ' CO C5 © I 1-1 iH C5 tH CI CO Tfl lO CI CI CI CI CI DR TauL?: II. — STIKLTXfj's TxTERPOLATIXfi COEFFICIENTS. 221 El, M O O CO -f 50 I- CO Ci — (M C^ G-l (M C-\ CC O 50 O CO l« O -+ -^ C5 O C-O O C5 r> X yj cc X o o o o s s o o o o o o O >0 C-l 35 ^ I- -Tt 1-1 l_-- -J o 5 5 o o + CO 0-1 -* lO lO O t^ CO C^ 'O w o »o -t> -t< o o C; O o o s o o o + ' ' ' ' CO rt 00 -C CS — I I- o CC CO -t CO CO c-1 :■ 1 o o o o o o o o o o + ' ' ' ' CO O 30 o o 05 -* C5 10 O tH -J O O O o c o o — o o s s o + + + ^ (M W CC O '^i' t- rH O t- t- 00 + »0 t- to CO t- o N iH C C5 I-- O O O O C-. C: C5 — , ^ -^ o o o o o c o o o O 00 CO to »o -t< iH C5 to CO o 05 CO cC' » c-1 t- 00 to tH O CO iM CO -* 2? t — '""' '~ o o 5 o o CO (M t- C5 t^ OJ -^ CO CI to "O O -1" -+ CO o c c o o o o o o o T-l C-l 00 IH © O CO >o cc o CO C-I »-^ o o c c o o o O o o c; O I + t' CO »0 tJ^ CO o w (M CO -r (W C^ (M C-J ^1 'M'-'OO o Oia>ooo o -i:? 1- oj c: cs o --H (7) C"l CI CM 7-1 C*l CM O? CO CO + en O -f 1-1 o o to lO C) 03 -^ o ■^ CO IM O C: OO O 10 »0 lO' -^ — t' O O O O O' c CO t- Tjl (M tH -* t- O IM CO to ^ CO -rt Oi (M LO CV Tf c: CO C-l O t» IQ t- 10 CO 'O CO CO CO CO CO CI O C O O C' t- lO -* T)< ^ O lO C3 (M -* to CO O OO lO CI CI CI 1-J -rH o o c c o ■* lO t~ CO c 10 10 ^ CI o CI CJ to CO c ▼-< O ■— O' o C ex C: O C + ic:i iQ ifo ifo lo «o ICO >o I--OOCi O — i(MCOTt< 1— t- t- X iX CO . CO CO liO ifO iO »o O t- CO Oi CO CO CO CO lO tO o o --■ Q O ^ C^ CO ■'t lO O >0 >0 lO + O O lO lO o r- CO C5 iCT' ifO ifO »iO »0 O lO »c uo ,-. (M 00 -^ O CO CO CO ^ CO iCO ifO lO t.'O »f3 CO i^ CO OS c: CO CD CO O t- 1- t- r- r- + © lO © lO © lO © 1-0 © lO © lO © LO © LO © © C) ■* 00 CI OD -t< CI © © © CI -+ CO CI LO © LO © 10 TH © CI 00 •* '© © CI CO -f -^ CI CO CO -* ^ LO LO © © t- CO oc © © © "-1 T— (tHtHtHt-It— I 1— It— ItH^Ht-* i— It-Ht— (CICl © 10 © LO ©' CO '^ CI © © 1- -Jl ^H C» LO —I CI CO CO 'Td CI CI CI CI CI LO © lO © LO © CI -t< CO CI C) © ■© CO "H LO lO © I- OO CI CI CI CI CI + + + + + Interval © -rH CI CO '^ lO lO lO LO lOl LO LO © t- CO © © LO lO lO lO © rH CI CO ^ LO © © © © © © t- OO © © © © © © l- tH CI CO -^ >0 t- t- t- l^ t- 222 Table III. — Bessel's Intekpolating Coefficients. o b. H \A E O w m W 25 > -1 to 5 t-" ^ ^1 ^ tM C^ 05 5 OS O iCi M (M d -^ '^ Tji -^ ^ CO + CO o -^ ^ o CO CO CO CO CO GO CO CO Ol O CI d (N C^ Ol J- CO CO CI Oi r- CO CO rt + I t S 05 00 '^ 05 (N ■* C >0 O -* 05 CO I- t- OO CO CC 05 T-t 1-H 7— 1 1-H ^^ -rH o c o o s o CO 1-1 CO © tH t- i-( ^ CO 1-1 © © © © -^ 1-i CI CI CI CI o © © © © 1-1 © 1.0 00 © 'J" © © iH ^ 1-1 M iH CI CI CI CI CI CI CI © © © © © © I- CO © oo © t- © © 1-1 CI CI C-l CO CO CI CI CI CI CI © c © © © t- ^ © CO "* C0 lO If5 lO lO CO *M 1-1 O c: o o >f^ ico »ro C» t- CO o ■* io iO kfi in CO G^ ^ 1 J ©» »n o lo o 'c o t^ C^J IC oo o o CO 5C 00 O C-l l-O C5 C5 C5 O C O O © O T-l 1-1 •rH lO © lO © lO © 00 lo CI t^ © CO © CI CO © © 1-1 1-^ 1-1 © lO © lO © CI lO CO © © lO © l- CO © 1— 1— T^ ■rH CI LO © lO © LO © 00 lO CI t^ © iH CI CO CO CI CI CI CI CI 1— ( 1— ( 1— 1 1— i 1— 1 © LO © LO © C) LO CO © © ^ Tji -* ^ lO C^l CI CI CI CI 1— 1 1— I iH iH 1— 1 1 1 1 1 ■ 1 1 Interval s lO O I- CO Ci o C-l CI 11 CI CI CO 1-1 CI CO -* lO CO CO CO CO CO © t~ 00 © © CO CO CO CO -c iH CI CO ^ lO © t^ 00 © © 1* Tt -^ Tt< U5 — © a o t. g •A O O cc » H C3 > in 5 O) CO r- i^ o I- 1 ir: »C kO O rji CO ^ CO Ol CI CI CI ^ o ^ 1 1 O O -H —1 + + I 1 1 O CO O CO O 'O • O C i-l CI CO CO o o o o o o O S' C' o o o O C' o o o c ...... CO 00 00 CO CO -)< -* UO 10 © © © © © © © © © © © ©©■©©© 1 -.00067 .00070 .00074 .00077 .00079 iH CO LO. © © CO CO 03 00 00 © © © © © © © © © © © c © © © ..... I- t^ t- © LO 00 CC CO CO oo © © © © © © © © © © © © © © © ..... > (5 CO (M 1-t O CI OD 00 X CO 'X 1-- t- + t' CO -* -t iM 1- I- 1- 1' t— O O 00 CO o r- t- CO CO CO 't CI O Ol t- co CO CO o iro CO -* CI o O >0 »£0 >0 1 s O CO »c tc -o o C- CO O ^ CI o O O iH C) CO '^ O C: O O O O c c o o o c CO © © © r)< l» © CO -rt 00 -* 10 © t- t- © © © © © © © © © © © © © ^ © lO CJ © © CO CO © © © 1-1 © © © iM 1-1 c © © © © lO © 1-1 tH © © 10 CI 00 ^ 1-1 CI CO CO Tf r-* 1-H 1— ' ^H 1— 1 © © © © © t^ CO t- © © © lO © lO © -* LO © © t- iH 1— t 1— 1 1— ( 1— 1 © © © © © + + + + + + T 5 ^ CO •-< «0 (M GO 00 t- t~ CO to ira CO C» lO O CO »o ^ ■* -I' CO CO 00 iO oi a» CO Ol CI CI ^ -t* '-H X i.'T' CI + 1-1 t t- 05 1 1 T I to O iH t- OO -* CO O CO iO CI 05 o O O tH CI CI CO o o o o o o o o o o o o -* t^ lO © © iH © ^ © © •i< -f iTj 1.0 © © © © © © © © © © © © © t- CI Tf CO © © CI Tf © © © t- t~ © © © © © © © © © © CI © t-- lO © © I- 00 © © t~ t- t- t- CO © © © © © © © © © © CI T-l t- © 1-1 © © © © 00 00 a; t- i^ t^ © © © © © © © © © © + + + + + + •i *3 5 »0 tC ira »0 »C5 lO OS 00 C' CO lO '^ ^ tfi r)< r)i ■* -)< ira »o m »to wo CO N 1-1 O -* ^ Tf -t CO iQ to lO lO iCi 00 t- CO «5 '^ CO CO CC CO CO UO l£0 O tra >0 CO CI ^ o o: CO CO CO CO CI in o 110 lO O) t- CD ira CI CI CI CI T d O "C O >.0 O "O O C5 X 1-0 CI l- c ■* c; ^ Ci CO O O O T-l ,-1 CI o o o o o o © 1.0 © >o © CI lO CO 05 © 00 C) © © 10 CI CO CO -+ -* © © © © © 1.0 © 1.0 © lO © CO 10 CI t- CO CI © © CO 'I' lO lO © © © © © © © © lO © lO © CI LO 00 © © I- © CO © © © t^ t- t- CO © © © © © 10 © lO C LO © CO lO CI t- CI 10 00 ■r- CO oo 00 oo © © © © © © © 1 1 1 1 II Interval s O iH CI CO ^ lO o o o c © o © I- CO © © © © © © tH 1-1 CI CO -* lO ^H 1— t 1— 1 iH iH © I- CO © © tH 1— ( 1— * 1-H CI ■r^ CI CO -+ LO CI CI CI CI CI © => Table III. — Bessel's Ixterpolating Coefficients. 223 £ H •A W (d O r, X > 5 + + * 7 "' ^ -M cq CI -M 1 CO -1" CO Tl< ,.0 1.0 lO iO I- CO i- r- X CO 1 I % 5 o 10 50 l^ t^ I- O CO a> CO Gc cc oc o c c o o o O O O O O O' o o C' =; o o CO in CO th C5 CO' CO CO 00 I- O O C' c o O O' o o o o o o o o t- "ct* © t- CO t- I- t- CO CO © © © © © © © o © © © © © © © 00 CO 00 CO CO in in •* -* CO © © © © © © © © © © © © © © © © CO © CO © CO Cl 1-1 © © © © © © © © © © © © c © © c © + 4- + + + + > la (5 O CI -i^ CD t- OS lO iQ t.t. ir: o o 1 o cq "* 1» o O O CD CD CO CO O O IM •* CD t- It- t- 1^ -+ CO (- CO Cl I- t- t- r- I— O " * © © © © © © © © © © © © in CO © Cl -* © CO © CO Cl tH © © © © © © © © © © © © + + + + + + T 5 O t- -t^ ^ ?! O 1 1 + X ^ rt* CC -M ^ ^ r-. (M lO 00 CO CO o (M (M CO CO -* .0 00 00 CO Ol Tti -t LO lO CO ^3 I- I^ CO + T s X T to T-l O t^ 1-1 C-l o 00 OS 05 o o o l^ I- t- OD CO CO o o o o o c c o o o o o in t- CO CI -t O 00 l^ O "* t- t- t- t- t^ o c c o o o C' o c; o Cl t- O CO © Cl C5 CO CO © t-^ CO CD CO CD © © © © © © © © © © © in t- T* CD © -rH © 1-i in in i.n -t< -* CO © © © © © © © © © © -t CO t- tH © © Cl O CO © Cl Cl 1-1 © © © © © © © c © © o © 1 1 1 1 II ^ lei i5 o lO o o lO »:^ O CO IT- 00 O O iri o lo »o « 0 i.O lO to CO 1- 00 Ci Tt Tji i# 1^ + 7 c» O O lO O l-O o t-- CI in CO Ci o ct th CO 'O CI o C2 c: 00 CO CO CO O C' o o c o in o in o m ci CO m CI I- CO CO O I- CO 1- t^ 1- CO CO o o o o © © in © LO © Cl O CO © © © CO Cl 00 in CD in in -* -f © © © © © in © o © in © CO in Cl t- © © Cl OC CO -t< CO CO Cl Cl © © © © © © in © in © Cl in 00 © © ©-*©-*© iH 1-1 © © © © © © © © 1 1 1 1 II Interval s lO O I- CO c^ o t^ I- t- t- t- CO ^ CI CO -* in CO CO CO CO CO CD t- 00 © © CO CO CO CO © th Cl CO Ti( in © © © © © © l^ 00 © © © © © © © o 1— ( O t. -/. H ?; H O (d « > (5 1^ -^ >0 lO ^ iO + Til -^ O rtl -t Tti CO -t CO -rP CO Ol CO 01 Ol Cl e-l rH rt + 1 + o O lO CM 'f Oi CO O O O -rH tH CI O O O C' c o o o o o c o c» CI CO T-i in C) CO CO T* -+ © © © © © © © © © © © © o © © © CO CO © CO ^ in in CO © © © © © © © © © © © © © © © © 1- © Cl lO t~ CO t^ t- t^ t- © © © © © © © © © © © © © © © © ^^ CO -^ in t- 00 CO CO 00 © © © © © © © © © © © © © © © + + + 4- + + > 5 i-H C--: CO I- C5 (M 1 « CO t- o « CO CO 00 O — ' Ol IM Ol CO CO -i' »c^ c» O Ol CO CO CO CO Tt CO \Ci CO Cr. -* Tt< -t ■* 1 + ^ CO C -^< t- CO -t" -t> -^ CO ci — 1 CO CO CO CO CO CO C) CI CI CI CI CI o o © o o o CO CO t^ © o © C3 1-- CO Ti< CO CI CI CI C1 CI CI CI CI CI © © © © © 02218 02195 021(39 02141 02111 © © T-l CO ^ CO T* 1-1 1- CO © © © © © Cl Cl Cl iH 1-1 © © © © © Cl © "* 00 © © ^ © in © CO CO CO t^ t^ 1— 1 -rH 1— 1 iH 1— 1 © © © © © + + + + + + T la 5 C-J ^ (M ^ O O ^ "* -f -t 't 'l^ O OS CO I^ CD 'H CO 00 CO CO lO -* CO r-< Cl CO UO 0? CO C>l CO t- -W CO ^ C^l !M Ol Ol Ol 00 r- -f i-H ""7 1 X V. C CI CO "S o o O -* GO CI 'O o O C O tH ^ CI o o o — o o o o c o o o ■ 1 CO CO m CO © ^ OD CI CO © CI CI CO CO -f © © © © © © © © © © 1 ' ' ' ' CO th m CO © CO t- © CO CO -* -)< in in in © © © © © © © © © © 1 OO © CO t- © © Cl in t-- © in © © © t-- © © © © © © © © © © 1 1-1 © © © 1-1 Cl CO in t- 00 I- t^ i^ t- t- © © © © © © © o © © ..... n la 5 iO »C5 lO »ra lO xd T^ — 1 lO >ra »fO to lO ^ c-i CO -p If: lo in lo >o lo CO t— 00 Oi o r-. rt ^ ,-1 Cl lO t.O to in iH Cl CO -t Cl Cl Cl Cl 7 ei o o o o o »o O C5 CO O CI t- »0 ^ -t Tf -f CO CI CI CI CI CI CI © in © in © ci in 00 C3 © CO Cl i-H © © CI Cl Cl Cl Cl T^ r-< -r^ T^ -^^ in © in © in © CO m Cl t- 00 t~ CO in CO © in © in © Cl in CO © © Cl © CO CO in iH rH © © © in © in © in © (x> in Cl t-- ci © CO © CO © © © © © 1-1 -rH © © © 1 1 1 1 1 1 Interval O tH CI CO "* 1.0 lo lo in lo o lo CO t^ CO Ci © in in in in CO 1-1 Cl CO Ml in CO CO CO CO CO © i^ CO © © CO © CO CO t- 1-1 Cl CO T)< in t- t- t- t- t- o © 224 Table IV. — 1*^ewton's Coefficients for F'{T). 1 U, 1 5 c;i--f(MO CO oocoff^ o G a )005QO 1- T-t-^Tj*-* CO COCCCOCO CO COO?(M(N 1 -f OC 5 OC5CSOO CO 1-t^r-i— oocoin-p Q7 w— 'oos COCOCOCO CO C0«C?C^1 + 1 1:1" C -* C-. t- CO » -* -H C5 C» O C^l t- CI -^ CC ■-H-f'X-Cr-O C:S-rH-rrC5 '1<0(»COCO lO C-. ^t CO CC 00 CI 03 CO CZ) 00 O O C^ CO l- 50 >-■: 1- -f -t oo CO c-1 CI rJ -H OOOOO OOCOOO OOOOO OOOOO LO LO ^ CO iH C~. CO CO O t- Cl O I- CO 00 1-1 -C CO I- CO — 10 00 CI lO C5 CI LO CO 1-1 -H -H tH CI CI CI CO. CO CO ^ OO'OOO OOOOO 1 1 11 + + + + 'I fa' Q »o iO in lO ira o 1.-^ U5 lO o lo o lo io >ra ».o -^COOl-HO O; GOf-OiGt -f COC<1-hO CS t-t-t-t-t- CO OOCOO CO OCDCOCO >n i + T COOOOOOOCOOO OOOOOOCOOO COCO)COO0CO 1(5 -H I— L-O OO 00 CO lO l^ i-< O -H l~ l-O; CO CO -* I- O CI lO CO tH -It t- rt -ti CO -tH LO C5 CO -H O C3 02 CO t- t-- CO lO LO -* CO CO C1 -H — I -H-HOCOO OOOOO OOOOO 00 CO CI t- CI t- CI t^ M t^ CO lO CI CO -i< CO C5 ^ CO CO t--H-tC:LO OCOiHCOi-i OOOO-H CICIOOOOrtt OOO. C.O OOOOO + + + + + 1 1 1 n '.0'*C0C I- cc c: o ^ o Coefficients for > (5 i/;t-ocot- o ^co^co 1-1 iiroioo lo OlOOit-iO rjt vNOCrit- CO tPCO^O CO CCOOt-t-l^ 1- t-l^-COCO CO OOCDCO lO I-H t- CO Ol \Ci CI CC lO CM 1 + 1 OOOOCOLOO) CO-tOluOCJi cococooooo O 1- O I- O -t< O CO t^ CO O ^ O I- »o l-O O-HCOl-OOCO COlOCWiHlO OOCIlOCiOO O' O CO t- CO -O 1.0 -* CO CO M -H 1-1 O C5 es Cli-Hi-liH-^rH T-.,H--liHiH lH^i-(00 CO CI lo CI CO a; CO. CO CO 1-1 I- O •* O l- "0 lO CO 05 CO t^ CI CO .r^ >0 O LO O 10 -^ GO. CZ) 1- t- CO CO LO LO -f -f OOOOO OOOOO + + + + + + T n in c:3ooi— coin -^ cotNi-io o ooi— coo ** C505005CS CS <35CSOSOS CO OOOOOOCO CO in in in in m in in in m COo LO -!<-*-♦<-)<-(< ^n^-*"*--)* COCOCOOOCO -* CO. CI 1-J O CS 00 1- CO 10 CO CO CO 00 CO CI CI CI CI CI d ■ ■ 1 o 1 Interval ~ o-ncico-fio coi-a;c50 i-(Cicoi*>o OOOOOO OOOO-H -HT-li-li-lT-i COt-QOCSO i-IClCO-fO T-i 1-1 1-1 T-l CI CI CI C) CI CI d o Table IV. — jSI"ewton's Cokffioients for F'(T). 225 « o u. (n h Z Ed »-« It o O > 6: a »-H O O O OS o 7 1 + " O CO X C^ CO »-< C^ C^ CO CO O -P 1^ 71 t-O T i^' 't lO ..O ■X Ol i.O x — tro CC CO w I- -r CO o 01 I- I- :c X + + "si" 1 1-0 O "H tH CO t~ C) CO -f --f CO C-I o o o o o o CO to CO to CO «o o o o o o o C^l CO O CI o tH C3 I- -(< ^ O C3 C5 C-. o CD O iCi O i.O O O' o o o -* -* O CO ■rH l~ CO C3 -* C5 00 00 I- I- CO lO 10 LO 10 >o o o o o o ■O X CO 1-1 CO CO t- 1-H lO X CO 10 10 ^ CO lO 10 LO LO o O O O O 'O 05312 05238 05162 05082 05000 1 1 1 1 1 1 5 CI CM TJH I— w M O CO lO '^ Tp cc + + 1 CD CO O lO rt -H C<1 CI CO x CO x in 05 CC T 11' iC lO CO C CO -H CD t- t- X 1 r T "el" ^ CO "O O CO t- C5 O (M t- CI CO >o CO t- t- CO 00 00 CO 00 OO CC CO O O O O O O O CO CO OO l-O O CI -* O CD C5 0-. C3 O C5 00 00 CO CO 20 o o o o o ^ CO LO IC o CO LO' 'SI CI o C5 C5 05 O CS 00 CO 00 00 00 o o o o o Ci 1^ X O LO CO CO X Tf X X X t- t^ CD X X X X X O O O O O 08626 08560 08490 08414 08333 + + + + + + ^ 5 lO »0 lO iO O lO -ii CC C-1 '-' O C5 CI 0 »f3 »fi CO Cra m lO CO 5 CO I- t- I- CO CO CO r- o .o >* CO (M (N N ra o lio O) t- CO LO -^ CO CO CO CO CO »0 «0 wO to lO CO CM — O ~. CO CO CO CO Ol to O to o X t- CO i^ CI (N CI (N 1 + T t- M t- CI l~ C0 O O O O o o CO t- 00 C5 O O O O O 1-f ^ C-1 CO '^ LO 1— t 1— 1 tH 1— 1 1— 1 CD t- X c; o 1— 1 TH 1— 1 T-1 CI ^ CI CO TjH LO d CI CI CI CI o o + o + Interval s O ^ CI CO -* o »0' IC O »C LO iO CD It- CO C5 O LOi LO »0 LO CO tH C) CO -I* LO CO CO CO CO CO CO t- X C5 o CO CD CD CO t- tH CI CO t IC' t- t- t- t^ t- o - 226 Table V. — Stirling's Coefficients for F'{T). trj so ■^ o I— c: C-1 « »« C- X o C'l tt t' 1- I- I- 00 CO 00 CO 00 03 ro =: + 00 I- ^ C5 CI J1 50 O Tt t- rt --*■ o "n •* o 0-; 01 CI I-l CI CI CI CI o o o o o o tH t- ■rH Tf 1,-5 I- 35 CI ':»< ^ l-^ O O O X CI CI CI rt >-H c: c o s o cooioxc ccc'soe^ CO O T-H CI I- L- '~C' « -:)< lO »~ O I- I-- CO CJ rt C' C3 7-1 ^^ ^H 1-H O O O O C' o in o 1-0 en OS t^ t- l~ l~ o 00 t- «> o ^ O O O O' o O O O C' o + + + + 4- + -t o t- -^ M ^ T-^ CC C: to ^ + + 1 M-*Clt-00 t--T-l-.H.XC1 ClOid-rHt- XlOXt-i-H CI I- CI X X C^ ~ C C: th 1-1 i-H -^ CI CI o o c o o s X CI l.~ 1^ o C' ^-^ T-l ^ C) CI CI CI CI CI CI CO lO CO o CI CI CI M CI CI CI CI CJ CI O O 'O o o CO CO o '^ CO C) CI CI Cvl CI M C) CI CI C^l o o o o c iH CO t- CO CO T- X 1-0 CI X CI — 1— T-l O C1 CI CI CI c^ C C C: O O m o lo ir3 lA O ?D t- 00 OS (M -M (M -N (M + O f-H (N CO -^ CO CO CO CO CO lO if5 »c^ to »n o >o lO ira in •n lO >re ira to 1- 00 Ol o -H CJ CO ^ lO m 1— en re CO 00 00 CC ■* TJH TJH -* ■* M< •*• ■(t •* '^ + ^kl' CI I- CI t- CI l~ I* X CI -* CO CO 'O CI C t- f T-i CO CO :o CI CI ci c 1 5 I— O CD CO >0 -P ^ d C-l ^ OS — -^ -< -H O 2'=g3 a rH c; CO »o -r c; =-. o C-. S -H Ol t- lO cr. X CO CC 1 1 + 1 "sis O t^ CO C5 o © OO C2 tH CI '^ CO CO -* O t^ CO C3 Cq CI CI CI CI CI O O O 'O s o 1 ■* CO o en CO t^ CO O 1-1 Cl O -r^ CO '* la CO CO CO CO CO O C' o o c 1 C-l 1-i OO ^ CO CO -* -* 10 i.O CO l~ OC CJ 'O CO CO CO' CO -t c s o o o 1 rH Cl rH 35 r:)< CO © © lO lO 1-H Cl CO -f 1-0 -* -f -*<-)> -t< ©©©©'© 1 CC Cl CO IC © -t CO Cl ^ © © l- 'X c © -f -f -T -f LO © © © © © ..... > Id 5 (N 3i t- O M O O C -^ 0-1 CO -1< (M c-1 I?! CN fM fN + Ci t- ift rt< (M ^ »c CD 1— a: 2^ tM c-1 C^ (M rH O OO t^ CD C5 O O rH (M c-1 CO CO CO 00 »0 1(3 tP CO CO 00 Tt^ lO O t- O? CO CO CO CO CT (M (M ^ CO CO ■^ -f + a 12 1 iH CO CI C5 '^ t- OO CC C3 O CO CO t- C-. i-( ';)< CO CO O O tH T-H ^H 1— ( q o q 'O o o + t- CO CO 00 en O 110 1-1 t- lO •^ CO CO CO 1-1 Cl M Cl en CO q c o o o + ' ' ' ' -* iO lO CO o CO c-1 Cl CO 10 r:t< t~ C CO CO 00 CO rt rt r-Ji O c: c o o + ' ' ' ' + .04976 .05311 .05656 .06010 .06373 + .06746 .07128 .07520 .07922 + .08333 T i5 »r5 lO »ft »0 tra lO O O I— OO Ci o i^ t- t- t- ir- GO + O »0 lO lO o ^ IM CO -* O 00 CC 03 CO '-/3 lo o lo in o CO t- CO Ci o X CO CO 00 o m i.o if3 ip ic -H c-1 CO ^ in O C3 O C2 o iO lO >0 >0 + -to 1 00 CO yj CO 00 CO lO -rH t- W CO CO -* CI C5 t~ lo CO ^ CI Cl CO -* 1.0 + CO CO GO CO CO CO "O l^ tH o 1-1 C5 t^ CO -* CO 10 t^ CO o> ^^ lH 1-4 1—1 1—1 + ' ' ' ' CO CO CO CO CO rH t- lO CO CO CO rH O C5 CO O rH Cl ei CO en Cl Cl en en + ' ' ' ' X CO 03 CO 00 CO 10 t- rH 1.0 t- © LO i-O -*< -* 1.0 © I- X en Cl ei ci ei + ' ■ ' ' + .29413 .30378 .31353 .32338 + .33333 Interval s lO CD t~ 00 C3 O I- L- t- t~ I- 00 r-^ Cl CO -^ l-O CO CO CO CO <» CO ,1^ C«' C2 O CO CO CO 00 C5 TH en CO -:)< l-O C5 Ci Cl Cl Cl © 1^ X c. © '© © © © © o r^ o a; H 15 O b Ii. W O Q > to 5 L^ -^ r;; 00 o o ^ Ol OC M* -^ i.o CD CO CO r- t- CC t- X CO X X 1- X 1 1 ^ Ice + 'l 1:1^ era -i< 00 1-I CO t^ CO CO >-0 l-O' -* CO tf CO CI -rH O O O O 'O o o o O' O 'O O C5' O t- 00 O CO i-^ t- a; o -^ Cl ^ Cl -* lO CO O C' o o o O 'O 'C: 'O O iH CO en CO ^ -* 1-0 t- OC o t^ 00 C5 O Cl O 'O O rH rH o o o o o rH X © 00-rH en CO o t- Cl CO' -* LO © 1- y-\ -^i T^ y-i T^ O '© o © © o t- lO en © © Cl rt CC X © © 1-1 Cl CO rH Cl ei en ci ©©'©©© + + 1 1 1 1 II > -* OS »0 Cl ?o o Tf ■r:^ O o CD I:- + t- C O lO c-1 C3i ^ (M O rH r- (N CO t- IC rH 00 CD CO ^ »0 O CD en oi t- It* t- t- X OS + = 12 1 CO C5 O LO CO O OO 00 CS CO t^ iH O 'O' C5 Cl CO 00 Cl CI 1-1 1-1 ^ 1-1 o c:; c 'C c o O CO iH ^ O -* CO OO C! O t- CO 1.0 -t< ^ T— t 1— < rH 1— ^ 1— ( C O 'C 'O o O l-O CO ■* C' O C2 CO CO -t CO 1-1 O Ci CO rH -rH rH O O O O CO o o CO ^ © LO t^ © 1- Cl t^ rH t- LO •* en 1-^ © © c © © © © © © © © rH © t- rH -* en © X CO © en It LO t- © © c © © © © C © '© 1 1 1 1 1 + + T la 5 lO iC irt »ra 1^ lO O ^ CM O'J -t »0 O »C O i--^ tr? ^^ + O if5 o >o o CD 1- 00 C5 O O O O O CD O O 1-0 IC 1/0 rH c-1 CO rtl lO CD CO O CD CD \0 O lO ICO iO CO t^ CO Cl o CO CO CD CO t- lo "A >f2 IC 1-H (M CO ^ t- r- t- I:- + -to 1 t- Cl t- CI t- Cl CO CO ^ Cl CO -* tH CO rH CO O 1.0 ^ CO CO Cl C> 1-1 O O C' C C 'O t- Cl CO CO CO as Cl 1-0' CO CO C5 ^ iH t- CO C O O O rH 00 CO 00 CO oo CO LO t^ rH lO Ci lO rH £0 "^ rH en CO CO '^ o c © o o CO so CO CO 00 1-1 t- LO 00 CO 1-1 t- ^ rH 00 LO lO © t-- t- © © © © © X CO X CO X CO LO t^ 1-H LO LO Cl © t-- -* X © © © tH © '© © rH rH 1 1 1 + + + + + Interval S O iH Cl CO ^ »n lO lO lO 1-0 lO lO CO C- 00 C5 o lO O iC lO CO -^ en 00 -f lo CO CO CO CO CO © t- x © © © © © © I- rH Cl CO -f 10 t- I- t^ t, t- o "-^ to o o S o 228 Table VI. — Bes.sel's Coefficients fok F'{T). ■f. V, 3 b 'A O O > fa' Cl 1-- -O Kt. -^ -rt* ^1 Tl M G^l ?J o c o o o o o + + .00249 .00271 .00292 .00311 .00330 00 '* o •* t- •>*• CD CO OO o CO CO CO CO -t< o c:> o o o o o o o o + ' ' ■ ' CO C5 CO CD CO i-H Cl eo -^ LO ^ -* -* -t •* o o o o o o o o o o + ' ' ' ' 05 eo CD t/o 05 LO CD CD CD CD -* -* -f "Cf ^ © © © © © © © © © © + ' ■ ■ + > fa' i5 CO « CO K-? X CS r- X CC X CO X* 7 '^""" T- CO »« CD CO OS C5 O 05 c: Oi ^ Cl fM ^ CS O O O CO ^ Cl O !M Cl LO lo ^- CO CO o cc o o o Cl Cl Cl Cl Cl 1- X O: X (N CN (N C^ 1 1 00 O O O tH CO 't t- cc c CI CO C: l^ '0 -f CI O -+ -t- -f ^ Tf -1< o o o o o o -* CO O 13 05 ■* lO CO CO CO OO CO -t" Cl o CO CO CO CO CO o o o o o T-i Cl tH 05 t- t- l^ t- CD CO CO CO -l< Cl o Cl Cl Cl Cl Cl o c o o o CO OO eo CO o CO lO lO -^- -*< CO CO "-f Cl o T-^ T^ T^ T^ T^ O CD o o o Cl LO I- 00 © CO Cl r^ © © CD 1* Cl © © © © © © © © © © © + + + + + + n fa (5 O to O JfS »o »o ^ CO C^ ^ O Oi (M CJ CN !M 3^ T-. 1 lO lO lO o »^ 00 t- CO O rP »0 lO ic^ »o to CO Cl ^ O OS lo »ro >o to to 00 t- CO to '* to to to to eo Cl 1-1 "12 + 1 0>1 t^ CI t^ C1 t- -f ->: CI T -o o C Cl 1.0 I- c3^ <-^ — 1 r^ ^ T-- T-l CI o o o c o o Cl t- Cl t- Cl CD ^ Cl OO -:1< CO O t- CO o Cl Cl C^l Cl CO o o o o o t^ Cl t- Cl t- 00 Cl -* CO CD ■r-l CO -* O CO CO CO CO CO CO o o o o o Cl t- Cl t- Cl CD -i< Cl OO TjH t^ CO C-- C-. o CO eo eo CO -* o o o o c t- Cl I— Cl t- £» Cl -f CD © O i-H 1— ' 1— 1 1— t ^ -ll -f •* -* © © © © © 1 1 1 1 1 1 n 1 S LO -^ CO CO tH O C) CI Cl Cl C) Cl d 1 C5 00 t- CD lO •* CO Cl i-( o 1H T-l tH tH T— 1 Oi 00 t- CO lO q CD- o q q -1* eo Cl 1-1 © © © © © © ■ ©■ © 1 Interval ^ LO CD t- OO O O Cl Cl CI Cl 00 CO 1-H Cl CO ■* lO CO CO CO CO CO O I- CO o o CO CO CO CO ^ ^th Cl CO ^ lO © t- 00 05 © 'Cf Tj< 'Jjl >* LO o © o w H 5 b O > 1 5 — ■ -M — 71 ^ v- ^ -^ -IH -r -^ -i^ + ^ ^ C^' o--^ o ^ -i^ -r CO Ti^ GO CO 00 1^ CO CO CO CO o^ CO CO -t to CO o^ CD CO CO or 00 -H ^ O CS 00 CO 00 Cl + CO Cl o c; 1- 'O CO Ci LO O -O Cl CO t- t- t- o o o o o o o o o o o o o o 00585 00544 00504 004G4 00425 O t- C5 i-H -)< CO -* O I- CO . CO CO CO Cl Cl o o o o o o o o o o <» Cl CO 00 o C3 CD Cl C: CD 1— 1 ^H 1— i O O o o o o o o o o o o r^ ^ ic L5 Tji Cl © CO © Cl c © © © © © © © © © © © © © © 1 1 1 1 1 + + > 5 »r^ .-■ -^ O «i^ O X Gl O O O O ■n< X !M CO -. rH .-H fN G^ CO -* CO Cl LO CS CO CO ^ -^ •* d to CS Cl to to O to CO CD CO O CO CO CO r- L- t^ 1 T II 1 1 CO C» t- ^rH iH CD CO rl* O CO o lO CO C-1 rH O O OO OO CO CO CO I- I-- c o o o o o t- CO >0 CO b- -* CO 1-1 C3 CD l^ CO lO CO Cl t- I- t^ t- t- o o o o o CO Cl TiH Cl t- CO O CD Cl t-- i-( o a; t~- lo I- t- CD CD CO o o c o o OO CO i-l Cl o Cl t- Cl CD O ^ Cl ^ C5 OO CD CO CD lO LO O C CD O O lO t- t- -* 00 CO © © Cl ->< © -)i Cl 1-1 © IC 10 LO lO -^ © © © © © + + + + + + T fa' 5 1.'^ »f5 lO iC >0 lO Ci X i— O lO "^ rji -^ Tp -^ Tfl ^ 1 lO »o O i.O itO CO Cl ^ o c» ■* rjl 1^1 TJH CO »0 lO »0 ».C ICO 00 t- CO Ui ^ CO CO CO CO 00 to to to to to CO Cl T-H O <3S CO CO CO CO Cl to to to to 00 t- CD to W Cl d Cl i CO OO CO OO CO OO CO CO '-0 t- tH >o CO (» CO CO '^ o 03 b- t- cr> CD lO o o o o o o eo 00 CO CO CO tH t- «0 00 CO lO O CO Cl OO LO O '^ '* CO o o o o o 00 CO OO CO OO CO LO t- T-l lO -* O CO CO 03 CO eo Cl Cl th o o o o o CO CO CO 00 CO iH t- LO eo 00 CD Cl C5 CD CO 1-1 -rH O O O © o o c o 00 l^ Cl l~ CO CO ^ Cl 00 -* © Cl LO t^ © © © © © tH © © © © © + + + + + 1 1 n Hot O Ci OO l^ CO UO 1.0 ^ -* 'ri* -* tH '^ CO Cl iH o ^^ '•^^ "^ ^T ^T 05 00 t~ CD lO CO CO CO CO eo -* eo Cl ^ © eo CO 00 CO 00 © CO t- © lO Cl Cl Cl Cl Cl © 1 © 1 Interval S O — 1 Cl CO -^^ "O c o tr o o o CO l- CO Ci o C O O O --1 iH Cl eo 'Ttt lo tH ^H 1-H 1-H tH CO t- CO as © tH r-l 1-1 1-1 Cl 1-1 Cl eo 't lO Cl Cl Cl Cl Cl o © Table YI. — Bessel's Coefficients for F'{T). 229 O h a; « O o > (5 05 O rH rH O^ CO (N CO CO CO 05 CO 1 lO -t CD CD t- CO CO CO CO CO Ol CO CO O OS CO CO CO If CO O O rH -H rt -^ Til It 11* ■* Ol rH Ol rH '^ -)< -t •* 1 r + "sis -* »0 lO '^ t- o C5 'O CO O C-l CO c o o c o o o o c o o o o o o o c o CO a> 01 00 •* C3 Ol O C3 CO O r- rt -^ M O O C' o o c o o o o •pH C3 1- 1.0 lO l^ O -t OO Ol Ol CO CO CO -+ o o o o o o o o o o icH ^ T)< l.O CO O O -t< !» Ol -* lO 1-0 1-0 CO O O O C' o O O CD o o t- Cs O 01 CO CO O 10 C^ CO CO t- t- t- aj o o o o o o o o o o + + 1 1 1 1 1 1 > la 5 O CO O 00 >o M t- t- I- CO as o c:5 ira (M Ci no O K5 lO -* •* !M 00 Ttl r-1 03 '* CO CO 00 o^ (M CO tC oa .n (M rt rt o O 8§SS „^„^^^„„„ ill -IS + s|2 1 ■^'sl-' 00 -^ t- t^ "rt o ^ C-1 C^ O CO o C5 -n 1-1 •* CO 00 ^ UO lO lO "O lO o o o o o o 01 ^ CO CO t- CO ^1 i~ oi i^ C5 ^ 01 -f o O CO O CO CO o o o o o Ol ^ Ol CO 1- Ol CO O CO CO t^ CO O 1-1 Ol CO O L- t- t- o o o o o 00 l-O CO t- CO C2 tH CO -)< CO o CO t- CO [_ I- l^ l^ t^ o o o o o 1-1 ^ t^ OO CO CO CO LO rjl CO C5 C rt 01 00 t- Oj CO O) CO o o o o o 1 1 1 1 1 1 n Q »f2 »f5 lO o ifr uD iC :0 IT- CO Oi O CM (M Cl Ol (N CO + lO lO lO lo in rH C-1 CO "^ >0 CO CO CO CO CO lO >0 >C5 iO o CO r- CO o; o CO CO CO CO '^ »r5 >c >o »o lO ^ Ol CO i* no -t Tl< -Jl •* -* m in iO >ft CO t- X) C2 -* Tl* rt* -I- SIn 1 01 t- c-i t- cc CO -* CO Ol -* CO CO O I- O C-1 O CO tH o o o o o o o o o o o 00 CO 00 CO CO CO lO t^ 1-1 lO O C2 Ol CO 05 O O iH -H tH o o o o o CO wo CO OO CO T-l l^ lO CO CO CO CO O ^ CO 01 Ol 00 CO CO o o o o c 00 00 CO CO CO CO lO t^ T-l l-O 01 CD O 1-0 C5 -+ -ti LO lO lO c o o o o CO 00 CO CO CO tH 1-- lO CO 00 -t OO 00 00 CO CO CO t-- t- OO o o c o o 1 1 + + + + + + n >0 CO l-~ CO C3 o CI C^l C-I 0) 01 CO o + ■rH 01 CO '^ O CO CO CO CO CO CO t- a; 0-. o CO C0_ 0^ 00 Tt* ^ Ol 00 ^ lO •^ -* ^ ■* -* CO 1^ CO C5 O ^ T)< -I* ^ lO ■ ■ ■ ■ o + Interval s lO CO t- CO OS o t- I- t- I- t- 00 iH Ol CO -f lO CO CO OO CO CO CO t- CO C5 o OO OC OO CO 05 1-1 Ol CO -f lO '^ 1^ Gi Oi Oi CO t-- 00 C5 o O CS Ci Oi o o lH o b. H Z 5 b fa H O o > -1 In 5 rt « D5 -* CO t- 1 CO Ol — — 1 05 •+ CD CO O) c; O 1— 01 -il -t* lO CO 1- 0-. C-I Ol CI Ol 1 -A 1 'sis O CO CO CO O CO CO CO CO O lO lO ^ ^ -* 'I* -* -Tfl o o o o o o o o o o o o O CO O OO t- -* 00 01 T-< o ^ -* -* -f< -H o o o o o o o o o o -* O -f 00 o C: OO CO •* 00 CO CO CO CO CO o o o o o o o o o o 1-1 01 1-1 O lO 1-1 C2 t- -* Ol CO Ol Ol Ol Ol o o o o o o o o o o iH CO O 00 T* t- LO Ol o 01 1-1 iH -rH O o o o o o o o o o o + + + + + + > fa 5 CO C» l» t- O) CO o o o o o o 0^ O (M tM (N in ^- »o lo 1* 0 O lO to >ra CO 1- 00 c:i o »0 lO O lO lis w Ol CO '^ tf5 lO »0 lO 4iO no CD r- CO o o rH — t T-H T-l Ol lO o »o to .-H CM CO Ml CN CI iM C^ + -12 + 1 t^ Ol t- Ol t^ 01 CO CO -+ 01 00 -^ ^H T-( T^ 1— 1 O O ^ -* -H -t< -+ -t< O O O C' o o t- Ol l^ Ol t- CO 01 ^ O CO C5 O OC' I- CO CO CO CO CO CO o o o o o Ol I- Ol t^- 01 CO -* 01 CO -c)< O -)< CO i-^ o CO CO CO CO 00 o s o o o t- 01 t- Ol t- OO Ol -!f CO CO 00 l— LO CO 1-1 01 01 Ol Ol Ol o o o c o Ol I- 01 t- 01 CD -t Ol CO -* 0> t- IC Ol o lH iH iH 1-* iH O CD' o o o 1 1 1 1 1 1 n 1 C — t Ol CO 'f lO o o o o o o CO t- CO C5 O O O O O 1-1 ■r-l Ol CO 'cf »C ■r-t T~> T-t ^-i T^ to b- OO O! O 1-1 iH 1— ( 1— ( Ol iH 0-1 CO "* lO Ol Ol Ol Ol 01 o + o + Interval s O 1-1 IM CO -f lO lO i-0 »0 lO O 1.0 CO t— (» 05 O 1-0 l-O UO O CO t— 1 Ol CO "^ lO CO CO CO -O CO CO t^ OO CS o CO CO CO CO t- ^ Ol CO 1* LO t^ b- t- t- I- o o DR. Gc 230 Table VII. 0) ■73 s o "-J CI ^■: -t >-•: o s c = o o O O C: C: C: O OOf) 007 008 001) 010 rH CI CO -»■ 10 c © S © © 010 017 018 019 020 021 022 02;; 024 025 O © o K b O s ■< O d c o c '^ CI ^•5 -* 1- © X © rH CI rj- t- © CO rH T-l rH CI CI © © CI © © Cl Cl CO CO rf 't OO CO GO CO rt< TJH lO l-O © d © O O •rH C1 CI « »0 to 00 © rH rH -* © © T^ TH rH rH rH CI rl< t^ rH ^ OO CI CI CO CO CO CI © © lO © ^ -* >0 lO lO r-t CO O O O T-l rH CI CO T(< © t- ©1 rH CO LO OO © rH rH rH rH CI CO © © CI © CI CI CJ CO CO © rJ4 CO CI © ■* -*< rf lO 1.0 00 r- O O O tH tH CI CO rf lO t- © O CI rf t- C5 T^ r^ r-i tH T-i CI lO 00 rH rj< CI CI CI CO CO I- rH lO © CO CO TJH r). rt lO t* ^ O O O tH 1-1 CI CO -* lO CC CO © CI -H © GO © Ct © © CI CI CI CI CI CO 1.0 © CI © © CO CO "* Tl* lO CO 1-1 lO O O O tH i-H CI CO rf J-O' © X © rH CO lO t— j-< r-' r~i T~^ © CI -* t- © r- d C^l CI CO CO © © CO I- CO CO ■* -* •* 1.-5 rH -* O O O i-H rH CI CO CO -f © I- CO © CI -* © T-t J-< T-i T-i CO © CO lO CO rH CI CI CI CI rH -^ l^ © .^ CO CO CO -^ ■* Tf CO C: O O tH ^ C1 CI CO Tjt 1.0 t- 00 © rH CO 10 rH T-1 rH t^ © rH CO © rH rH CI CI CI © rH rj* t^ rH CI CO CO CO ■* CO 1-* * K5 © © t- GO © © rH rH C^ CO r* © rH rH rH rH rH g o ©©©©©© rH rH rH CI CI CI CO CO rX rj< lO © © t^ 00 © © rH CI CO y~t T^ T-{ T^ rP q § ©©©©©© CI C-l CO CO CO rf rji lO 1.0 © t- t~ CO © © 05 q o ©©©©©© © © rH -H rH rH rH C) CI CI CO CO CO rjl rH -* LO 10 © © 9 q ©©©©©© © © © © © »^ rH CI CI CI CI CI CO CO CO q o o d o © © © © © © © o © © © © © © © © © © © © © © © © © § d H O rH CI CO rj* ira o © © © © © © © c © © © ©■ CD t- 00 C5 © © O © O rH ©© © © © rH C? CO '^ »0 rH rH rH rH rH CO©©© © t- 00 © © rH rH -H rH CI © © © © © rH CI CO rH 1.0 CI CI CI CI CI © © © © © ' ■ ■ ■ © Giving ij: To he used in finding u avhen J^„, ls given. XoTE. — The ([uantity ;/ has the same sign as argument K. Table VU. 2'.n T-H -tJ %/ i;; o t- :^ — O CI 01 CI CI CI :■: © o c o c o T- c) CO -t< "O CO CO CO CO CO c o o o o CO t- CO' CJ o CO CO CO CO ■* o o o c o 1-1 CI CO' -t< 1-0 -* -^ -t -t -t O' o c o o O I- CO CV o -* -t -t -* 'O c o o o o o > o s s a K o d CO oo eo 00 '^ o CO O t- I- 00 05 CO CI 05 CO CO 05 O O 1-1 c? 1-1 tH tH -M O t- ■* CI o CO CO ^ lO CO iH 1-1 iH iH iH CO CO 1.0 -* CO CO t^ CO C3 o 1— 1 1— 1 T-l 1— 1 C t- CO OO ^ t-- CO O CO O Ci O T^ T-t ^H 1— < tH CO C t- 1* CI CI CO CO Tt< lO ■r^ 1— ( iH iH 1-1 O 00 CO -♦1 CI CO CO 1-- CO o iH O 03 00 00 C 1-1 —1 CI CO CI CI CI CI CI Ol 00 CO tH CD tH CC -^ O CO CO t- t- CO CO C1 CO -1- o CO O C5 O T-l T-l 1— 1 t- CO O t- 1* 1-1 CI CO CO ^ ■rH 1— 1 1— 1 1— 1 iH •rH C3 CO -* C-1 o lo CO I- CO tH T-l iH 1-1 iH O C5 t^ CO lO C-. O O 1-1 O) tH -H CI CI CI CO 1-f i-H CO t- N t- T-( CO lO lO CO CO I- r- C) 1-- 00 O) -* CO CO 02 C5 o O CO CO Ci O iH T^ CJ CI CO 1— ( 1— ( tH rH 1— 1 CO O I- LO C<1 1* IS la CO t- IH T-l 1-1 1-1 T-l O 00 CO ^ CI CO CO c; o T-l tH 1-1 tH C^l C-1 t- o O '^ CO CO t~ C^l O lO >0 CO CO t- t- CI t- CI CO t- 00 CO crs C5 -f C CO CI CO O I-H 1-1 CI CI iH 1— 1 iH 1— 1 iH -* th CO lo CI CO '^ -* 1-0 CO T— 1 T— 1 1— 1 1— 1 tH 03 t f CI o CO i^ a> cc o 1-1 1-1 T-l 1-1 CI CO lo t- T-i >n 05 CO 00 TJH lO lO lO CO CO C-1 t~ CI t^ CI t- t- 00 CO 05 t- CO CO ■* o 05 o o ■^ cq iH -^^ ^H iH CO C-l C: lO CI Ca CO CO -rti lO T-l T— 1 tH T— 1 1— 1 C3 CO CO O CO LO CO t- CO CO ▼H iH rH 1— 1 T-l lO ■^ t~ -rH lO 05 CO rt( Tt lO lO lO CO l- CI CO T-l CO CO t- t- 00 00 1-1 CO ^ CO CI C3 05 O O tH ^ T-l 1-1 CO CO Ol CO CI •rH CI CI CO '^ 1— 1 1— 1 iH 1— 1 iH CO lO T-H OO la 1* lO CO -.o t^ -t « •t^ ^ t- 1-1 lO CO -^ t1< Tj< lO "O »o CI t^ 1-1 iO o CO CO t^ t~ oo 1* C3 -T)l 03 -* CO CO 03 OS o 1—1 C5 O O CO CI O iH CI CI 00 1— 1 tH tH T— 1 T— 1 CO -f O CO CI CO -*• O l-O CO ^^ tH ^^ T-l 1— 1 Vi (M CO iH -# t- O -* CO -* -* 'it O "O CO 1-1 »Oi 05 '^ O CO O CO t^ OO C-1 t-- •^ CO 1- CO CO 03 05 tH CO tH CO C<1 O O -^ TH CI tH 1— 1 tH T-l T-l t- CO 00 -^ o CI CO CO -* lO tH tH T-l iH tH lo c; ^ a> CI en C^ O O 1-1 tH IH iH ° 00 q lO t- 05 i-l -* CO CI 01 C-1 CO CO CO CO 1-1 -* CO S3 CO "^ "^ "^ 1^ CI O 00 -r^ -t< lO lO >0 CO o t- 1-1 -)l t— T-l CO t- I- I- CO lO CO CI CO o CO o) o a: o T— 1 q q C-1 ^ CO t-- C2 M C^l CI CI CI C\ CO -t CO CO O CO CO CO CO -* '^< 1-0 CO T-l CO CO -* -* lO liO o CR CI O CO iH IC CO CO CO t- ^ t- 1-1 -^ 00 i^ t- CO oo CO s 9 05 O M ^ >o t~ T-H C? Co iH CI CI CI CI CO t~ 03 O 0^1 C^ 01 C^l CO CO T)l LO t- C5 O CO CO CO oo -* CI -* CO' CO o Tf ^ 1* ^ lO o § C5 O T-( CI CO -i* -* lO O L- CO iH 1— 1 ^H -rH 1-H C3 r-H CI CO -* iH CI C^l M CI lO O 00 03 o 01 C^ C< C