m ':■_■. \- (Btoraell Hmwjsiitg |f ib«g BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF lUettvy W. Sage yKogaSM a urn* *#?/.$«? Cornell University Library QA 281.R49 The theory of practice of interpolation; 3 1924 001 560 949 V DATE DUE I CAYLORD PRINTED IN U.S. A. ^ & Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001560949 THE THEORY AND PRACTICE ; / / I. \ OF INTEEPOLATION 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 Efhemekis, and professor of astronomy in the corcoran scientific school, washington, d. c LYNN, MASS. The Nichols Press — Thos. P. Nichols. 1899. EM ' "111." <&Q i A,\Mt>4-3 Copyright, 1899, by HERBERT L. RICE, Washington, D.C. PUEFACE. 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 thus far no distinct treatise covering the entire ground. As a consequence the author 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, the author offers the present volume. But while viewing the matter in this practical sense, the writer regards his work as no mere compilation. Many of the processes and developments are original, so far as he is concerned, and possibly 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 have been employed 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, Chatjvenet, 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 (which includes the foregoing names), it is but proper to state that references to several of the earliest writers — such as Briggs, Wallis, Motjton, Cotes, Stirling, Mayer, Walmesley, Lalande — have purposely been omitted because of the general inaccessibility of their 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 PREPACK. Special care has been given to the preparation and printing of the tables, with the hope of securing absolute accuracy. At a considerable cost of labor, 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. Rtjebsam, of the Nautical Almanac Office, and to Mr. M. E. Porter, 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. Washington, D.C., December, 1899. CONTENTS. CHAPTER I. OF DIFFERENCES , •> Section. Pago. 1. General remarks concerning tabular functions and the construction of mathematical tables, ......... 1 2. Fundamental definitions and notation. General schedule of functions and differences, .......... 3. Method of checking the numerical accuracy of differences. Theorem I, 4 4. N functions yield N—n rath differences. Theorem II, . . f> 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 are not mathemati- cally exact, . 7 8. 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 J[ n) in terms of the nth and higher derivatives of F(t)\ — equation (4), . . 15 12. Determination of the coefficients B, C, D, etc., in equation (4), . 18 13. Remarkable formal relation between the expressions for J;,"' and J' , . 21 14. The rath differences of any rational integral function of the nth degree are constant. Theorem V, .23 15. Converse of the foregoing proposition. Theorem YI, ... 24 16. 17. Convergency of differences. Magnitude of tabular interval and char- acter of function the principal elements involved. Numerical illus- trations, .25 18. Expression of wF'(t), ui 2 F"(t), etc., in terms of tabular differences, . 28 19. Change of the argument interval from ho>: effect upon the mag- nitude of the successive differences, ...... 30 20. Practical result of the foregoing investigation. Theorem VII, . 34 21. Numerical example — reduction of tabular interval, ... 34 22. Expression of any difference in terms of tabular functions, . . 35 23. Expression of any tabular function in terms of F , A'^, z/„', J^", etc., 36 Examples, 38 VI CONTENTS. CHAPTEE II. OF INTERPOLATION. Section. Page. 24. Statement of the problem, ......... 40 25. Bigorous proof 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 Newton'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, .... 61 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. Backward interpolation by Stirling's Formula. Example, ... 65 36. Further example in the use of Stirling's Formula, . 65 37. The algebraic 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. Beduction of a given tabular in- terval, ............ 78 49. Interpolation to halves. Practical rule, ...... 80 50. Precepts for systematic interpolation to halves. Schedule showing arrangement of quantities. Numerical example, .... 81 51. Derivation of general formulae for reducing the tabular interval from a) to tow, m being the reciprocal of a positive odd integer, ,. . 83 Sectlou. 52. 53. 54. 55. CONTENTS. Systematic interpolation to thirds. Example, ..... Systematic interpolation to fifths. Example, ..... On the best order of performing successive interpolations to halves, thirds, etc., ........... Interpolation, with a constant interval n, of an entire series of functions. Example, Examples, ............ Vll Fage. 88 89 91 91 94 CHAPTER III. DERIVATIVES' OF TABULAR FUNCTIONS. 56. Concerning the close relation between differences and differential co- efficients, 97 57. Practical applications of formulae resulting from this relation. Impor- tance of tabular derivatives in Astronomy, ..... 97 58. Derivation of the required formulae in general terms, .... 98 59. Formulae for computing derivatives at or near the beginning of a series. Examples, ........... 101 60. Formulae applicable at or near the end of a series. Examples, . . 105 61. Derivatives from Stirling's Formula. Rule for computing F'(t). Examples, 109 62. Derivatives from Bessel's Formula. Simple expression for F'(t-\-^u}). Applications and examples, ........ 115 63. Interpolation by means of tabular first derivatives. Example, . . 121 64. Application of preceding method when second differences are nearly constant. Practical rule for this case. Examples, . . . 124 65. Regarding a choice of formulae in any given case, .... 127 Examples, 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 the fundamental relations of §67, such that integrals whose limits are non-tabular values of T are expressed directly in terms of interpolated values of 'F, F, J 1 , z/", A"', etc. Formulae and examples, 140 72. Formulae for single integration as derived from Stirling's Formula. Schedule of functions and differences. Examples, . . . 146 73. Generalization of preceding formulae to include integrals of any limits. Example 151 74. Formulae for single integration from Bessbl's Formula. Extension to any limits. Examples, ......... 153 75. Double integration. The conditions involved, 160 76. Derivation of formulae for double integration from Newton's Formula. Introduction of the series "F. Schedule of functions and differ- ences. General formulae and relations, ..... 160 77. Value of the first integral at the lower limit. Introduction and defi- nition of the quantity Jff . Collection of formulae for double integration covering all possible cases. Examples, . . . 166 78. Derivation of formulae for double integration from Stirling's and Bessbl'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 H, ......... 188 Examples, 189 CHAPTER V. miscellaneous problems and applications. 80. Introductory statement, .......... 191 81. Problem I. — To find the sum of the ftth powers of the first r inte- gers. Application to S= l 4 + 2 4 +3 4 +. .. .+r\ 191 82. 83. Problem II. — Given the series F_ 2 , F_ ls F , F 1} F 2) etc., and an as- signed value of F n ; to find the corresponding interval n. Two solutions. Examples, ......... 192 84. Problem III. — To solve any numerical equation 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_ 2 , F_ lf F , F lt F 2 , -etc., of some function F(T) analytically unknown; to find an approxi- mate algebraic expression for F(T). Examples, .... 198 87. Geometrical problem, 200 88. Concluding remarks, 202 Examples, 203 CONTENTS. IX APPENDIX. ON THE SYMBOLIC METHOD OF DEVELOPMENT. 8cctlon. Page. 89. Introductory remarks, 205 90. Definition and operation of the symbols A, A 2 , A 8 , etc., . . . 205 91. Definition and operation of D, D 2 , D 8 , etc., 206 92. 93. Proof that the foregoing symbols of operation obey, in general, the fundamental laws of algebraic combination, ..... 206 94. Consideration of negative powers of A and D, . . . . . 208 95. Remark concerning results established in the preceding sections, . 209 96. Demonstration of Theorem III, 209 97. Fundamental relation between A and D, 210 98. Expression of A, A 2 , A 8 , etc., in terms of ascending powers of D. Demonstration of Theorem V, 210 99. Expression of D, D 2 , D 8 , etc., in terms of ascending powers of A, . 211 100. Eeduction of the tabular interval a>. Expression of d, d*, d 8 , etc., in terms of ascending powers of A, . . . . . . . 211 101. Effect of the operator 1 + A. Newton's Formula of interpolation, . 211 102. Definition of the symbol of operation V. Its relation to A and D, 212 103. Derivation of Newton's Formula for backward interpolation, . . 213 104. Expression of any difference in terms of the given tabular functions, 213 105. Derivation of the fundamental relations of mechanical quadrature. Single integration, 214 106. The fundamental formulae of double integration, 214 TABLES. Table I. Newton's coefficients of interpolation, Table II. Stirling's coefficients of interpolation, Table III. Bessel's coefficients of interpolation, . Table IV. Newton's coefficients for computing F'(T), Table V. Stirling's coefficients for computing F'(T), Table VI. Bessel's coefficients for computing F'(T), . Table VII. Giving y : For finding n when F n is given, . Table VIII. Coefficients for interpolating by means of tabular first derivatives 218 220 222 224 226 228 230 232 Bibliography, 233 CHAPTER I. OF DIFFERENCES. 1. In many applications of the exact sciences, and of Astronomy in particular, it is often necessary to tabulate a series of numerical values of some quantity or function, corresponding to certain assumed values of the element or argument upon which the functional values depend. In the more purely mathematical tables, the function is analyti- cally known ; the argument is then the independent variable of the given expression. The common tables of logarithms, trigonometrical functions, squares, cubes, and reciprocals, are examples of tabular functions of this class. A second and larger class includes those functions which are not related analytically to the argument, but which are either determined directly by experiment, or based wholly or partly upon observation. The final results are usually obtained from the fundamental obser- vations by suitable mathematical transformations or reductions, which frequently include the process of adjustment known as the method of least-squares. Empirical values are also occasionally introduced in the development of functions of this class, to supply some theoretical deficiency. In the great majority of such cases, the time is the argument of the tabulated function. This is particularly the case in astronomical tables. Thus the Nautical Almanac gives the right-ascensions and declinations of the sun and the planets for every Greenwich mean noon ; in the case of the moon, these coordinates are given for every hour, because of the rapid motion of our satellite. The moon's horizontal parallax is tabulated for every twelve hours ; the sun's for every ten days. In like manner, the readings of the barometer and thermometer A THE THEORY AND PRACTICE OP INTERPOLATION. are recorded for certain hours of the day, and therefore may be regarded as functions of the time. The velocity of the wind, the height of tide- water, the correction and rate of a clock, are further instances of a large number of physical quantities which are tabulated as functions of the time. As examples of tabular functions of the physical or observational kind, whose arguments are elements other than the time, we may mention : (a) The force of gravity (determined by pendulum experiments), as a function of the latitude ; (b) The atmospheric pressure (determined by the barometer), as a function of the altitude ; (c) The angle of refraction in a particular substance, as a function of the angle of incidence. Although differing thus fundamentally in the character of their respective functions, all mathematical tables are alike in giving the numerical values of the functions for certain assumed values of the argument, so chosen that intermediate values of the function may readily be derived by the process of interpolation. For this purpose it is convenient, though not essential, to have the assumed argument values proceed according to some law ; and since as a rule the greatest simplicity is attained where the argument varies uniformly, it is nearly always so taken. The interval of the argument is decided in general by the rapidity with which the given function varies. We shall assume throughout these pages that the given values of the argument are equidistant. The present chapter will be devoted to the subject of differences, as defined below. The student should become thoroughly and practi- cally familiar with this fundamental portion of the work before entering upon the chapters that follow. 2. Definitions and Notation. — If we have given a series of quantities proceeding according to any law, and take the difference of every two consecutive terms, we obtain a series of values called the first order of differences, or briefly, first differences. THE THEORY AND PRACTICE OP INTERPOLATION. 3 If we difference the first differences in the same manner, we form a new series called second differences. The process may be con- tinued, if necessary, so long as any differences remain. We shall apply this process of differencing to the tabular values of functions given for equidistant values of the argument. Let T designate the argument; w, its interval; F(T), or simply F, the function ; t, t -f- w, t -\- 2w, t -\- 3w, , the given values of T; F , Fi, F 2 , F 3 , , the corresponding values of F(T); J', J", J 1 ", A iy , . . . . , the successive orders of differences. The arrange- ment is then shown in the following schedule : Argument Function 1st Diff. 2d Din*. 3d Diff. 4th Diff. 5th Diff. 6th Diff. T F(T) J' J" J'" Jiv Jv Jvi t F n t + O) F x a h * + 2(0 F> a i ^ c o d t + 3o> F 3 a 2 \ c i F, « 3 h c 2 d 2 «i t + 5o> F, « 4 h c 3 t + 6o> F a «5 where a =F l — F 0i a x = F 2 — F u . . . ; & = #i — « , &i = & 3 — a i> • • •'■> c = \ — b , c x = b 2 — & n . . . ; and so on. "We shall also find it convenient to represent a , a t , a 2 , . . . . by 4/, /*/, J 2 >, . . . . , respectively; b , 6,, b 2 , .... by J a ", ,J t ", j*, , etc., Thus, generally, J[ n) denotes the (s-f-1)* 11 value in the column of n th differences. As an example, we tabulate and difference several successive values of F(T) EE T'— 10 T 2 — 20, thus : THE THEORY AND PRACTICE OF INTERPOLATION. T F(T) J' J" /!»' /]iv J-" - 20 - 9 1 - 29 - 15 - 6 + 36 2 - 44 + 15 + 30 + 60 + 24 3 - 29 + 105 + 90 + 84 + 24 4 + 76 + 279 + 174 + 108 + 24 5 +355 + 561 + 282 6 + 916 The differences are in all cases formed by subtracting (algebrai- cally) downwards, as in the above examples. It will be noted that the even differences (j", A iv , . . . . ) always fall on the same lines with the argument and function, while the odd differences (j 1 , J 1 ", J v , . . .) 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 A\ we find -9 - 15 + 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 4/" +4/" +4/" = +36 + 60 + 84 = +180 = +174 - (-6) = A 3 "-A Q " These relations may be expressed generally as follows : Theorem I. — The algebraic sum of any s consecutive values of A in) , is equal to the last, minus the first, of the s-\-l consecutive j(n-i) terms used in forming the s values of A (n K To prove this proposition, let the differences be as below : ^ ■ K k K *_! *. THE THEORY AND PRACTICE OF INTERPOLATION. 5 Then, from the definition of differences, we have \ = A 2 — \ , /c 2 = A, — A 2 , , &,_! = A, — /*,_! , /;, = A J+1 — A, Hence, by addition, we find k, + *, + A 8 + + /,-„_, + A. = A, +1 - A, which is the algebraic statement of Theorem I. This theorem may obviously be applied as an independent check upon the numerical accuracy of the differencing. 4. Theorem II. — If the differences of N values of F(T) are taken, N~ — n values of J M are derived ; it being assumed that JV>n. For, JV functions evidently yield _ZV — 1 values of J 1 , N—2 values of J", N— 3 values of j>", etc. ; hence JSf values of F(T) yield JV — n values of //<"». 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 1 F(T) J' J" J'" J IV ' Jv 6 5 4 3 2 1 + 916 + 355 + 76 - 29 - 44 - 29 - 20 -561 -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 j» v (the even orders) remain unaltered in sign. 6 THE THEORY AND PRACTICE OE INTERPOLATION. To prove that such an effect is true generally, we consider the two series below, the second series being an inversion of the first : F(T) J' A" Jin Jiv F„ F b «0 «2 d F(T) A 1 J" j"' zjiv F r F t «o ft F, «i ft y So F 2 «8 /So yi s, F, «3 ft y 2 F Cl i Comparing the first differences, we find = F t - -F 5 = -(*;- -F>) = — a 4 1 = F s - -F A = -(F t - ~F S ) = — a 8 2 = F 2 - -F s = ~(F S - -F,) = — a.. Hence, for the second differences, we obtain ft = «1 — «0 = — «8 — (~0 = «4 — ft S = ft = «2 — «1 = ( — a s) = J" jiii Jiv 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 16.00 19.46 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 by Theorem IY, 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 unit of the second place, f may have any value from — 0.5 to — j— 0.5, in terms of the same unit. Moreover, the values of f do not follow any law of progression, but proceed at random, with arbitrary changes of sign. Hence, the differ- ences of f 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 IY that the differences of F must equal the sums of the corresponding dif- erences of F and / ; 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 ; whence f follows, in units of the second decimal, and also the differences of f to the fourth order : THE THEORY AND PRACTICE OF INTERPOLATION. T F(T) F(T) f=F-F J' J" J'" Jiv 2.0 16.00,00 16.00 0.00 + 0.19 + 0.25 -0.85 + 0.65 -0.49 + 0.49 -0.65 + 0.85 2.1 19.44,81 19.45 + 0.19 + 0.06 -1.16 + 2.60 -2.64 + 2.12 — 2.12 + 2.64 2.2 23.42,56 23.43 + 0.44 -1.10 + 3.76 2.3 27.98,41 27.98 -0.41 + 1.50 -5.24 2.4 33.17,76 33.18 + 0.24 -1.14 + 4.76 2.5 39.06,25 39.06 -0.25 + 0.98 -4.24 2.6 45.69,76 45.70 + 0.24 -1.14 + 4.76 2.7 53.14,41 53.14 -0.41 + 1.50 2.8 61.46,56 61.47 + 0.44 We now bring together, from the above tables, the fourth differ- ences of F and /, denoting these quantities by (J W )F and (z/ lv )/, respec- tively. The fourth differences of F then follow, since we have shown that (J iv )F = (J iv )F + (4 lv )f; thus we form the table below : (A*)F (A lv )f (jiv)F + 0.04 -0.05 + 0.05 -0.04 + 0.05 + 0.03,76 -0.05,24 + 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) are absolutely uniform, — that is, the irregularities in (J iv )F and (z/ iv )/ ex- actly correspond, or balance. 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 valuable exercise, the student should now difference the function F directly, and find the fourth differences exactly as above deduced. 8. Detection of Accidental Errors. — We have just seen how 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 appreciable quantity, an in- spection of the differences will indicate definitely the location and magnitude of the error sought. 10 THE THEORY AND PRACTICE OF INTERPOLATION. To investigate the principle that underlies the method, let F a , F lt F„ F t , F t , F t , denote the correct values of any function F (T) (tabulated for equi- distant values of T), and let the differences be as shown in the schedule below : F(T) A 1 A" J 1 " Jiv Jv F a K F„ «i h Co d. F s F, F, F 6 F, F a F, F l are brought into practical uniformity. In the column c x & c 2 are given the corrections to A m , according to the binomial numbers, 1, 3, 3, 1. In the next column, the sum z/'"+c 1 +c 2 is written, which evidently requires a third correction, tabulated under c 3 . The differences are now sufficiently smooth. Since c 3 corresponds 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° 54' 49".8, respectively. It occasionally happens that some order of difference clearly THE THEORY AND PRACTICE OE INTERPOLATION. la indicates a correction corresponding to the binomial coefficients of a lower order than that of the difference in question. This means the existence of an error in some earlier order of difference, rather than an error in the column of functions. For example, if d" requires a correction of the order 1, 3, 3, 1, it follows that an error exists in J", since J y is the third difference of A". More generally, when A (n) requires a correction according to the binomial coefficients of the m th order, an error exists in A ( "- m) . These remarks imply the necessity of some caution on the part of the beginner. It will be observed that when either the first or 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 error. 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-compute the values in question, when the data for the calculation are available. General Properties of Dieeerences. 11. Let F(t), F(t+o>), F(t-\-2a)), represent any series of tabular functions, whose differences are taken as in the schedule below : Function, F(T) i' j" j'" j<«i J(n+D . . . F(t) 4/ F(t + m) 4' 4," 4/» F(t +2o>) ■ Ai A" z//" F(t+3a) A" A'" 4T n 1 A («+l) z/ 2 <" +1 ' F[t+S(o'] • 4' Flt+is + l)^ ^' !+ , 4," 4'" if[if+(s + 2)(o] ^" s+ i ^'".+i J(») •^s+l jt i+i) ^s+1 16 THE THEORY AND PRACTICE OE INTERPOLATION. We shall assume that F (T) is a finite and continuous function, and that F(t-\-sco) is capable of expansion in a series of powers of sco, within the limits of the given table; then, denoting the successive derivatives of F(T) by F\T), F"(T), etc., we have, by Taylor's Theorem, the following expressions : F(t) = F(t) F(t + «) = F(t) + o,F (t) + £ F» + £ F>» (t) + £ F» (t) + . . F(t+2») = F(t) + 2^(t)+ 4|V"(0+ 8^F'»(t)+ 16 £ F» (f) + . . , (()) F(t+3u>) = F(t) + 3«>F' (t) + 9^F" (t) + 27 ^F'"(t)+ 81^F iy (t)+. F(t + 4 (t) '+ 256 ~ F iv (t) + . c. 12. li Differencing these values of the functions in the usual manner, we obtain successively the expressions for A 1 , A", A 111 . . . . , as follows : A > = « *"(Q + £ F"(t) + | F'"(t) + £ F»(t)+ .... 4' = u>F'(t)+ 3 ^. ^''(i) + 7 -j^ J"''(f) + 15 ^ F*(t) + . AJ - ^'(J) + 5 ^ F"(t) + 19 ^ F'"(t) + 65 -^ ^ iv (J) + . I |3_ L_ <,l 2 G) 8 u 4 4/ = *F"(t)+ 8 .F '»(*) + ff ^F^(t) + . (1) (2) 4/" = ^F ln {t)+ %w i F i "(f)+ .... 4"' = U) 8 F '"(<) + fa) 4 i^ iv (0 + .... \ (3) It will be observed that all terms of the expansions (0) are of the general form, K = Au r F^(t) + Ba> r+1 F^ l) (t) + Cu> r + 2 F< r+!i \t) + ZW+ 8 F^ s) (t) + . . . . Let us now assume what appears from (1), (2), and (3) to be the general law; that is A = 1 r = n leaving the coefficients B, C, D, We therefore assume undetermined for the present. .JJ" 1 = a)"^'" (*) + Bo,p+i.F<»+i>(*) + CV'+ 2 J F , <"+ 2 > (t) + Z>a>"+ 8 2^" +8) (if) + (4) Since the value of t is arbitrary, we may write t -\- w for t ; by making this substitution in the right-hand member of (4), we evidently get the expression for the n tb difference immediately following J\"\ — that is, the value of ,/^\. Hence we have J('0 X = o)»i?' ( " ) (< + (o)H- J Ba)"+ 1 jP'<"+ 1) (« + o))+C'a ) * , + 2 i? , ("+ 2, (i + a ) ) + 2V'+ 8 .F<"+ 8 > (* + a/) + . . . Developing the functions of the right-hand member by Taylor's Theorem, we find F^ (t) + uF^" (t) + ■£- F^'™ (t) + — F^+v (t) + [2 [S + Ba> n+1 + CV+ 2 + Da>"+ 8 F'"+ 1] (t) + o)i?'<"+ 2, (t) + ~F<"+ S) (t) + F (n+*l(t) + 0>F<» +S) (t) + ^(,,+8) (^ + _ _ + Collecting the coefficients of F (n '\t), F (n+i) (t), . . . ., we obtain joo = m -F^ (t) + (B+l)w" +l F^+ 1) (t)+ I C+B+-U«+ 2 F<»™(t) (5) + (D + C+^+±y»+*F^(t)+ . . . . Subtracting (4) from (5), and observing that .J<£- /;."> = ,./y +1 \ we get + fx» + - +— + -) u'^FW (t)+ . . . . . Viiiiii/ 18 THE THEORY AND PRACTICE OF INTERPOLATION. If, therefore, we put B> = £+± C '= C+ I + 1 ) (6) we have / J<"+»= u> »+W+»(t) + B'o, n+i F ("+"(<) +C' 0i " +S F^ +S) (t) + D'a> n + i FC+ 4i (t)+ . ... (7) Hence, if the general form of expression assumed in (4) is true for the index n, it follows from (7) that it is also true for n -\- 1 ; but we see by equations (1), (2), and (3), that the law obtains for n = 1, 2, 3, respectively; 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. We have now to determine the coefficients B, C, D, . . . . , of equation (4). These quantities are evidently functions of n and s, and will be determined in the following manner : ■ First, we take s = 0, and determine the constants for j^, which we shall denote for this purpose by B n , C„, D n , .... These values are found by induction, thus: the relations (6) give B n+1 , C n+1 , D n+l , .... in terms of B n , C n , D n , . . . . Making n = l, we take B x , C 1; D v , . . . . directly from the first of the equations (1) ; a continued application of (6) therefore gives succes- sively the values of B 2 , B 3 , B i} . . . . B n _j, B n . Similarly, we derive G n , D n , . . . . Hence, the coefficients of (4) become known for s = 0. Second, the coefficients of J[ n) easily follow from those of z/<,"'; for it is clear from the schedule of §11 that z/<»» is related to JF(t-\-sa>) in precisely the manner that z/{,"> is related to F(t). Hence, if for brevity we write we shall have, since the value of t is arbitrary, THE THEORY AND PRACTICE OF INTERPOLATION. 19 Then, expanding yr(t-\-sa>) in a series of powers of sw, we arrive at an expression of the form (4), in which the coefficients are fully determined functions of n and s. To perform the steps indicated, we take from the first of the equations (1) the following values: A = i C x = i A = ,V • • • • (8) To find B n : By repeated application of the first of (6), we have A = A + i B s = A + i A = A + £ A-i = B„_ 3 + i B n = B n _ x + i Hence, by the addition of these n — 1 equations, we get A. = A + i(n-l) = i + *(n-l) = ^ (9) To find C n : Using the second of (6), we obtain c. = c, + i^ + j C 3 = C t + i A + i A = C„_ x + i A-> + i whence, by addition, we find C a = C x + HA+ A +....+ B n _ x ) + i («_1) Since C x = I , this gives c u = *(A+A + ....+ A_0 + 1 = i 2 B - + But, from (9), we have B r = ^ ; hence we get r=ii— 1 = i Zj '• + a = i n C. 6 24 To find D„: Again, from (6), we derive A = D x + iC x + $B x + & A = A + * c 2 + * A + A A = A-! + * G n _ x + i A-! + *V + H=^(3»+i) (10) 20 THE THEORY AND PEACTIOE OE INTERPOLATION. whence r=\ r=\ r= 1 From (10), we have ~n(n — 1)" + 24 or l>. = Jg(»+1) (11) In like manner, the process might be extended to the values of E n , F n , . . . . ; but the results already obtained are here sufficient. Substituting in equation (4) the values of B n , C n , and D n , given by (9), (10), (11), (remembering that these values suppose s = 0), we have (12) J<"> = (0 + -je (» + l) a."+ 3 ^("+ 8) (0+ • • We now obtain from (12) the expression for z/<">. As already proposed, we .write JJ, n > = *(*) = " +1 F( n +»(t)+ C„"+ 1 Fl»+» (t) + C n o. n+2 J ?'<"+ 2 » (t)+D n m n+s F^'^ (t) + + su (w"F^+V(t) + B n u n + 1 F^(t)+C n o> n + i Fi"+»(t)+ . . . . + —(^F^+v (t) +B n a, n + 1 F( n +v (t) + + £Y(< u »i^>(0+. . . .) +. Upon arranging this expression according to ascending powers of &>, we get + lD„ + C n S + _S- + — j ««+**■<*+« (*) + . . . . Hence, substituting the foregoing values of B n , C„, and D n , we THE THEORY AND PRACTICE OP INTERPOLATION. 21 find that the values of B, C, D, .... in equation (4) aiv as follows: 11 c j> '- i+2 < 12 Wn + 1) These results are easily verified by substituting special values of n and s, and comparing with the coefficients in equations ( 1 ) , ( 2) , ( 3) ; thus, putting s = 1, and taking n = 1, 2, 3, successively, we ob- tain the numerical coefficients in the expansions of J t ', ./",, and //", respectively. 13. Remarkable Formal Relation between the Expressions for Jj") and J >. — The coefficients B n , C n , D n , in the expression for ./;", may also be determined by the following method, which not only is shorter 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 // l " ) = m"F^ (I) + B„ or+'i^'+D (t) + C„a ) "+ 2 ^"+ 21 (t) + . . . . (15) We now let +2 yit+S 7/"+ 4 ?„ +1 (y) = r +1 +^+^-+^- + - ■ • • (yJl+S ytl + i yTl+5 v" +2 + — +^— + — + ■ ■ ■ y li li It / 7/ M + 4 7/ n + G 7/«+f» + £„U"+ 3 +— +— + — + • • ■ V li li li / 7 .n+5 /wn+6 7/ n + 7 ^ n \ y 15. li li + ^Vlilili T nj V li li li / ?/ 2 « 3 V 4 + Hence, by (15«) and (17), we have -qp>.+i = r h ■ i 9> B ?3 = «_2 % = l)" (18) Therefore, by (17), we have *M-(»+HHr + -----M 1+ £ + £ + T + ••• (0+ ^ by J , J*, J*, . .J;;; and for <»i?"(<), w 2 F"(t), a> 3 F'"(t), .... ^F^ (t) let us write the symbols Z>, X> 2 , !>*, .... Z>", respectively; then we shall have IP l) 3 ./> 7/ ,./. = /;+ — +—+—+ — + .... 12. 12. li li / D- I) a D* Y 2 7 1 ' f --(* + F + t + r + ) -'"+^ +I5 ^+i^+. ■ • ■ /■ Z> 2 7/ D* V II 11 li 14. Theorem V. — T/te n th differences of any rational integral expression of the n th degree are constant. If the general form of the function is F(T) = aT*+/8T*- 1 4-yT"- 8 + . . . . , the constant Value Of J'" 1 is a>"a \n . For, from the nature of the function, we have, evidently, & U) F and *<"+»(*) = F<»+*\t) = .... =0 Hence, from (4), we have z/i"' = oi n F {,,) {t) — u>"a\n (22) The theorem is therefore true, whatever the value of the constant interval w. Several examples have already occurred: in §2 we have 24 THE THEORY ANP PRACTICE OE INTERPOLATIOK. the differences of F(T) = T'— 10 T 2 — 20; here » = 4, a = 1, to = 1. Hence, by (22), we get Jiv = H = 24 — the value already found by differencing. In Example I of §9, F (T) = .T 3 , a. = 1; we there obtained for the value of the third difference z1"> = 6 which agrees with the theorem. Again, in Example I of §10, F (T) = 2T 3 — 25 T— 40, a = l; whence the theorem requires J'" = «|3 = 211 = 12 which is the result already obtained. 15. Theorem VI. — If the n th 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"+/32 7 "- 1 + r 2 7n - 2 + .... This proposition is the converse of Theorem V, and is proved as follows : Let F(T) denote the function whose true mathematical values, tabulated for the given values of T, form the given series of quantities. From (4) and (5), we see that the expressions for J< n) and j^ agree only in their first term, o n F (n) (t) ; the remaining terms of like order in o) having unlike coefficients. Hence, the conditions necessary in order that z/ (n) shall be constant throughout are as follows: First, that a) n F in) (t) does not vanish; Second, that ( o n+1 F^+ 1) (t) = (o n+2 F in+2) (t) = . . . . = 0; But, since w cannot vanish, these conditions reduce to the form — (23) F<-"> (t) > < F«*»(t) = F(«+»(t) = .... =o If now we put T = t + t (24) then, by Taylor's Theorem, we have THE THEOBY AND PRACTICE OF INTERPOLATION. 25 F(T) = F(t. + r) = F(t) + rF'(t) + — F"(t) + . . + - F<"> (t) + — 7<<"'+» (t)+ . . |2_ |» |n-j-i By (23), this gives F(T) = F(t) + T F'(t) + . . . .+ — Fi-»it)+ — Fi"(t) (-'<">) |n— 1 In in which, we observe, the coefficient of t" cannot vanish. Substituting in (25) the value of t given by (24), we obtain WH (t\ V^(f\ F{T) = F(t) + (T-t)F'(t) + (T-tf^—^-+. . . . + (T~t)"~ ^ li 12 Since ^ has a fixed value, the right-hand member of this equation is an -expression of the n th degree in the variable T, and hence may be written in the form F(T) = uT" + pT"- i + y T'>-*+ .... (26) which establishes the theorem. 16. Convergence of the Differences in Practice. — In the discussion of Theorems Y 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 order of difference progresses. The explanation of this tendency follows readily from equation (4), thus: for smj given function, the derivatives F w (t), F {n+1) (t), ^ n+2) (t), 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, except the first. When this condition obtains, the value of J l,,) is sensibly constant, and equal to o n F (n) (t). The differences of F (T) are thus practi- cally brought to a termination at the n th 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 THEORY AND PRACTICE OP INTERPOLATION. to the obvious statement that, when a function is to be tabulated so as to difference readily, the interval of the argument must be decided by the manner in which the given function varies. To exemplify these principles, we take the following table of seven-figure logarithms : r LogT 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, a> = 0.01, t = 1.00, t + a> = 1.01, t+2co = 1.02, etc. To serve our present purpose, we here transcribe from (1), (2), and (3), the following expressions: A> = a F'(f) + 2- F" (t) + -g F"> (f) + 2j ^ iv (0 + z/ " = a*F"(t) + w*F'"(t)+^ < u 4 J F ,lv (0+ . . • J >" = a s F'"(t)+§ < u 4 i?' iv (*) + .... (27) z/ iv = u> 4 F iv (t) + Since F(T) = log T, we have F'(t) = + Mt- 1 , F"(t) = -Mt' 2 , F'"(t) = +2Mt-* , F w (f) = -QMt^ , . . . . where Mis the modulus of the common system of logarithmSj = 0.434294. Hence, with t = 1 and w = 0.01, we find uF' (t) = +0.0043429,4 v'F'ilt) = -0.0000434,3 a *F"i(t) = +0.0000008,7 oAF* (t) = -0.0000000,3 Substituting these numerical values in (27), we obtain, in units of the 7th decimal, A J = +43214 J " = -426 d»> +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. to be due to the small value of the interval w, which makes the term a> 3 F'" (t) appreciable, but renders w 4 i^ lv (t), ^F\i), .... quite insensible; accordingly, J'" is the last difference which we need take into account, the remaining- differences being practically zero. We may add that if the values of T in the present table were 100, 101, etc., instead of the given values, the interval to would become 1 instead of 0.01, and hence to, or, to 3 , to*, .... would not converge as above. We should then, however, have t = 100 instead of 1, which would cause the successive derivatives to converge rapidly, as is obvious from the general expression F^{t) = (-1)'- 1 M\n-\ . - Furthermore, the differences of F(T) contain only terms of the form K«rF^ (t) (-ly^KM [n-W - where K denotes a numerical factor; hence, since the values of to and t are both increased one hundred-fold by the assumed change, it is evident that the general term Kto n F 8 «£ In this example we have o) = 0.05 a = 1 and hence A"< = (0.05) 8 X6 = 0.00,075 which is insensible when only two decimals are concerned. Thus, in the approximations so frequently used in practice, the differences generally terminate (either absolutely or approximately) at some order earlier than would occur if the true mathematical values of the function were employed. 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«T 2 + /3T+y Thus, in particular, if the student tabulates the function F(T) = 16 (T 2 -5.3325 T+9.476975) for T = 5.16, 5.21, .... 5.46, and retains all decimals involved, he will find his tabular numbers identical with the above series. 18. To Express w n F (n) (t) in Terms of z/;,">, j%*», z/<"+ 2 », etc. — The problem consists in reversing the series (15), which expresses jp> in terms of w n F w (i), w"+ 1 i^ (B+1 >(^), .... Let us denote io r F Kr) (t) by x r ; then, writing successively, n, w-f-l, n-\-2, .... for n in (15), we have THE THEORY AND PRACTICE OF INTERPOLATION. 29 ,/{,"> = a-„+B„^ +1 +0, l+2 +D„.r„ +s + •••■ \ from which we obtain, by transposition, ■'•„ = H"' - ■«„■'•„+! - C n x„ + „_ - Z)„r„ +n - X n+l == ^o" — -»„+! •<■„+:> — C, l+1 X, |+8 — -D„ +1 .r, 1+4 ^. /(n+2) /} r r" t J~> t (29) The second of the equations (29) gives a value of x n+l , which, substituted in the first equation, gives*aj„ in terms of ./;/", z/i" +u , .r„ +2 , * B+ .,, . . ; substituting in the latter expression the value of x n+2 given by the third of (29), we find x n in terms of ./<•", ^»+ 1 », j$-h>, a;„ +s , x n+i , .... Con- tinuing this process of elimination indefinitely, we arrive at an expres- sion of the form ,,•„ = „**-<« + K // , '+- + D n i/ n + s + . . . . \ •r.n-i — v T/> .+ij ~ W+iy "-^11+12/ ~ • • • • r „ 1 m — „»+2 4- R ,,»+s 4- n ,M1 + 7) »,»+•■■ + / v'' 1 ) 'Pii+2 — y ' - D »+2?/ ~ ii+2y ~ ■ U n+S'j ' • • • ■ Comparing (28) and (31), we observe that the latter group may be obtained from the former by writing q r and y r for /J, r) and x r , respectively; the algebraic relations in both groups are otherwise identi- cal. Hence, if from (31) we seek to express y H in terms of q>„, (f n+ i, q> H+2 , . . . ., the. process of reversion will be identical with that which gives ,r n in terms of /;/", J;; ,+1, J . . . . ; hence we must find >/" — (9 = Ji-'-^4-+"+ 2 * (3n+5) ^<"+ 2) - ^ (w + 2) ( w + 3) ^"+ 3 » + . . (44) Using the symbolic notation adopted in (21), we have the follow- ing expressions : D = J.-iJ^ + i^-i^ + i^- . . . D*= (J -i^+ i J^-iJ^+ . . .y = 4 2 -4 3 +H4 4 -Mo 6 + d 8 = (j t -ij*+ij*-ij*+ ...)» = 4,»-i4«+i4»-¥.4, , + (45) 2)-= (J -iJ 2 +^ 8 -iJ 4 + . . .)" = ^ 2 JJ+ 1 + ^ (3^ + 5) zf »+ 2 - Ji (»+2) (n+3) z/ »+« + . . . 19. ^fec^ o/ a Change in the Argument Interval to, upon the Magnitude of the Several Orders of Differences. — Let us now suppose THF THEORY AND PRACTICE OP INTERPOLATION. 31 that a second tabulation of F (T) has been made, differing from the first only in the value of the interval, to. Let to = raw 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' V t + iiiu> F(t + una) 8„" V" 8/" 8."' t + 2mto F(t + 2m n+1 F("+»(t)+ C n m» +2 ta"+*F<"+ 2) (t) + . (46) If, as above, we write ,v r for to r F (r) (t), this equation becomes 8V" = m".r„+ B n m"+*x n+1 + C„m»+ 2 .r„ + , + . . From (30) we obtain, in succession, Xii+1 = 4v +I) + ft„+i 4" +2) + «„+i^" +8, + *.+. 4" +4, + (47) (48) Eliminating x n , x n+1 , .... from (47), by means of (48), there results an equation of the form 8<»> = >»•• j<»> + p„ Ji»+ 1, + y* / i" +2> + • • • • ( 49 ) which, for n = 1, becomes 8' = «J ll ' + AV' + r 1 J.'"+ • ■ • • ( 60 ) 32 THE THEORY AND PRACTICE OP INTERPOLATION. Now let «,, = »>">/' + -B„wi" +1 ?/" +I + C„m"+ 2 2/ n+2 + . ... (51) be an auxiliary expression, such that the coefficient of y' ft+r is the coefficient of x n+r in (47). From (33) we obtain, in succession, y" = yl + Ktf +1 + Cntf + * + dntf +s + .... | y n+1 = ?i' +1 + &» +1 <#« + c „ +1 <#« + rf„ +1 <^+* + . . . . V (52) Now, to eliminate y", 2/ M+ S • • • • from (51) , by means of (52) , we must perform precisely the same algebraic steps as in the deriva- tion of equation (49) from (47) and (48) ; we shall therefore obtain z, = W'cpl +A,g);' +1 + y„j(my)j" = ®» Hence, from (57), we find (59) (GO) *„ = i , m(m — 1) „ m i 2 + >(m — !)(/»— 2) |2 |8 Expanding and factoring, we obtain «,. = m n <' + g m" (m - 1) qtf+i + — (3re + l)m-(3w + 5) 91 + 48 rt(9i + l)w' 2 -2(9t 2 +3re + l)m+(M + 2)(?i + 3) <7i + • ■ • TO"(?tt — l)gij+ s in"(m — 1) g>;* +8 + (61) Equating coefficients of like powers of ^ in (53) and (61), we have P« = 2 w "( m - 1 ) . V» = 24 wt " ("*-!) (3« + l)w.-(3w+5) (62) Substituting these values in (49), the latter becomes 8J») = m" jy> -|- - !«,» (m — 1) J, ( ," +11 + ST to" (jh - 1) (3» + l)?ft-(3» + 5) + _ m -(„ { _l) J{," +2) (63) J<"+ 3 » + n(n + l)w--2(n 2 +3n + l)m + (n + 2)(n + 3) Finally, we may symbolize these results by the following expres- sions: (64) X - „, / j_ m ('"— 1 ) / *x '"('»-!) 0~ 2 ) w(m-l). .(m-3) w (»t-l). .(/n-4) 6 ° ~ '" J °+ j» '■'» + jj[ ° + ~ [i ° + j[ 4. + v = (,j + ^ii ^ k»-ik«-3) . ,. + y = »i'J «+ '" 2 ('«-l) ■ V+ J2 C'"- 1 ) (7'" -11) J,i 4 + jo (w*-l)(»-2)(3»»-6) . V+ 8/ = («4,+ " (W ~ 1) .V+ ) 8 = '"U*+ I m 8 (m-l) J/+ ^(m-l)(5m-7) J/+ . . . . V = ( W J t + W(W ~ 1) J. , + ) 4 = '«H 4 + 2w«(«-l) J/+ £ (»«-l) (13m-17) J/+ / , «*(»*- 1) ,, Y V = ^j + v £ ; J 2 + j = w 6 J„ 6 + « ?» 6 (>»-l) ^ 6 + ir »« 6 (»»-l)(4»i-5) J„'4- 34 THE THEORY AND PRACTICE OE INTERPOLATION. 20. Theorem VII. — If the n th differences of a given series of functions are numerically large as compared with all the following differences, then, if the series be re-tabulated with the argument interval m times its original value, the n th differences of the new series will be approximately m n times the corresponding n th differences of the original series. The theorem is a direct interpretation of equation (63). For, if z/{, n+1) , J< n+2 >, . . . are all small in comparison with ^<»>, then the ap- proximate value of S (n) is m"z/ (n) - Corollary. — If the n th differences of the given series are con- stant, then the n th differences of the new series are also constant, and equal to m n times the original n m differences. For, if J("> is constant, S H+1) , ji"+*\ .... are all zero, and hence (63) gives, rigorously, S (n) = fflM 1 ' 1 21. To illustrate the foregoing results, we take the following table of cubes: T F(T)=T S J' J" j'" ' 100 103 106 109 112 115 1000000 1092727 1191016 1295029 1404928 1520875 + 92727 98289 104013 109899 + 115947 +5562 5724 5886 + 6048 + 162 162 + 162 Here the interval V" = A4/" (65) From the first of the above tables, we take 4/ = +92727 J " = +5562 J/" = +162 Whence, from (65), we derive 8 ' = 30909-618 + 10 = 30301 8 " = 618 -12 = 606 S "' = 6 which agree exactly with the values found in the second table above. It will be observed that S ' and S " come within 5 V part of equaling ^ J ' and £J ", respectively; while S "' = ^\J V '", exactly. These rela- tions are in accord with Theorem VII. 22. To Express the Differences of F(T) in Terms of the Given Functions only. — Let the given series be F Q , F x , F 2 , F 3 , . . . . ; then the first differences are F x — F , F 2 — -F\, F 3 — F 2 , . . . .; the second differences, F 2 — 2F 1 -\-F , F 3 — 2F 2 + F x , . . . . ; the third differ- ences, F a —SF a -\-3F 1 —F 0i ^4 — 3^3+3^—^, . . . . ; and so on. The coefficients evidently follow the binomial law. Thus we have generally (66) zT = F n - vF„_ 1+ !fc^ F„_ 2 - . . + (-l)\C,.F n _ r ± . . + (-l)»-hiF, + {-1)'F in which, according to the usual notation, we put n C r for the co- efficient of af in the expansion of (1-f-*)"- To prove (66), let us assume it true for the index n; then the expression for the n th difference immediately following J ( y" (i.e., z/<»>) will be obtained by increasing the subscripts of F„, F n _ ly .... in (66) by unity. We therefore have 4»> = i r n+I _^, i+ !fcl)^_ 1 - . .+(-iy+\C r+1 F n _,.± . . + (-l)vF, (67) Subtracting (66) from (67), we find (n + l)n JU+v = j { n)_ j(») = Fn+i _^ l + l)F n + K —^F n _ 1 - .... + (_l)^( n C r+1 +„(7 r )^_ r ± . . .+(-iy{n + l)F 1 +(-!)-» F 9 36 THE THEORY AND PRACTICE OE INTERPOLATION. But, as proved in Algebra, we have and hence the preceding equation becomes Jfr+» = F Mi -(n + l)F 1 ,+ (n-\-V)n F„_ (68) . + (-ir\ +l C,. +1 F„__ r ± . . + (-l)»+ 1 -F It follows from (68) that if the law expressed in (66) 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 = 4, and so on indefinitely; the equation (66) is therefore true for all positive integral values of n. 23. To Express Any Function of a Gfiven Series in Terms of Some Particular Function (F ), an d of the Differences (a , b , c Q ,- . . . .) which Follow that Function. — As before, let F , F x , F 2 , F 3 , . . . . denote the given series, the differences being taken as in the schedule below : F(T) F x F 3 F A F n -r F n F„,, J" jiii Jiv Jv Jvi A Let it be required to express F n in terms of F , a , b , c , d , . From the nature of the differences, we have F, = F + a F 2 = F 1 + a, = (F + a ) + (a +b ) = F + 2 % + b F s = F 2 + a 2 = (F +2a + b ) + (a + 2b +c ) = F + 3a + 3b + c THE THEORY AND PRACTICE OF INTERPOLATION. 37 and so on. The coefficients again follow the binomial law, which sug- gests for the form of the general term — n(n-l) 1 n(n — l)(n-2) F n = F + na + A__J j o+ J A > c + . . . . (69) To prove (69) by induction, we assume that it is true for the index n. Moreover, we evidently have K+i = K + a„ We may now find a n in terms of a , b , c , d , . . . . from (69), — since the relation is here the same as the relation of F n to F Qi a , b , c , . . . . ; thus we obtain , n(n— 1) a, = a<,+ nb +-±-- — '- c + .... Adding this value of a n to that of F n given by (69), we find* ,, „ (n + l)n , (ii+l)n(n—l) ,_ n . K-h = F % +a K = F +(n+l)a +^— ±- b + ± LA 1 c + .... (70) .Thus, having assumed the relation (69) to be true for the index n, 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 THEOEY AND PRACTICE OE INTERPOLATION. 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 the table. 3. Verify the accuracy of both the functions and their differences in Examples 1 and 2, by noting the degree of regularity in A y , accord- ing to the method of §8. 4. Also, rigorously check the differencing in the above examples, by taking the algebraic sum 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 z/ v are the sums of the fifth differences of the other series, according to Theorem IV. 6. Correct the errors in the following tables by the method of differences : id) (*) («) l T r(T) = ^, 0.21 4.762 .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 / // 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 o' 0.00 2 48.02 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 THEORY AND PRACTICE OF INTERPOLATION. 39 (d) W (/) T f ( t ) = r 8in r 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 o t n 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 58 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 computed values. (a) F{T) = T e - 50T 4 + 100T 2 . (Tabulate for T = -8, -6, -4, -2, 0, +2, +4, +6, +8.) (ft) F(T) = 2T 8 -7T-400. (T = 8.0, 8.3, 8.6, .... 9.8.) (c) F(T) = 0.16T 4 -0.3T 8 - (T = 2, 3, 4, 5, 6, 7, 8.) 8. By means of the first of equations (1), compute the value of J' which immediately follows log cos 56° in the table of Example 2. The value of eo (= 3°) must be expressed in circular measure. Com- pare 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 differ only in w, the interval having now been doubled. Then, in the second of the equations (64), put m = 2, and substitute from A the values of J ", J "', J iv , 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 tT and i^ 5 (~log50), by (66) and (69) respectively; compare the results with the values found in the table. CHAPTER II. OF INTERPOLATION. 24. Statement of the Problem. — Given a series of numerical values of a function, for equidistant values of the argument, it is required to find the value of the function for xiny intermediate value of the argu- ment, independently of the analytical form of the function, which may or may not be given. Interpolation is the process or method by which the required values are found. Without certain restrictions 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 day. 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 interpolated with accuracy. The Nautical Almanac gives the heliocentric longitude of Jupiter. for every 4th day; but, because of the slow, continuous, and syste- matic character of Jupiter'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 Almanac for every twelve hours; 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 appears that, to render the problem of interpolation determinate, the tabular interval (w) must be sufficiently small that the nature or law of the function will be definitively shown by the tabular values in question. The condition thus imposed will be satisfied when, in a given table, the differences become either rigorously or sensibly constant at some particular 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 sensibly true, according to the foregoing distinction. 25. Extension of Formula (69) to Fractional and Negative Values of n, Provided the Differences of Some Particular Order are Constant. — We have shown (Theorem Y) that the differences of a rational inte- gral function vanish beyond a certain order. We proceed to prove that, for any such function, the formula (69) is rigorously true for all values of n. Let F(T) denote any function whose differences become con- stant at the order i, and let j (,) = l ; F(T) and its differences are then shown in the schedule on following page. ♦Excepting, of course, any periodic function whose tabular interval (cu) differs but little from some multiple of its period, P. An example of such a series is the following : Date, 1898 Day of the Tear Heliocentric Longitude of Mercury A> J" J"' Jan. 4 Apr. 4 July 3 Oct. 1 Dec. 30 4 94 184 274 364 / 93 105 33 117 40 129 14 140 10 / +12 33 12 7 11 34 . +10 56 / —2(5 33 —38 —7 —5 where P (the time of one revolution of Mercury) = 87.97 days; and hence m = 90 d = P + 2''.03. The differences A' 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, I = 93°.0 + (MX12 .6) = 94°.4 which bears no relation to the truth. The possibility of thus committing serious error through fail- ing to account for completed periods or revolutions, suggests the necessity of caution in this direction. 42 THE THEORY AND PRACTICE OF INTERPOLATION. T F{T) J' A" . . . JO t K t + 2o, t+3a F, F s «0 a 2 h h h t+(i + 2)+»(t) = //('+» + b i+l Jtf™ + ffl Wi?('-H)(t) = ^'+« + .... With the condition assumed, these equations give m i+1 FW (t) = < 1 )' +2 J' ( ' +S> (*) = . . . . = Hence, in this case, the expansions (0) end at the (i-)-l)th term. It follows that, under the present assumption, the expansions (0) are valid; in other words, F(t-\-nF' (t) + — F"(t) + . . . . + ^ F™ (t) (71) li Li Let us now consider the expression 11 12 Substituting, successively, ??, = 0, 1, 2, 3, get, according to (69), (72) *+3, in (72), we Q = F , F lt F 2 F t , . . . . F i+3) respectively. Substituting these same values of n in (71), we evidently obtain the same results, namely — F n = F , F lt F t , F s , . . . . F i+i , in succession, THE THEORY AND PRACTICE OF INTERPOLATION. 43 Hence, F n and Q are equal to each other for more than i values of n. But F n and Q are both expressions of the degree i in n. Now, when two expressions of the degree i in n are equal to each other for more than i values of n, they are equal for all values of n. There- fore, for all values of n, fractional and negative, we have *= *<*+»•) = K+n %+ n ^ bo+ + »(»-*)• ■ •(»-<+!) , (73) EL Li provided that J<" = Z = constant. This is the fundamental formula of interpolation, and is known as Newton's Formula. 26. Second Proof of Newton's Formula, for Constant Values of /<•>. — Formula (73) is readily proved by means of equation (59), in which to may have any value. The only condition necessary for the validity of (59) is that the expansions (0) are themselves valid. But since we assume that the differences beyond J fi) vanish, it follows (as proved in the last section) that the expansions (0) are valid. Hence (59) gives, rigorously, „, , . m(m — V) ... m(m — 1) . . . (m — i+1) ,,,., V = mJ '+ I > J "+ . . . .+-i 1 -i >-J^ \L Li From the definition of S ' (see schedule, p. 31), we have V = F(t + m„)-F(t) = F m -F • •• F m = F(t + m») = i^ + 8/ _ . m(m — l) ... m(m — l) . . (m — i+1) ,,.. = F 0+ mJ > + K ' d<< + . . + -* J —- A l -V" ll LL which is the same as formula (73), except that to is written for n. 27. To Find n, the Interval of Interpolation. — The binomial co- efficients of Newton's Formula are given in Table I, for every hun- dredth part of a unit in the argument n. The quantity n is called the interval of interpolation, and in practice is always less than unity. To obtain an expression for n, suppose that we are to interpolate the value of the function corresponding to the argument T, whose value lies between t and t-\-ca; then we shall have F n E F(t + n being supposed constant as before : T F(T) J' J" J"' Jiv J<" ..t — 3to ^-s &- rf , 7o t -2u> t Hi F-i "•-a a_„ 5 2 r -4 C-8 <2 3 ^0 t t+ to t + 2w F F x F, CLo "l 6-1 C -2 C-, Co d ., ^0 ■t + 3w F, «2 h C l <>l • • • u We might substitute — n for w. in (73), and find directly, '-. = *.+ (-.)s + ( -- x -- 1 \ + ( - ) <-V )( --' ) * + • • ■ • 11 li But this formula, while true, is inconvenient from the fact that its coefficients neither converge as rapidly as the binomial coefficients for -\-n, nor can their numerical values be taken from Table I. To avoid the negative interval, we have only to suppose the series inverted, thus making F 3 the first, and F_^ the last of the tabular functions. Then, by Theorem III, the signs of J 1 , J'", J v , . . . . are changed, while the signs of J", J 1V . .... are unaltered. Now the value of F_ n is obtained by interpolating forward with the interval -\-n "i the inverted series; hence the differences to be used in Newton's Formula are- Z-i, +*_ a > — c -s> +d-i> »(n-l)(n-2) p n(n-l)(n-2)(n-3) d (75) We therefore have, by (73), F^=F(t- n ^ ^ p the differences being taken as in the above schedule. The coefficients, as before, are taken from Table I with the argument n. An immediate and important application of (75) is in finding the value of a function near the end of a given series. Thus, in the pre- ceding schedule, suppose the series ended with F , and it were required to interpolate a value of F between F_ x and F : since the differences &_!, c_ n d_i (required in interpolating forward from F_^ are not 46 THE THEORY AND PRACTICE OF INTERPOLATION. given in this case, the formula (75) must be used; n being the inter- val of the required function from F toward F_±. Example. — From the table of T* given on page 44, find the value of (13.26) 4 . Taking t = 14, we find 14-13.26 n = = = 0.67 Z which is the interval counted 'backwards from F = 38416. Hence, from Table I, we obtain A = + 0.37 C = +0.06333 B = -0.11655 B = -0.04164 for the differences required by (75), we havi a_! = +17680 c_, = +2112 b-> = + 6944 d_i = + 384 ifore, by (75), we derive F = +38416.00 -Aa_, = - 6541.60 + Bb_., = - 809.32 -Cc_z = - 133.75 +Dd_ i = - 15.99 .-. F n = (13.26) 4 = +30915.34 By direct calculation, we find (13.26) 4 = 30915.34492 + 30. Application of Newton's Formula, when 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 order 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 F , F 1} F 2 , F 3 , . . . . F r , . . . . denote a series of approxi- mate tabular values of any function F(T), given for equidistant THE THEORY AND PRACTICE OF INTERPOLATION. 47 values of^ T, and true to the nearest unit of their last figure; let F , F ly F 2 , F 3 , .... F r , . . . . denote the corresponding true mathe- matical values of the series, which we shall designate generally as F; also, let F r = F r +f r ; f r 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 /„,/,, f 2 , f 3 , are now defined by the two schedules below: T F(T) A< J' A'" AW /jc+i) t K a„ t + O) t + 2m F x F 2 c o t + 3m t + 4o> t + 5o> 1\ F, F, a„ a 4 K K k m m 2 (A) T / j' A" j'" J(0 J('+l> t /. t + CO /, "o ft t -f-2a> f. «j (8, Vo t + 3u> A «2 ft yi /"o t + 4(0 A «S ft Vj /"•I t + 5o> A K i ft ys /*2 (B) Then, since F = F -\-f, it follows from Theorem IV that the differences of F are as given in the appended table : T F(T) J' J' J"' J<0 JC+l) ■ • t F = F +f a n + a„ t+ <0 t + 2o> F 1 = F 1 +f 1 F,= F 2 +f, a x + aj a, + a 2 *o + ft *i + ft f o + y ''i + yi c 2 + y 2 c s + y 8 Z + t + 3o> F, = F t +f, a s + « s &2 + ft l, + K »'o + ho t + 4o) F t = F<+A a 4 + « 4 *3 + ft h + K »h + /"i t + 5w F, = F,+f, &4 + ft '"2 + H ■ ■ (C) Let us now suppose that the differences j« +1 > in Table (C) are either alternately -f- and — , or that -f- and — signs follow each other 48 THE THEOET AND PRACTICE OF INTERPOLATION. irregularly. Moreover, the foregoing definition of F requires that the terms in J< i+1) are sufficiently small to indicate that no errors exceed- ing half a unit in the last place exist in the functions F(T). The values of J ( " are then approximately constant, and therefore Table (C) represents the typical case in practice. We proceed to investigate the accuracy of Newton's Formula as applied in this case; assuming that n is always taken within the limits and 4-1, and that terms beyond J (i > are neglected. Applying (73) to find F n from Table (C), and omitting the terms beyond z/<°, we have F„ = (F +f )+A («„+«„) + B (b +^) + C(c +y )+ .... +L(l + \,) (76) in which A, B, C, . . . . L denote the binomial coefficients of the nth order. Let us now examine the approximate formula (76), to dis- cover its maximum error when all conditions conspire to that end. The formula (76) may be written F n = (F + Aa +Bb + .... +LQ + (f +Aa +Bp + .... +LX,) (77) For brevity, let us put Q = F + Aa + Bb + .... +Ll ) B=f + Aa + Bf3 + .... +L\, V (77«) ••■ F n =Q + B ) It will be observed that Q is the value obtained for F n when (73) is applied to Table (A), terms beyond J (i) being neglected. We leave the discussion of Q for the present, to consider the quantity R, which evidently expresses the error of interpolation due to the un- avoidable errors, f, contained in the tabular functions F. Applying the formulae of §22 to the differences of Table (B), we have "■a = A -fa Po =/ 2 -2/i+/ la = ft — 3 /a + 3/i — /„ . So =/*-¥,+ 6/ 2 - if+f I *o = /. - 5/« + 10/, - 10/, + 5/ - f THE THEORY AND PRACTICE OF INTERPOLATION. 49 Hence, from (77a), we obtain B = f + Aa a +B$ a + Cy + BS + Ee + .... +LX, = /. + A (f x -f a ) + ^(/ 2 -2/ 1 +/ ) + C{f 3 -Zf,+Sf x -f a ) + -DC/.- W+W-ifi +/o) + • • • • .-. R = f l) (l-A+B-C+D-E+ . . ±L)+f 1 (A-2B+3C-4D+5E- . .)) +f 2 (B-3C+6D-10E+ . . . . )+f s (C-4D+10E- . . . .) V (79) +MD-5JE+ . . . . )+/ 6 (2?_. ...)+.... J Now the binomial coefficients A, B, C, . . . . are connected by the following relations: '-• • * = (^ ■ °-+x*+ . . +x<+ . . )(1-Ax+Bx 1 -Cx i + . . ±Lx<* . . ) = (1-s)- 1 50 THE THEORY AND PEACTICE OE INTERPOLATION. Upon equating the coefficients of x i in the two members of this identity, we find 1-A+B- C+ 0-1)0-2)0-3) . . ■ (n-i) ±L = (-1) Now, the first member of this equation is the coefficient of /„ in (79) ; and since the final member contains only positive factors, it follows that the coefficient of /„ in (79) is a positive quantity. Ac- cordingly, if we take f = +0.5, in conjunction with the values of /i 5/25/3 ? designated in (80), the value of It given by (79) will then be the greatest possible under the assigned conditions. We now append a table of the quantities f , f x , f 2 , f 3 , as above determined, with their differences : T / A' A" j'" Jiv A* Jvi JtU t t + t + 3o) t + 4(0 t + 5(o +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 + 8 +15 -16 + 16 -31 + 32 -32 +63 -64 (BO The special values which must be assigned to the quantities /o) "o? A; 7oj • • ■ • of Table (B) are, therefore, /. A, yo +0.5 0.0 -1 +3 -7 +15 in units of the last place of the tabular quantities F. Substituting these values in the original expression for R given in (77a), namely, R = /„ + Aa + Bp + C7 r „ + . . . we obtain R= +0.5-B + ZC-1D + 15E-31F+6ZG - . . which gives the maximum value possible to JR for n>\. (81) '< ] THE THEORY AND PRACTICE OE INTERPOLATION. 51 To evaluate (81) for different values of n between and -f-1, we make use of the following abridged table : n = A B c D E F + — + — + — + 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 .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 0.50 0.60 0.70 0.80 0.90 1.00 + + + + + + .120 + + + - B .000 .045 .080 .105 .120 .125 .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 -31^ .000 .409 .632 .722 .713 .635 .521 .384 .245 .115 .000 +63 G .000 .699 1.065 1.197 1.159 1.014 .813 .592 .372 .170 .000 If, now, we let R 2 , B 3 , i2 4 , . . . differences beyond the 2d, 3d, 4th, .... lected, then, from (81), we find R t = 0.5 -B B 3 = 0.5-B+3C B, = 0.5-.B + 3C-7Z> denote the values of R when . order respectively are neg- (82) From the last table we obtain, by successive additions, the values of R 2 , B 3 , S 4 , . . . . 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 B 3 0.50 0.55 0.58 0.60 0.62 0.63 0.62 0.60 0.58 0.55 0.50 B s 0.50 0.63 0.72 0.78 0.81 0.81 0.79 0.74 0.68 0.59 0.50 B t 0.50 0.78 o.de 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 ■^-6 0.50 1.42 1.97 2.23 2.27 2.13 1.89 1.57 1.21 0.85 0.50 B 7 0.50 2.12 3.04 3.43 3.43 3.14 2.70 2.16 1.59 1.02 0.50 52 THE THEORY AND PRACTICE OE INTERPOLATION. Whence it is seen that the greatest possible values of R, under the assumed conditions, are — Ii 2 -tl s Ji i Zl 6 -fi/ 6 My .... I (RV\ 0.6 0.8 1.1 1.6 2.3 3.4 .... \ K J 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 practice. In such cases the errors (R) in functions interpolated by Newton's Formula may be a considerable part of the values given by (83). These values show that when the differences beyond /P are neglected, the error R cannot be greater than 1.6, in units of the last place in F. 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 amount, we are justified in neglecting the error R, provided that fifth differences are practically constant. Beyond R 5 , the limiting values of R increase rapidly. We there- fore conclude that, aside from the inconvenience involved, it is im- practicable to interpolate by Newton's Formula when the differences beyond J T are too large to be neglected.* We now consider the expression Q of (77a), that is — Q= F + Aa + M + . . . . +Ll (84) Now, because the differences of F in Table (C) become approxi- mately constant at J (i> , notwithstanding the irregularities they contain; so, a fortiori, must the differences of F in Table (A) become sensibly constant at z/ ((> , the quantities of this table being mathematically exact. Hence the differences J« +1) in Table (A), namely, m , m lf m 2 , m 3 , . . . . 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 m + /A , m 1 + fr, m 2 + ^ 2 , .... ♦Excepting the case where F(T) is a rational integral function of T, whose tabular values are mathematically exact. THE THEORY AND PRACTICE OP INTERPOLATION. 53 of Table (C) either are alternately -f- and — , or that + and — terms succeed each other irregularly. It follows that the quantities m must be numerically less than the maximum value of fj, in the series For, otherwise, if the quantities m exceeded the greatest of the quantities /*, 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 TO o + Mo> »»i + /*i> ™ 2 + H But this is contrary to our assumption that the differencing in Table (C) has been carried to an order j«+« which does exhibit a general alternation of signs. We therefore conclude that m is numeri- cally less than the maximum value of /u,. JS"ow, from Table (B'), we observe that under the conditions assumed, The maximum value of «(= J 1 ) is 1 = (2)°; « « « u p ( = jii) u 2 = (2) 1 ; " " " " y(= J 1 ") " 4 = (2) 2 ; " " " " ^(=z/< i + 1 >) = (2)\ Hence, m 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 ^<"; 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 n for m in (59), we have V = Aa + Bb +Cc + .... +Ll + Mm + Nn + .... in which as many terms should be retained as accuracy requires. But we also have* V = F(t+n») - F(t) = F n -F and therefore F n = F + Aa + Bb + Cc + . . . . +Ll + Mm + Nn a + .... *See §26, where the same relations were similarly employed. 54 THE THEORY AND PRACTICE OE INTERPOLATION. Now, by (84), this equation may be written F n = Q + Mm + Nn + . . . . or F n - Q = Mm + iVX + . . . . (85) The series Mm -f- Nn Q -\- .... therefore expresses the differ- ence between the true mathematical value of the interpolated function and its approximate value Q. But since, as above observed, the differ- ences m are nearly constant, it follows that the differences n are small in comparison. Hence, Nn is small as compared with Mm ; in brief, Mm represents, very nearly, the value of the rapidly converging series Mm -)- Nn -(-.... in the right-hand member of (85) . The latter equation may therefore be written, without sensible error, F n -Q = Mm (86) From (82) we derive B 3 -R 2 = + 3C = (2 2 -l) (+C) _B 4 -.K S = - 7D = (2 8 -l) (-Z>) B,-B i = +15^= (2*-l)(+E) '- (87 ) B i+1 -B t = (2'-l) (-l)W From the last of these, we obtain ±2 are quite small, with -(- and — signs following irregularly. Under these conditions, it follows from (94) that the computed value F n can never differ from the true value F n by more than the quantity R i+1 . One point further, however, must be considered. In computing F n by (76), we should, in practice, obtain the values of the 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, F n , the extra decimals are dropped, the result being taken to the nearest unit, as in F. Thus we actually use, not the quantity F n ob- tained rigorously by (76), but a close approximation to that value, which we may denote by (F n ). Accordingly, the relation F n -(F n ) = ±0.5 expresses the maximum discrepancy between F n and (F n ). Combining this expression with (94), we finally obtain K-(K) = -B i+1 ±0.5 (95) The quantity JR^ + 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 B 6 given in (83), we find that when zT is nearly constant, the limiting error is +2.8 units. Since it is highly improbable that all the necessary conditions will conspire to produce this maximum 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 A 1 may be THE THEORY AND PRACTICE OE INTERPOLATION. 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 differ- 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 differ- 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) - j^w 606607.920 - 199841.772 T + 50804.968 T* + 5645.715 T a - 2169.395 T* + 116.817 T 6 + 1.507 T 6 5 (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 2 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 J" + 1.50987 1.15032 + 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 v + .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 peculiar from the fact that the 58 THE THEORY AND PRACTICE OE INTERPOLATION. last three decimals of each differ only slightly from the quantity 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 with its differences : T F(T) J' A" A"< /Jiv Zlv Jvi 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 (co Table (C) corresponds to Table (C) of §30. It will be observed that zP and d«, in (C), represent J« and j«+»>, of Table (C). The differencing in (C) is not carried beyond J Ti , 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 column therefore represent the residual terms denoted by / 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 OE INTERPOLATION. 59 T / J' J" J'" J It jv Jvi 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 + 2.955 -3.936 + 3.936 -3.936 + 3.941 -3.940 + 3.914 -6.891 + 7.872 -7.872 + 7.877 -7.881 + 7.854 + 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) values given in Table (B'), of §30. Let us now apply Newton's Formula to interpolate the value of F which corresponds to T= 0.40, in Table (C) . Neglecting differ- ences beyond z/ v , we take from Table I (for n = 0.40) , and from Table (C), the quantities to be employed. The result is as follows: F = +8.43 A = +0.40 a = -2.02 Aa = -0.8080 B = -0.12 b = +1.50 Bb = -0.1800 C = +0.064 c = -0.33 Co = -0.0211 D = -0.0416 d = -0.37 Dd = +0.0154 E = +0.02995 e = +0.38 Ee = +0.0114 F. = +7.4477 Whence, we write for the value of the interpolated function, (F„) = 7.45 = 7.44,77 + 0.00,23 = ^,,+0.00,23 (97) Computing the true value F n from (96), we obtain F.. = 7.4320416 + (98) Hence the value (F n ) =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 n from Table (A'), neglecting differences after J\ Thus we determine Q as follows : 60 THE THEORY AND PRACTICE OE INTERPOLATION. ^ = +8.425110 A = +0.40 a = -2.02003 ^a = -0.808012 B = -0.12 b = +1.50987 £5 = -0.181184 + C = +0.064 c = -0.35955 Cc = -0.023011 + D = -0.0416 d ±= -0.30109 Z« = +0.012525 + E = +0.02995 e = +0.23237 Ee = +0.006959 + .-. Q = +7.432387 + The value of _S g is computed from Table (B") in the same man- ner that Q has just been obtained from (A'). Thus we find /„ = +0.489 A = +0.40 « = + 0.003 Aa = +0.001 B = -0.12 ft = - 0.987 £/3 = +0.118 C = +0.064 y = + 2.955 Cy = +0.189 D = -0.0416 8 = - 6.891 Z>S = +0.287 E = +0.02995 e = +14.763 -ffe„ = +0.442 .. (In units of the second decimal) J? 5 = +1.526 [Cf. (83)] Now, from (91) we have F n = g + _R 5 (99) Substituting the above values of Q and R 5 , we find ^ = 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 n differs from the above value of Q only by the term in d* of Newton's Formula. Now, the coefficient of zf 1 is found from Table (D) of the last section to be approximately — 0.0230. Hence, with J n = -|-0.01507, we derive W n = Q- (0.0230 X 0.01507) \ = Q- 0.000346 ( = 7.432387-0.000346 [ ( nearl y) = 7.432041 ) which agrees with (98). The second of these equations gives Q = F n + 0.000346 + Substituting this value of Q in (99), we have F n = F n + E 5 + 0.0346 THE THEORY AND PRACTICE OP INTERPOLATION. 61 where the numerical term is now expressed in the same unit as i? 5 . With the above determined value of B 5 (= -j-1.526), the last equation becomes F n = F n + 1.56 Finally, since we were obliged to write (F n ) greater than F n 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. — From the appended table, find the sun's right-ascension for April 20 d 0". Date 1898 Sun's E.A. A' A" Jin Jlv April 1 6 11 16 21 26 May 1 6 h ra s 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 m s + 18 13.77 18 18.92 18 26.60 18 36.25 18 47.24 18 59.28 + 19 12.38 8 + 5.15 7.68 9.65 10.99 12.04 + 13.10 8 + 2.53 1.97 1.34 1.05 + 1.06 8 -0.56 0.63 -0.29 +0.01 Letting t = April 16, we have _ 20-16 5 = 0.80 Then, from Table I, and the above differences, we find m s F = 1 38 19.59 A = + 0.80 a = +18 36.25 Aa = +0 14 53.000 B = -0.08 b = + 10.99 Bb = - 0.879 C = + 0.032 c = + 1.05 Cc = + 0.034 D = -0.0176 d = + 0.01 Dd a = 0.000 . Sun's E.A., A .pril 20 d h = 1 53 11.75 which is the value given in the American Ephemeris for 1898. 33. Since the value of n in the preceding example is only 0.2 less than unity, it is more convenient to interpolate backwards from 62 THE THEORY AND PRACTICE ,OE INTERPOLATION. April 21, by means of (75). Thus, from Table I (for n = 0.20), and the tabular differences, we find m b h m b F = 1 56 55.84 A = +0.20 a_! = +18 36.25 -Aa_ x = -0 3 43.250 B = -0.08 b_ 2 = + 9.65 + Bb^ = - 0.772 C = +0.048 c_^ = + 1.97 -Cc_ s = - 0.095 D = -0.0336 d_i = - 0.56 + Dd_ l = + 0.019 .-. Sun's E.A., April 20 d h = 1 53 11.74 which agrees within 9 .01 of the first result. Whenever a check is considered necessary, the interpolation may be performed by both methods. Transformations oe Newton's Formula. 34. Modification of the Foregoing Notation of Differences: Stir- ling's Formula. — In Newton's Formula of interpolation we use differences which depend only upon the functions F , F t , F 2 , . . . .; the functions preceding 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 preceding notation of differences is modified: the even differences which fall on the horizontal line through F are now denoted by the subscript zero, as b and d ; all differences above this line are indicated by accents, as a', V, c," etc.; while all differences below the horizontal line through F are indicated by sub- scripts, as a x , b lf c 2 , etc. The new schedule of differences will then be as follows: T F{T) A 1 A" A"< Jiv A* t — 2a) t — (O t t + a> t + 2u> t + 3u> *-* F* a" a' + . . . we obtain «! = a + $b , c' = c-id , c a = c + id , e 1 = e+ . . . (102) Using the equations (102), together with the relations given in §23, we find a, = a +ib h = K + e+ ^ X + -^ j^ ; e + • . (104) which is known as Stirling's Formula. The even differences em- ployed in this formula are those falling on the horizontal line through 64 THE THEORY AND PRACTICE OP INTERPOLATION. F ; 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 d 0\ Date 1898 Sun's R.A. A> A" j'" Jiv April 1 6 11 li m 6 43 20.30 1 1 34.07 1 19 52.99 ID. S + 18 13.77 18 18.92 18 26.60 s + 5.15 7.68 8 + 2.53 1.97 s -0.56 16 1 38 19.59 9.65 0.63 18 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 16 (as in §32) , we have n = ?2rl 6 = 0.80 5 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 J7 , A", and A iT are those included between two lines; (2) The required values of A 1 and A 111 are the means of the quantities separated by a single line. As before, we shall denote the coefficients of A', A", A"', .... by A, B, C, . . . . Taking their values from Table II, with n = 0.80, and forming the required differences as indicated, we obtain m s F = 1 38 19.59 A = +0.80- a = +18 31.425 Aa = + 14 49.140 B = +0.32 b = + 9.65 Bb = + 3.088 C = -0.048 c = + 1.66 ■Co = - 0.080 D = -0.0096 d — — 0.63 Dd = + 0.006 .: Sun's E.A., u ipril 20 d h = 1 53 11.74 which agrees exactly with the result found in §33. THE THEORY AND PRACTICE OE INTERPOLATION. 65 35. Backward Interpolation by Stirling's Formula. — When the forward interval approaches unity, it will be more convenient to pro- ceed backwards from the following function by the formula F-n= F -na+-b„ w(n 2 — 1) » 2 (re 2 — 1) e + 24 i(ra 3 -l)(» 2 -4) 120 e + (105) the coefficients of which are taken from Table II with the argument //, as before. It will be observed that (105) is derived from (104) by merely writing — u for u in the latter; or, by supposing the given series to be inverted, and hence ^(Theorem III) changing the signs of a, r, and c. Example. — Solve the preceding example by (105); that is, find the sun's R.A. for April 20 d h by backward interpolation. Taking t = April 21, we have n = 2 J=?° = 0.20 The differences are formed for the date April 21 in the same manner as found above for April 20; thence, taking the coefficients from II, with n = 0.20, we find m s F = 1 m s 56 55.84 A = +0.20 a = +18 41.745 — Aa = — 3 44.349 B = +0.02 b = + 10.99 + Bb = + 0.220 C = -0.032 c = + 1.20 -Co = + 0.038 D = -0.0016 d = - 0.29 + Dd = 0.000 .-. Sun's R.A., April 20 d 0* = 1 53 11.75 36. Example. — Use Stirling's Formula to compute log sin 9° 22' from 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 n = 0.36667 66 THE THEOKY AND PEAOTIOE OE INTERPOLATION. and we therefore obtain ^0 = 9.19433 A = +0.36667 a = +4805.5 Aa = + 1762.0 £ = +0.06722 h = - 543 Bb = - 36.5 C = -0.05289 c = + 124.5 Co = - 6.6 D = -0.00485 d = - 45 Dd = + 0.2 E = +0.01022 e = + 18 Ee = + 0.2 .-. Log sin 9° 22' __ ).21152.3 The true value to six decimals is 9.211526. 37. The Algebraic Mean. — It may be well to observe that in taking the mean of two quantities having like signs, and of nearly the same magnitude, it is easier to add one-half their difference to the lesser number, than to take one-half the sum of the two quantities. That is, we proceed according to the identity in which we suppose y numerically greater than x. Thus, in the last example, instead of taking a = i(a> + aj = £(5077 + 4534) = £(9611) = +4805.5 it is easier to follow the equivalent formula a = ai _£( ai - a ') = ai -ib = 4534 + £(543) = +4805.5 Similarly, we find e = 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) A' d" -4226 + 5088 + 9314 a = £(5088-4226) = £(+862) = +431 With these precepts, the required mean differences of interpolation are very readily taken. THE THEORY AND PRACTICE OE INTERPOLATION. 67 38. Bessel's Formula. — "We now pass from Stirling's Formula to another, somewhat similar, wherein we employ the odd differences a n Cj, e n which fall on the horizontal line between F and F lf and the means of the even differences falling immediately above and below this line. Using the schedule on page 62, let us put b = h{K+h) , d = iK+d.) (106) Then, since b x — & =d, and d x — d = e x , these equations give b = b-ic, , d = d-ie, (107) Let us write the formula (104), for brevity, F n = F +na+tfb + Cc+ Dd + Ee + .... (108) where »(n'-l) n' (»»-!) F _ n(n«-l)(n'-4) C = g , D = gl - ' * ~ 120 (1Uy; Now, by means of (102) and (107), we derive a = a x — \ b = «! — £ (J— | Cn) = «! — -J-5 + i C! *o = * — i c i c = Cj-ido = rf = + 336 .. 10 4 = +10000 40. Backward Interpolation by Bessel's Formula. — To find F- n by Bessel's Formula, we conceive the series given on page 02 to be inverted; the required function is then found by interpo- lating foivanl from F toward F_^ with the interval n. Hence, the differences to be used in (111) are — -«', +t(b + b>), -c,', +i(d +d'), -e>, .... We therefore have , n(n-l) b D + b' n(n-l)(n-l) , (Ilia) 2 2 the coeflicients, as in (111), being taken from Table III with the argument n. Example. — Find 10 4 from the table of §39, by means of (lllo). Taking / = 12, we find n = 3t» = 0.40 The differences are here the same as in the last example; thus we obtain F Q = +20736 A = +0.40 a> = +18335 -Aa' = - 7334 B = -0.120 b " + b - +30200 + B. b 4^ = - 3624 C = +0.0040 c' = +28500 -Cc> = - 114 1> = +0.0224 ^'= +15000 + Z>. <'. + ''' = + 336 . 10 4 = +10000 11. Property of Bessel's Coefficient*. — If we take from Table III the coefficients for ./", ./'", J iv , J v , with the argument // = 0.30, and also with n = 0.70 (=1.00 — 0.30), we find the following values: n B C D E 0.30 -.10500 +.00700 +.01934 -.00077 0.70 -.10500 -.00700 +.01934 +.00077 70 THE THEORY AND PRACTICE OF INTERPOLATION. It will be observed that the coefficients are here numerically the same for the arguments n 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'", J 1 ", z/ v , .... taken with the argument n, by B, C, D, E, . . . . , respectively; and the corresponding values taken with the argument 1 — n by J?!, Ci, Di, i?!, . . . . An inspection of Table III then shows that we have B x = +B Cj = -c A = +D ) (112) E x = -E To establish these relations generally, we write (111) in the form F n = F + wffli + Bb + C Cl + JDd + Ee 1 + . . . . (113) Now, the value of F n may also be obtained by interpolating hack- wards from F x with the interval 1 — n; the differences thus involved will be exactly the same as in (113). Hence, after the manner of formula (Ilia), we have F n = F 1 - (1-w) a t + BJ> - C& + Djd - E A + . . . . (114) But we have, also, F 1 — (1— ri)^ = (F 1 —a 1 ) + na x = F + na 1 Whence, (114) becomes F n = F + na x + B x b - Cj x + D t d - E& + . . . . (115) which, subtracted from (113), gives = (B-B 1 )b+ (0+0,)^+ (B-B 1 )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 e x = d = e t = . . . . = The equation (116) then becomes = (B-BJb THE THEOEY AND PEACTICE OF INTERPOLATION. 71 Hence, since b cannot vanish, we have J3f x = +B This result reduces (116) to the form = (C+C 1 )c 1 + (D-D 1 )d+ (E+EJe^ .... (117) Again, we may suppose J'" constant; that is, we may put d = e x = ..,.= The equation (117) then becomes = (C+C^c, or C t = -0 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 n by Bessel's Formula is identical whether we interpolate forward from F or backward from F 1; except for the terms in F and J'. Hence little or no check is afforded by performing the interpolation by both methods. When such a check is deemed necessary, Bessel's and Stirling's Formulae should both be used. 42. Relative Advantages of Newton's, Stirling's, and Bessel's Formulae. — In practice, the only important application of Newton's Formula consists in interpolating functional values near the beginning 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. Eegarding 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 perhaps a trifle more accurate in practice than Stirling's form, particularly for values of n in the neighborhood of one-half. When n is quite small, however, Stirling's Formula will probably be found more convenient. 72 THE THEORY AND PRACTICE OF INTERPOLATION. Suppose we have given a limited table of functions, as follows F(T) J 1 d" A'" Jlv F-, F F, F, F s a" a 1 «i a 3 V h c' c 2 Assuming that fourth differences must be taken into account, and that fifth differences are to be neglected, the value of F n should in this case be computed by Bessel's Formula, which employs the mean of the quantities d and d x . If, however, the function F 3 were not included in this series, then the term d t would not be given, and we should proceed by Stirling's Formula, which involves d directly. Bessel's Formula is particularly simple and convenient when n = \ , that is, when it is required to find the function which falls midway between F and F^ ; 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 differences may be neglected. Bessel's Formula gives in this case F n = F + na, (118) To interpolate oackwards from F , that is, to find F_ n , we obtain from (Ilia), by neglecting second and higher differences, F_„ = F n (119) Upon these formulae the process of simple interpolation is based. The first difference to be used in either case is the value falling between F and the function toward which the interpolation proceeds. Frequently, where great accuracy is not required, it is sufficient to obtain F n by simple interpolation even when the second differences are considerable. In such a case, supposing that the third differences THE THEORY AND PRACTICE OF INTERPOL ATION. 73 are insensible, we observe from Bessel's Formula that the error of the approximate value of F n will be — BF n = "("-V j» (120) The maximum value of — g - ^, which obtains for v = |, is — i; whence we have the following result : When second differences are sensibly constant, the maximum error of functions obtained by simple interpolation is 1 zl" . Thus, in Tables I, II, and III, the values of the coefficients for J" (designated above as J3) can never be in error by more than 1 of 10 units, or 1.2 units in the fifth decimal, when found by simple interpolation. 44. Interpolation Involving Second Differences, by Means of a Corrected First 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 n obtained) by the follow- ing simple method : Since third differences are supposed insensible, Bessel's Formula becomes F n = F + na x + v 2 > -b which may be written in the form (121) Now, because third differences are negligible, we may write b for b in (121) ; then, putting we have ' ) (122) F„ = F + nc h The value of F n 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 n is ordinarily required. The _value of o^ , the corrected first 74 THE THEORY AND PRACTICE OE INTERPOLATION. difference, is thus found by an easy mental process amounting almost to mere inspection. Example. — Find (8.2) 2 from the following values of T 2 : T Ji2 A 1 A" 4 7 10 13 16 49 100 169 +33 51 + 69 + 18 + 18 Here we have t = 7 n = 0.4 F„ = 49 ij = 51 6 = 18 Hence, by (122), we find 1-n 1_0.4 = 0.3 2 2 «, = 51 - (0.3 X 18) = 45.6 . F n = 49 + (0.4 X 45.6) = 67.24 This result is exact, because the second differences are rigorously constant. 45. Backward Interpolation by Means of a Corrected First Difference. — From (Ilia), neglecting differences beyond A", we obtain „ -,-, , n(n — V) b/. + b' _, , n(n — 1) , F_ n = F -na'+ \ 2 > . -^- = ^ - na'+ -i_J J o or Hence, if we put we have /••.., /•'„ -„(»'+ 1-5 * Q ' - -M-feU (123) (124) ^_„ = F n -na> Example. — From Hill's Tables of Saturn, the following pertur- bations are taken; find the value corresponding to the argument T = 30682.38. T F(T) A' A" 28800 29760 30720 31680 32640 12.5751 12.1998 11.8315 11.4700 11.1148 -3753 3683 3615 -3552 + 70 68 + 63 THE THEORY AND PRACTICE OF INTERPOLATION. 75 Taking t = 30720, we have F = 11.8315 n = 960 = °- 03919 (backward from F ) T = 30682.38 a 1 = -3683 «d = 960 b = + 68 Using 0.04 as a sufficiently accurate value of n in determining a, we find by (124), !___„ = 1^04 = o 4g a 1 = -3683 + (0.48 X 68) = -3650 .-. F_ n = 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 A 1 to be corrected is the difference of these two functions, or 8683; the sign we disregard. The correction to be applied to this number is 0.48 X 68, or 33. Again neglecting sigus, we simply apply this quantity to 3683 in such a manner as to obtain a result falling somewhere 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, na! = 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 x and a' have opposite signs ; that is, when j> changes sign in passing the function F . In this case the sign of a must be noted; we then have, as in (122) and (124), F^= F - na' $ K > 76 THE THEORY AND PRACTICE OF INTERPOLATION. For example, given the values below : T 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 = 638, and interpolate backwards with n = 0.20. To obtain a, decrease 100 by 0.1X300, or 120; whence a' = — 20, and therefore F_ = F„ net 1 = 638 - [0.2 X (-20)] = 642 We remark in passing that the value of the corrected first differ- ence, either in forward or backward interpolation, is always contained between the limits a^ and a . The number of instances in practice where the differences beyond J" may be neglected is very large. The precepts given above are therefore important, and should be practiced by the student until their application becomes rapid and mechanical. 16. Correction of Erroneous Functions by Direct Interpolation of the Values in Question. — When an error has been detected in some one function of a series by the method of differences, as explained in §8, it is often possible to find the true value of that quantity by direct interpolation. To accomplish this, we have only to omit from the given series every alternate function, the incorrect value being one of the number rejected. We have then to make but one interpolation, midway between two functions of the new series, to obtain the value required. It is necessary, however, that the given series shall include a sufficient number of functions to furnish an adequate schedule of differences in the abridged table ; furthermore, the interval of the original table must be sufficiently small that the magnified differences of the abridged table will not be so large as to render interpolation impossible. We illustrate by means of Example III, §9. The value of /3 for May 11.0 was found to be incorrect ; hence, to find the true value, we omit from the given series the positions for every noon, retaining THE THEORY AND PRACTICE OP INTERPOLATION. 'ft only the values for each* midnight. Thus we obtain the following abridged series : Date 1898 J' J" J'" Jiv May 8.5 9.5 10.5 11.5 12.5 o ; n -1 59 54.2 -0 44 27.0 + 32 39.9 1 4G 12.4 + 2 51 51.2 o / It + 1 15 27.2 1 17 6.9 1 13 32.5 ■+ 1 5 38.8 / II + 1 39.7 -3 34.4 -7 53.7 / II -5 14.1 -4 19.3 / + 54.8 The value of /3 for May 11.0 is now readily found by interpola- tion ; for this purpose, we take * = May 10.5 F n = +0° 32' 39".9 0.50 Since but one value of . / iv is given, namely d = 54.8, we pro- ceed by Stirling's Formula (see §12); thus we find O / II F a = +0° 32' 39.9 A = +i a = +1 15 19.7 Aa = +0 37 39.85 B = +\ b = - 3 34.4 Bb = - 26.80 c = - T V c = - 4 46.7 Cc = + 17.92 D = -0.00781 d = + 54.8 Dd = - 0.43 .-. /? (May 11.0, 1898) = +1 10 10.44 The value found in §9 by the method of differences is -\-l° 10' 10".6. The result just obtained by interpolation is uncertain within narrow limits, because we have no knowledge of the value of p in the above table. The value 1° 10' 10".6 should therefore be taken as the more probable. Had the value of /3 for May 13.5 been included in the original series, our abridged table would have yielded two values of z/ lv and one of J v . We should then have used Bessel's Formula (see §42) to compute the latitude for May 11.0. Now, the moon's latitude for May 13.5, 1898, is +3 46' 22 ".2; including this value with the others above, and applying Bessel's Formula, we find /3 = -\-l° 10' 10".57. 47. When a series contains several incorrect functions, separated from each other by even multiples of the interval w, the foregoing 78 THE THEORY AND PRACTICE OE INTERPOLATION. method at once serves for the determination of the several values in question. Thus, in the series F a> F i> Fn F s> Ai> • • • • let us suppose that F x , F 3 , and F 7 are in error. Then, if we tabulate and difference the series ~TJ1 Tp Tp Tjl TJT A)> A) A) A) J #i, • • . . the required values are easily found by interpolation. Again, when two adjacent functions, say F 4 and F 5 , require cor- rection, we may proceed by tabulating every third function of the given series; thus we obtain the abridged series Fo> Q, F e , F t , . . . . from which the values of F± and F B are found by interpolating with n = i and f , respectively. Otherwise, if the differences of the latter series are too large for accurate interpolation, we may omit from the original table every alternate function only, as in §46. The resulting series, 7P TP TP TP TP *o> -Fa; -fa **} -P 8 > ■ • - • will therefore contain but one incorrect value, namely F±. The cor- rection to F± may then be found by the method of differences, whereas this method might be impracticable if applied to F± and F B simulta- neously. Similarly, we may correct F 5 by the differences of F u A) » F t> Ft, F g , .... or, by interpolation from the corrected series TP TP TP TP TP -PO) Al A) Ai J's, . . . . Systematic Interpolation — Subdivision op Tables. 48. Thus far we have considered interpolation as a process for computing the values of functions for occasional or special values of the argument, simply. We shall now consider the subject in a broader THE THEORY 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 large number of equidistant values of the argument, or when the tabular quantities result from a long and laborious calculation, it will be much shorter and easier to make the direct computation for a less frequent interval than is finally required, and thence to obtain the intermediate values by systematic interpolation. For example, suppose the function F(T) = 700".43 sin2T-l".19 sin4T is to be tabulated for every 10' from 30° to 60° ; 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 errors if neces- sary, the middle terms are then found by interpolation to halves. 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 third 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 with 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 interpolation 80 THE THEORY AND PRACTICE OF INTERPOLATION. just described. 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 additions 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 halves, which gives the function corresponding to the mean of two consecutive tabular values of the argument. This case will now be considered. * 49. Interpolation to Halves. — If, in Bessel's Formula (111), we put n = k, the coefficients of z/'" and J v vanish, and we get F h = F + i ai -ib + ^d- .... (125) Since F x — _F = a n we have F + i„ - F ° + F i Also, by (106), we have d = ^±^ Hence, (125) may be written in the form (126) which is the formula for interpolation to halves, true to fifth differences inclusive. The differences are to be taken according to the schedule on page 62. Supposing that fourth differences are so small as to produce no sensible effect, we obtain from (126) the very simple formula ,,_fi+£_ t ^Jt) cut, true to third differences inclusive. Hence, to interpolate a function midway between two consecutive tabular values, we have the following THE THEORY AND PRACTICE OF INTERPOLATION. 81 Rule : From the mean of the two given functions, subtract one- eighth the mean of the second differences which stand opposite. The result is true to third differences inclusive. To obtain the value true to fifth differences inclusive, add to the above result T -f ^ 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 differences may be neglected; accordingly, if we put 8 ' = F, -ft V = F,-F t F.-F, , 8 3 ' = F t -F,, (128) we obtain from (125), V = i«i - 1 V = *«* - * 2 2 2 (129) The quantities 8' defined by (128) are evidently the first differ- ences of the interpolated series ; the alternate terms, S ', S 2 ', 8 4 ', . . . . , are computed by (129) from the first and second differences of the given series of functions ; the values of 8/, S 3 ', 8 5 ', .... are not computed. The method and arrangement of the work are shown in the schedule below : T F(T) 8' 8" a P j' J" J'" t-w F-i a' e' t t +-J-0) t + co * + *«■> Fo F i F, 8„' V 8,' 8 " 8," 8 2 " "i a. ^0 f 2 t + 2co F 2 8s' 8/ 8," 8 4 » ±a* -u b -: b -) «3 K c 3 t+3 j' J" J'" O 30 9.69897 31 + 2524 32 9.72421 + 1190 1145 1103 1063 1024 988 953 + 920 -189 33 9.73611 -45 + 1167.5 + 22.4 2335 + 20 34 9.74756 42 169 35 9.75859 40 1083.0 20.2 2166 15 36 9.76922 39 154 37 9.77946 36 1006.0 18.3 2012 15 38 9.78934 35 139 39 9.79887 -33 + 936.5 + 16.7 1873 + 10 40 9.80807 -129 41 + 1744 42 9.82551 Since 4th differences may be neglected, only the two columns a and /3 are required for the computation of the differences 8'. All the quanti- ties actually used in the process are given in the above table. The computed quantities, together with the given 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 & to mco, where ^ is a Positive Odd Integer. — As particular cases of this problem, we may take m = -|-, |, ^, etc. Taking ra = £, we intro- duce two values' between every two adjacent functions of the given table; we thus derive the series F t , F it F v F x , F v . . . . in which the interval is £w. This process is called interpolation to thirds. To interpolate to fifths, we let m = | , thus introducing four functions between every two adjacent terms of the original series. "We then have the tabular values of F , F v F„ F t , F v F lt F it . . . . the interval being |w. More generally, let us take m = r , where k is a positive odd integer; we thus introduce k — 1 equidistant values of the function between every two adjacent terms of the given series. The resulting series will therefore be -*0> '»' 2>»> -^3m) F F F 84 THE THEORY AND PEACTICE OE INTERPOLATION. to in which the argument interval is ma>, or j. Now, the two adjacent functions of this interpolated series, which, as a pair, fall midway be- tween F and F x , are that is Hence, if we put '(¥)- a ° d .F (*?)" 'so ^ i* 1 V - ' m — J? (131) it follows that 8/ is the value of the first difference of the interpo- lated series which falls on the line midway between F and F : ; we shall designate this quantity a middle first difference of the required series. If we now let 1 + m —%- = n (132) we have 1 — m . ~2- = *-" and (131) becomes V = -F..-.FU (133) Hence, to express 8/ in terms of the differences of the given series, we have only to express the values of F n and Fi_ n by Bessel's Formula ; thus, abbreviating coefficients, we have, as in (113) , F„ = F + na^ + Bb + Cc x + Dd+ Ee^+ . . . . (134) Also, by virtue of the property of these coefficients established in §41, we have F^ = F +(l-n) ai + Bb- Cc x + Dd-Ee 1 + .... (135) The difference of these equations gives 8,' = F„-F 1 _ n = (2n-l)a l + 2Ce 1 + 2JSe l + .... (136) Now, by (132), we have 1 + m 71 = "2- hence, from (111), we find C = in(n-l)(n-i) = ™ (m 2 -l) q = T i T (n + l)n(n-l)(n-2)(n-i) = &(n+l) (n-2)C = ^ (m 2 -l)(m 2 -9) THE THEORY AND PRACTICE OP INTERPOLATION. 85 Substituting these values of n, C, and F in (136), we obtain the formula V = '««! + oi C'^ 2 - 1 ) ^ + tSo (« s -l)(»* 2 -9) * + 24 (137) by which the middle first differences may be computed in any case, provided — is a positive odd integer. m Let us now consider the schedule below : t-co t — u>-\-mo> t— Mis t t + t + co F{T) F-i F Fx 8'-, 8/ 8" 8^. 8„" 8i" 8{" j" v h j'" j |v d> dx Jv The quantities are here arranged in a manner somewhat similar to the schedule of §50. The given functions, F_ u F , F 1} . . . . , are separated, successively, 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 differences of the interpolated series ; and the differences of the given functions are written to the right, in columns A' to J v . The value of 8 S ' is now computed by (137) from the differences a u t\, and e u which stand opposite. In like manner, S'_j is computed from the differences a', c', and e'; 8/, from a^, c 2 , and e 2 ; and so on. We thus obtain a series of middle first differences, which are tabulated under 8' in the schedule above. Now it is clear that if we should interpolate the h — 1 inter- mediate terms between S'_j and 8/, between 8/ and 8/, etc., the resulting series would constitute the consecutive first differences of the interpolated series F (T) ; the required functions would then be formed by successive additions of these differences. The problem of 86 THE THEORY AND PRACTICE OF INTERPOLATION. interpolating the given series F(T) is thus virtually reduced to that of interpolating the computed values of 8' in precisely the same manner. Now, let S " denote the second difference of the interpolated series F, which stands opposite F ; 8/', the second difference opposite F 1 ; etc. It follows that 8 " is the middle first difference of the interpolated series 8', which falls between S'_j and 8' 4 ; 8/', that falling between S s ' and 8/; and so on. Hence, we may find S ", 8/', 8 2 ", .... from the computed series 8'^, 8/, 8/, .... , in precisely the manner that the latter quantities are derived from F_ x , F , F t , . . . .; that is, by application of the general formula (137), mutatis mutandis. For this purpose, we must form the differences of the computed series S'_p 8/, V, .... Accordingly, let us put, for brevity, and (137) becomes M> = 19^0 (m 2 -l)(m 2 -9) 8j' = ma x + Mc x + M'e 1 (138) (139) provided differences beyond z/ v are disregarded. We now form a table of the quantities S'_j, 8/, 8/, . . . . , and their differences, as follows : Function, = 8' 1st Diff. 2d Diff. 3d 4th S'.j = ma' + Me'+ M'e' 8j' = ma 1 + Mc x + M or, by (138), S " = m\+^(m*-l)d (140) by which the quantities 8"_ n S ", 8/', .... of the former schedule are computed from the differences A" and j lT which stand opposite. Again, we may suppose that the intermediate values of 8" have been interpolated between the computed values 8"_ 15 8 ", 8/', . . . . ; this completed series 8" constitutes the consecutive second differences THE THEORY AND PRACTICE OF INTERPOLATION. 87 of the interpolated series F(T). Finally, we shall denote by Sj" the third difference of the interpolated series F, which stands opposite 8/ in the given schedule. The quantity 8"' is therefore the middle first difference of the completed series 8", which falls between 8 " and 8,"; it bears the same relation to 8 " and 8,", that 8,' bears to F and F { . Hence, to find 8"', let us put and (140) becomes M" = jgK-1) 8 " = m\ + M"d The differences of 8"_, , S " , 8," , . . . (141) are therefore as follows: Function, = 8" 1st Diff. 2d 3d 8" t = mW + M"d> 8 " = m\ + M"d 8j» = m\ +- M "d, m a e' + M"e' m*d' m 2 d in}d x m 2 e x Whence, applying (as above) the general formula (139), we find 8{" = m{m\ + M"e l ) + M{in\) = m s c 1 +(mM" + m 2 M)e 1 Substituting the values of M and M", we have & 'i" m'^4- — (m 2 — l)e, (142) In practice, the values of 8 iv and 8 V are never required, and in many cases the column 8'" is not necessary. Supposing, however, that we have computed the (nearly constant) values of 8"j , 8{", 81", . . . by (142), the intermediate terms are then written in by mere inspec- tion. We thus complete the column 8'", — the consecutive third differ- ences of the required series F(T). Having also computed the quantities 8 ", 8,", 8 2 ", .... and 8'_ v 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 S 4 ', 8 ", and 8[" , in the order computed in practice, as follows : SI" = m^ + _(»»_ l)e, 8 „" = m 8 m *o + 12 (m2 - 1K (143) 8,' =«!«! + £ K- 1) *! + Jc^q K-l)K-9) «1 88 THE THEORY AND PRACTICE OF INTERPOLATION. which serve to reduce the tabular interval to m times its original value, m being the reciprocal of a positive odd integer. It will be observed that the differences required in computing each of the quantities 8 are always found on the same line with that quantity. 52. Interpolation to Thirds. — For this purpose, we take m = ^ in the formulae (143), and find 8J" = fro 8 „" = i S7 "l ' 5¥3 e l . _ 2 f ] These formulae are more conveniently computed in the form 8!" = .(V "W (144) (145) V = M-A<*,) V = ite-Si") 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 : T Log tan T 8' 8" 8'" A 1 J" j'" Jiv O 27 9.70717, + 5427 30 9.76144 + 3.1 5108 -319 + 85 33 34 35 36 37 38 39 40 41 42 9.81252 9.82899 9.84523 9.86126 9.87711 9.89281 9.90837 9.92382 9.93917 9.95444 + 1646.9 1623.8 1603.3 1585.3 1569.6 1556.1 1544.6 1535.1 + 1527.5 -25.9 23.1 20.5 18.0 15.7 13.5 11.5 9.5 7.6 - 5.7 3.0 2.8 2.6 2.5 2.3 2.2 2.0 2.0 1.9 1.9 1.9 + 1.9 4874 4711 4607 4556 234 163 104 - 51 71 59 53 + 51 -14 12 6 - 2 45 0.00000 + 4556 48 0.04556 The heavy type shows at a glance the given functions, and liker wise the computed middle differences. We observe that it is here THE THBOBY AND PBACTIOE OF INTERPOLATION. 89 necessary to compute five values of 8'", four values of 8", 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 error in the final additions. Having obtained for S'" the series + 3.1 2.6 2.2 1.9 +1.9 the intermediate terms are readily inserted, as shown above; it is necessary, however, to see that the completed series 8"' is consistent with the computed 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 are 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 series 8" agrees with the computed values of 8'. For we must have -(20.5 + 18.0 + 15.7) = 1569.6-1623.8 = -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 ^8' = 1646.9+1623.8 + 1603.3 = +4874.0 = JK 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. Interjiolation to Fifths. — Taking m = \ in the formulae (143), we obtain 8{" = T i*('i-ft*i) ) V = ih( b o-& d o) } (146) V = ite-A(Ci-T¥ir«i)l J In practice it will suffice to put \e x for both j 8 F e x and tW^; the formulae (146) then become, very approximately, 90 THE THEOEY ANTD PEAOTIOE OE INTEEPOLATION. 81" So" = t£f ( c i — i e v A(*o-A^o) •8i" (147) Example. — The following ephemeris gives the moon's R.A. for every ten hours. Obtain the value for every second hour, from Sept. 23 d 20 h to Sept. 25 d 12 h , inclusive. The details of the computation are as follows : Date, 1898 Sept. 23 d Sept. 23 10 Sept. 23 20 23 22 24 24 24 Sept. 24 24 24 10 24 12 24 14 Sept. 24 16 24 18 24 20 24 22 25 Sept. 25 25 25 25 25 10 Sept. 25 12 Moon's R.A. Sept. 25 22 Sept. 26 8 Urns 18 24 26.4 18 49 57.5 19 15 19 20 19 25 19 30 19 35 19 39 19 44 19 49 19 54 19 59 20 20 20 13 20 18 20 23 20 28 20 32 20 37 20 42 20 46 20 51 8.5 7.9 6.3 3.7 0.0 55.2 49.3 42.3 34.2 25.0 14.6 3.0 50.3 36.4 21.4 5.2 47.8 29.3 9.6 48.8 26.9 21 14 20.7 21 36 48.5 + 4 59.39 58.39 4 57.36 56.31 55.23 54.13 53.02 4 51.89 50.75 49.60 48.44 47.28 4 46.12 44.95 43.79 42.63 41.47 4 40.33 39.19 f4 38.06 -0.976 1.004 1.030 1.054 1.077 1.097 1.114 1.128 1.140 1.149 1.156 1.161 1.164 1.165 1.163 1.160 1.154 1.147 1.139 1.130 1.119 S'" .034 32 30 28 26 .024 23 20 17 14 .012 09 07 05 03 -.001 + .002 03 06 07 .008 09 11 12 14 + .015 + 25 31.1 25 11.0 24 46.7 24 19.4 23 50.6 23 21.7 22 53.8 +22 27.8 A" -20.1 24.3 27.3 28.8 28.9 27.9 -26.0 Am zjiv -4.2 3.0 1.5 -0.1 + 1.0 + 1.9 + 1. 1.5 1.4 1.1 + 0.9 THE THEORY AND PRACTICE OF INTERPOLATION. 91 Here we extend the computation of S"' and 8" two places of deci- mals ; one of which is dropped in computing 8', and the other in forming the required functions. The principle and method being the same as in the last example, further explanation is unnecessary. 54. Order of Interpolation to Follow, when a Series Requires 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 fifths, 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 a>, we may suppose the relation of these quantities to be — where Tc, I, and m are integers. It will then be found most expedient, first, to interpolate to halves, Jc times ; then to thirds, I times ; and finally to fifths, m times. For example, F being given for every degree, and required 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 Fthnctions. — Let the given series, with its differences, be as follows : T F(T) J' J" jm Jiv t t + O) t + 2 F 1 F 2 F s F< a x «4 hi h h "4 d 3 It is required to interpolate the values of F n , F 1+n , F 2+n , F 3+n , . . . . These functions evidently form a new series having the same interval as the old. Let us denote this new series by [F] ; also, let the dif- ferences of [F], denoted by \_A<\, [J"], [J'"], . . . . , be taken as shown in the table below : 92 THE THEORY AND PRACTICE OP INTERPOLATION. T [F] [J'] [J"] [j'»] [Jtv] t + IllD t + (1+71)0) t + (2 + w)o) t + (3 + »)<■> t + (4 + ra)(o F S+n «1 « 2 «S A ft A Yi Ys Y4 «4 Now, it was shown in §22 that differences of any order may be expressed in terms of the tabular functions. Thus, in particular, we obtain from the given series F, c x = F t - 3F 1 + 3F - F_ x = *(*) c 2 = F t — 3F 2 + 3F 1 - F = * (t+ ) (148) where *• (£) denotes, for brevity, the function of t expressed by F 2 - 3F X + 3^ - F_ x ; that is, * (t) = F(t+2w) - 3F(t+a) + 3F(t) - F(t-a>) Again, in like manner, the interpolated series [F~\ gives Yi = F 3F 1+n + 3F„ - F_ 1+n = *(t+na>) Y, = Fs+n-3F 2+n + 3F 1+n -F n = *(* + «,+»«,) (149) It follows, then, that the series [//'"] is simply the series A" 1 inter- polated forward with the constant interval n. Moreover, since the above reasoning is perfectly general, this relation holds for any order of differences. Hence, to perform the required interpolation of the series F(T), that is, to obtain the series [F~\, we have only to interpolate forward each value of A" with the constant interval n, thus forming the column \_d"~\. 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 every fourth degree of latitude ( + ™ (m 2 -l)z/''+ 2 + -^(m 2 -l)|(5i-2)TO 2 -(5i+22)^'+ 4 4- .... in which % may be either odd or even ; and where zf, J i+2 , z/ i+4 , .... symbolize the tabular differences which fall upon the same horizontal line with 8'. CHAPTER III. DERIVATIVES OP TABULAR FUNCTIONS. 56. It is often required to find certain numerical values of the differential coefficients of functions either analytically unknown, <- or complicated in expression. In the majority of such cases the function has been previously tabulated for particular (equidistant) values of the argument. The required derivatives are then readily computed from the differences of the tabular functions. We have already seen that — with certain limitations — particular values of a function, with their differences, practically determine the character and law of that function, thus enabling us to determine intermediate values by interpolation. The trend or law of variation of the function being thus defined by its differences, it is but natural to suppose that the successive derivatives are quantities closely related to these differences ; since the derivatives are themselves direct in- dices of the character of variation of the function. 57. Practical Applications. — The most useful application is in finding the change or variation in F(T) corresponding to an increase of one unit in T, supposing the rate of change in F to remain con- stant from T to T-\-l, and equal to the actual rate at the instant T; for this quantity is simply the first differential coefficient of F(T) with respect to T, which we shall denote by F\T). For example, having observed that a freely falling body describes sixteen feet during the first second of its descent, forty-eight feet the second second, and eighty feet the third, its velocity at the end of two seconds is easily found to be sixty-four feet per second. This velocity of sixty-four feet is nothing more than the first differential coefficient of the sjmce with respect to the time, computed for the in- stant 2 9 .0 : it is the space which would be described during the third 98 THE THEOBY AND PRACTICE OF INTERPOLATION. second, supposing the action of gravity to have ceased at the end of the second second. The most frequent and important applications occur in Astronomy. An astronomical ephemeris contains a great variety of tables giving the positions and motions of various heavenly bodies, and of certain points of reference. From the given positions, tabulated for every hour or from day to day, are derived the motions per minute, per hour, or per day, according to circumstances. For instance, the Nautical Almanac gives the sun's declination for every Greenwich noon. The hourly motion in declination (also given for every noon) is computed from the dif- ferences of the tabular declinations : its value is the differential coef- ficient of the tabular function at the date in question. In the following sections the various formulae employed in com- puting the derivatives of tabular functions will be derived. 58. Development of the Required Formulae in General Terms. — The variables T and n are connected by the fundamental relation T = t + no> (150) in which t and w are constants for a given series. Accordingly, we have hitherto written, under varying circumstances, F(T) , F(t + nm) , F n as equivalent expressions of the same quantity. In like manner, we shall hereafter denote the successive derivatives of F (T) by the fol- lowing equivalent forms : ^ | F{T) £ = F> (T) = F> ) , FH, = Fi'it-nu) , F% = F'»(t-no>) , (152) THE THEORY AND PRACTICE OF INTERPOLATION. 99 Now, by means of any one of the fundamental formulae of inter- polation, we may express F n in the form F n = F + na + Bb + Co + Dd + Ee + (153) where, in any given case, a, b, c, . . . . are known differences; and where B, C, D, . . . . are definite functions of n. Let the successive derivatives of B, C, I), . . . . , taken with respect to «, be denoted by B> , B" , Em , . . c , C" , C" , . . r>' , D" , Dm , . . E' , E" , Ei" , . . Then, observing that the coefficient of A w is always of the degree i in n, we have dB dn d?B dUF d»B dn* = B' = B" = dC dn d*C dtf d*C dtf d"C dH* = Ci = C" = C"i = dD dn d 2 D dn 2 d»D dtf d*D dn* dW dn 6 = D' = D" = Di" = D lv = Reverting to (151), we have dF„ dF„ dE dn dm dtf d 3 E dn^ d*E dnF d h E dW d*E dn 6 = E' = E" = E'n = E' w = E v = F> = From (150) we derive whence dT dn dT dn 1 dn dT F'„ = - dn (154) (155) (156) (157) 100 THE THEORY AND PRACTICE OE INTERPOLATION. In like manner we obtain _ dF>. dF'. n JPIII _ dT dn dF" dF 1 ' dT dn dn 1 d*F v dT 1? ' dn 2 dn 1 d"F n dl 1 ~ ^ dn a dn 1 d*F n dT c 4 ' dn* dn 1 d 6 F dT 1? ' dn 6 dT dn m, = ^T dF* On £ dv^. J ( 158) dT dn ' "" ~" dF* dF? Therefore, using (153) and (154), we find F'„ = - (a + B'b+C'c+D'd+F'e + . . . .) (1) F» = ~(B"b+C"e+D"d+F"e+- . . . .) F>J< = 1 (C l "c + D'»d+F m e+ . . . . ) F* = -n (JD ty d + F lv e+ . . . . ) JfJ = \ (E*e+ . . . .) (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'!_ n , F"l n , . . . . , that is, to find the successive derivatives of F(t — nco), 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 F_ n , which has the form* -F_ = F - na + Bfi - Cy + Dh - Ec+ .... (160) where a, /8, y are, in general, different from the differences a, h, c, . . . . of (153). (2) In the present case, we have T = t —na> and therefore dn 1 dT = ~ Z which must be substituted for equation (156) above. * Compare (75), (105) and (Ilia) with (73), (104) and (111), respectively. THE THEORY AND PRACTICE OF INTERPOLATION. 101 Introducing these changes, and operating as before, we obtain the required formulae, namely, FL n = - (u-Bfp+C'y-m + JS'e-. . . .) to F'l n = — i (B"P-C"y + D"& -E»e+ . . . . ) F'" = 1 (C'"y-D"i8 + £:»' e - F l Z H = — . (Z> lv 8- J E' lv 6+ . . . (O FL n = I (^«- . . . . ) (161) It now remains to apply (159) and (161) specifically to each of the several formulae of interpolation, of which (153) is the general type. It is obvious that a particular set of- coefficients, B', B", . . . . , C, G", . . . . , etc., will result in each case. 59. To Compute Derivatives of F(T) at or near the Beginning of a Series. — The formulae adapted to this purpose are derived from Newton's Formula of interpolation (73), which is — where F n = F + n% + Bb + Cc +Dd Q + Ee» + . . . . (162) n(n — 1) n 2 n = ja = 2 ~ 2 ., _ n(n — l)(n — 2) n s n" n = js = ^~ ~2 + 3 _ w(»-l)(n-2)(n-3) _ n" n* 11 n ) (i 63 ) |4 24 4 + 24 4 _ »(»-!). ...(»-4) _J^_^_]_5_n |B ~ 120 12 + 24 12 + 5 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 OE INTERPOLATION. F> (t+nu) = ±(a +(n-i)b t +(f-n+l)e t +($'-in*+iin-$d + (£-/+ i" 2 -§n+lK+ ■ ■ ■ F>»(t + no>) = ^( Co+( »_j K+ ( f »_2n + i)a,+ . . . .) ^ (Ul > F*(t+nm) = i 4 ^ +(n_2)e + . . . . ) ^ v (' + »») = ^(«o+ • • • •) These formulae determine the derivatives of F(T) for any or all values of T between t and t-\-w, 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 tabular value of the argument, say t, we have only to make n = in (164) ; we thus derive the following simple expressions : F'(t) = ±( % -ib +ic -id +ie - *" = f E> n 2 + H » - i f i - f 8 + * »' - | n + i (166) have been tabulated in Table IV for every hundredth of a unit in the argument n. By means of these quantities, we readily compute F'(t-\-n) = - (a + 5'6 +CV +X>^ +£'e N ) (167) The formulae (164), (165), 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 F(T) =0.3T 4 — 2T 2 +4, compute F"(T) for T = 2.8. T F(T) A' A" A'" 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 + 1699.2 + 51.2 224.0 512.0 + 915.2 + 172.8 288.0 +403.2 + 115.2 + 115.2 Here we have * = 2 T = 2.8 - 9 = 0.40 a = + 48.0 b = +224.0 c = +288.0 d = + 115.2 Hence, using the second equation of (164), we find C" 1 = -0.60 D'i = | -fra + H = +0.39I 5 = +224.0 C"c = -172.80 Z»"rf„ = + 45.696 o 2 F" = + 96.896 Whence we obtain F" = 96.896 -M = +24.224 104 THE THEORY AND PRACTICE OF INTERPOLATION. This result is easily verified from the known analytical form of the function ; thus, since F(T) = 0.3T 4 - 22' 2 + 4 we derive F'(T) = 1.2T 8 -4T , F"(T) = 3.6T 2 -4 Substituting T= 2.8 in the last equation, we obtain F"(T) = +24.224 as found above. Example II. — From the table of the last example, compute F'(T) for T=0. Here we employ the first of (165) . Making t = 0, we have a = -3.2 b = +51.2 c = +172.8 d = +115.2 We therefore obtain F'(t) = ^(-3.2-5^+HM- 1 -^) = The result is obviously correct ; for we have F'(T) = 1.2T 3 -4r which vanishes for T = 0. Example in. — Given the following table of F(T)=sin 2 T: compute F'(T) for T=8°36'. T F(T) = sin 2 r A' A" j'» /)iv Jv O 4 8 12 16 20 24 28 0.004866 0.019369 0.043227 0.075976 0.116978 0.165435 0.220404 + 14503 23858 32749 41002 48457 +54969 + 9355 8891 8253 7455 + 6512 -464 638 798 -943 -174 160 -145 + 14 + 15 Here we have t = 8° o> = 4° = -np = 0.069813 + 45 T = 8° 36' 36 4x60 = 0.15 THE THEORY AND PRACTICE OP INTERPOLATION. 105 Taking the coefficients B ', C, D' and E' from Table IV with n = 0.15, and the differences a , b , c , .... from the given table, we find, in accordance with (167), % = + 0.023858 B> = -0.35 b a = +8891 B% = — 3111.9 C" = +0.19458 c = - 638 C'c = — 124.1 D' = -0.12881 d = - 160 Z>« = + 20.6 E> = +0.09358 «. = + 15 ^o = + 1.4 log(o.#'„) = 8.314794 ••■ «*". = + 0.020644 log a) = 8.843937 log F> = 9.470857 ••• J 7 . = +0.295704 This result is easily verified by observing that F'(T) = A( s in2T) = sin2T which, for T = 8° 36', becomes ^'(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 result would be, simply, F'(T) = 0.020644-^-4 = +0.005161 60. To Compute Derivatives of F(T) at or near the End of a Series. — In this case the requisite formulae are derived from Newton's Formula for backward interpolation (75), namely, F_ n = F - na_, + Bb_, - Co^ + Dd_t - 2fc_,+ (168) where B, C, D, . . . . have the values given by (163), as before ; and where the differences a_ l} b_ 2 , c_ 3 , . . . . are taken according to the schedule below : T F(T) j' J" J'" Jiv Jv t — 5a) F-+ «-6 «-4 ^ C-6 C_5 d_ 7 t — 4a) ^-4 *-r, d_t t — 3co F-, 6-4 «U t — 2a> F ., a_5 6_ C -i d-i 6-5 t — a> F-y "'-!! b-. C-s t F Cb-x 106 THE THEOBY AND PRACTICE OF INTERPOLATION. Comparing (168) with the general formula (160), we have a = »_! , fi = &_., , y = c_a , .... Therefore, substituting the previously determined values of B', B", . . . . , C, G", . . . . , etc., in the general formulae (161), we obtain F' (t-nm) = ~ («_!- (n-i) 6_ 2 + (| 2 -« + J) c_ s - (£ s -£ « 2 + ft n- i) d_, + (ft*-5'+ *»■-#» + *) «-.- • • • • F"'(t-n*) = ^ (c^-(n-f) rf_ 4 + (f 2 -2rc + |) e_ 5 - . . . .) > (169) 2? iT (t-nu>) = - 4 ^-(w-2)e_ 5 + J 2?" (i_n») = - 6 (e_ 5 - Making n = in (169) , we have F " (*) = - 2 (&_ 2 +C-8+H<*-4+!e-5 +. . . .) *""(*) = - I (e-.+ id_ 1 +i«_ B + • • • •) V (170) ^ iv (9 = - 4 (^-4+2e_ 5 + . . . . ) ^ v (0 = i( e -* + • ■ • •) As above, we emphasize the relative importance of the first deriv- ative in practice : thus, for brevity, we write the first of equations (169) in the form F'(t-nw) = -(a^-B'b^+C'c^-D'd^+H'e^- . . . .) (171) a) the coefficients B', C, D', W being taken from Table IY 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 THEORY AND PRACTICE OP INTERPOLATION. 107 Moreover, when the interval n approaches unity, formulae (169) and (171) are convenient for computing derivatives corresponding to the argument t -\- not, since they enable us to proceed backwards from the argument t-\- o-i _ 34 20 h 24 m 12 12* = 0.30 which is the interval reckoned backwards from t — Feb. l d h . De- noting the quantity sought by Ja, we then have J a = F'(t—nw) We therefore employ the formula (171) : thus, taking the requi- site differences from the given series, and their coefficients from Table IV, we obtain S m s a_ x = +25 47.79 B' = -0.20 b_ 2 = -17.07 -B'b_ 2 = - 3.414 C = +0.07833 c_3 = - 4.45 + C>c_ 3 = - 0.349 D 1 = -0.03800 d^ = + 0.92 -D'd^ = + 0.035 E> = +0.02009 e_4 = + 0.30 + E'e_, = + 0.006 .-. o>F'_ n = +25 44.07 Whence /la = i^'_ n = 25 m 44 8 .07 -^ 12 = 2 m S 8 .672. The change in a for one minute (j^) is simply Ja J^a = 60 = 2M445 108 THE THEORY AND PRACTICE OE INTERPOLATION. Example II. — From the preceding table of moon's JR.A., compute the hourly variation in A x n. for Feb. 3 d 12 h ; where, as above, z/ L « denotes the change per minute in R.A. Regarding one hour as the unit of time, it is clear that the value of F"(t) given by (170) is sixty times the quantity sought: the ex- pression for the required variation is therefore -£$ F"(t), where £ = Feb. 3 d 12\ Accordingly, using the second of (170), we find Hr. Var. in A x a, Feb. 3 d 12 h , = i X -^-, (-12.62-5.37 + \\ X 0.62 + £ X 0.54) = -0°.00196 Example III. — Given the following values of F(T) = \og e T: find F(T) for T=75. T F(T) = \og c T J' A" J'" A iv Jv 45 50 55 60 65 70 75 3.80666 3.91202 4.00733 4.09434 4.17439 4.24850 4.31749 + 10536 9531 8701 8005 7411 + 6899 -1005 830 696 594 - 512 + 175 134 102 + 82 -41 32 -20 + 9 + 12 Taking t = 75, and using the first of (170), we find F'(t) = -|~° (6899-£p. + 4/ -iy> + i 5 2) = +0.01334 Since F'(T) = ™, we observe that the true mathematical value of the computed quantity is — F'(t) = T V = +0.01333 J Example IV. — From the preceding table of natural logarithms, compute F"(T) for T= 67. We let t — 70, and proceed by the second of (169), observing that 70-67 5 " "'"" Thus we obtain 5_ 2 = -0.00594 C» = n-1 = -0.40 c_ 3 = +102 -C"c_ s = + 40.8 D" = f-* n +\i = +0.197 d_+ = - 32 + D"d_ A = - 6.3 K" = f-n 2 +ln-§ = -0.107 e_ 5 = + 9 -F"e^ = + 1.0 ... ^jpnj^ = _ 0.00558.5 .-. F'< = -0.00022.3 THE THEORY AND PRACTICE OF INTERPOLATION. 109 The true value of this quantity is — *"{?) = ~T* = - py, = -0.00022.27 . . . 61. Derivatives from Stirling's Formula. — When differences both preceding and following the function F(f) are available, formulae more convenient and accurate than the foregoing may be employed. The most useful and important of these are derived from kSTiRLiNG's Formula of interpolation (104), which is — F n = F + na + £b + Co + Dd + Fe + . . . . (172) where the differences are taken according to the schedule on page 62, a, c, and e being the mean differences denned by (101) ; and where B, C, . . . . have the values »-% C = D = E = n(n 2 — 1) n s 6 6 n 2 (n 2 -l) n* n 2 ( 173 ) 24 24 re (re 2 -1) (re 2 -4) 120 n 6 n? "24 n 5 n 3 n 120 "^^SO Whence, deriving the values of B', B", . . . . , C, C", . . . . , etc., from (173), and substituting these (with the above differences) in the general formulae (159), we get F> (t+nu>) = ^a + nb + &-i)c+ (f- T %)d + (£-tf + ^)e + F»(t+no>) = ± (b + no +(¥-&)d + (%*-«) e+ . . . 2?'" (*+««*) = ±- s (c+na +(t-i)e+ . . . .) . _ F* (t+no) = -fd +ne + . . . CO F- (t+nw) = -j(e + 110 THE THEORY AND PRACTICE OE INTERPOLATION. Making n = in (174), the latter become F> (t) = -( a - icWl5 e~ . . . .) F "(f) = ^,ft-A^ •■••) *""(*) = ^ («-*« + • • ■ •) i " v W = i ( e ~ ■ ■ ■ ) (175) Again, writing — » for w in (174), we obtain F' (t — na F" (t-nw F»'(t-no> F lv (t—nm F v (t—n = n C — " 2 D< it n „2 E 1 = » — a" -4- i (177) are given in Table V with the argument n. The quantity J?' (0T) is thus readily computed (for any value of T) by either one or both of the formulae F'(t+na>) = - (a+nb +C'e+D'd +Z;ie) F'(t-no>) = -(a-«5 +C"c-2)W +JS'e) (178) (179) in which the odd differences are algebraic means of the tabular differ- ences, taken as indicated below : THE THEORY AND PRACTICE OE INTERPOLATION. Ill , T F(T) A 1 J" Jill /Jiv JV t 0) F-i a' b> c' d> 6> t F (a) &o (0 d (e) t+ 0} *i «1 w <>1 d, «i The formulae (174) and (175) may also be obtained by the fol- lowing method, which reverses the preceding order of development by deriving first the particular, and from the latter, the more general of the two groups in question. Expanding F(t-\-not) by Taylor's Theorem, we have F(t+nm) = F (t) + nwF> (t) + — F» (t) + — F'»(t) + |2 [3 (180) Arranging Stirling's Formula (104) according to ascending powers of n, we find F(t+na>) = F + n(a— £c + -^ e — + -(e-ie+ + E. (181) Whence, by equating coefficients of like powers of n in the equivalent expressions (180) and (181), we obtain mF> (t) = a - i c + ^ e - . o> s F'"(t) = c -ie+ . . . . o'F»(t) = d W (t) = e - (181a) which agree with the formulae (175). Again, by Taylor's Theorem, we have F> (t + nm) = F> (t) + n<*F» (t) + — F'» (t) + li F"(t + n») = F" (t) + nwF'"(t) + — F* (*) + li 112 THE THEORY AND PRACTICE OE INTERPOLATION. which may be written in the form 1 / n 2 F' (t + «ft.) = ^ Uf' (t) + na>*F» (t) + T co s F»> (t) + 1 / n 2 F»(t+no>) = -J^F"(t)+ n w s F'"(t)+ — u'F* (t) + w \ |2_ ) Substituting in these equations the expressions for a>F' (i), oo 2 F"(i), . . . . , as given by (181a), we get F< (t+nw) F" (t + nw) F ir (t + n(t + a>) t + 2a> F, F'(t+2o>) *It is evident that F'{t-\-nu) can be derived from the column F' (T) by direct interpolation : moreover, when the tabular values of F'(T) are thus available, this method of computing F 1 (t-\-na) is more expeditious than the use of formula (178). THE THEOEY AND PRACTICE OP INTERPOLATION. 113 The first of the formulae (175) 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 reason we formulate the following Rule for computing the first derivative of a tabular function corresponding to one of the given functional values : From the mean of the two first differences which immediately precede and follow the function in question, subtract one-sixth (|) the mean of the correspond- ing third differences, and divide the result by the tabular interval. This rule neglects only 5th and higher differences. To include 5th and 6th differences, add to tlie above terms {before dividing by a) one- thirtieth (^\) the mean of the corresponding fifth differences, and divide by eo as before. 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 use of the preceding formulae. Example I. — Given the following ephemeris of the sun's decli- nation (S) : compute the hourly difference in 8 for the dates Jan. 7, 10, 13, and 16. Date Sun's Decl, A' J" J'" 1 r Diff. for 1898 S 6 C 1 hour Jan. 1 4 O / It -22 59 2.4 22 41 38.5 i it + 17 23.9 21 26.1 25 23.0 29 13.5 32 56.9 36 32.2 + 39 58.2 / n +4 2.2 // -5.3 6.4 7.1 8.1 -9.3 // n it 7 22 20 12.4 3 56.9 + 1404.55 + 0.98 + 19.52 10 21 54 49.4 3 50.5 1638.25 1.12 22.77 13 21 25 35.9 3 43.4 1865.20 1.27 25.92 16 20 52 39.0 3 35.3 + 2084.55 + 1.45 + 28.97 19 22 20 16 6.8 -19 36 8.6 + 3 26.0 The term ^ e in the first of (175) is here insensible ; hence, for each of the given dates we have only to compute the quantity F>(t) = ha-ic) Accordingly, in column a we write the required mean first differences, expressed in seconds of arc. The next column contains minus one- 114 THE THEORY AND PRACTICE OE INTERPOLATION. sixth of the corresponding mean third differences. Finally, since o) = 72 hours, we write in the last column y 1 ^ of the quantities formed by summing the corresponding terms of the two preceding columns. We thus obtain the hourly differences required. Example II. — Compute, from the ephemeris of the last example, the daily motion in declination for the date Jan. 6 d 13 h 30 m . We proceed backwards from Jan. 7, using the formula (179), and taking the coefficients from Table Y with the argument 7 d h m -6 d 13 ,, 30 m 10 h .5 72 h = 0.14583 Thus we find n = 0.14583 C = -0.1560 D 1 = -0.012 b = +236.9 c = - 5.85 d. = - 1.1 a = +23 24.55 -n5 = - 34.55 ■+ C'c = + 0.91 —D'd n = - 0.01 .-. wF'^ = +22 50.90 Whence, for the daily motion in 8, Jan. 6 d 13 h 30™, we obtain F 1 ^, = 22' 50".90-^3 = +7' 36".97 Example III. — The following table gives F (T) = e T , where e denotes the base of natural logarithms: compute F'(T) for T=0.30. T F(T)=e T A' J" jin Jiv Jt 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.000000 1.105171 1.221403 1.349859 1.491825 1.648721 1.822119 2.013753 + 105171 116232 128456 141966 156896 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 find -) A— 6 jf"(0.30) = ^-(135211-^ + ^) = 1.34986 It will be observed that our result is substantially equal to the value of F(T) for the same argument, 5 7 =0.30: this is required by the relation F(T) = F'(T) = F"(T) =....= e T 6 = +0.014930 e = +1496 „.<■ = + 927.5 d = + 152 D»d = + 16.6 e = + 14 E"e = - 1.6 THE THEORY AND PRACTICE OP INTERPOLATION. 115 Example IV. — From the table of Example III, compute F" (T) for T = 0.462. Taking t = 0.4 and n = 0.62, we obtain, by means of the second of (174), n = 0.62 D" = | 2 - T V = +0.1089 E» = f- » = -0.115 ... ^/'V = +0.015872.5 ... i?/' = +1.58725 The true mathematical value is — F"(T) = F(T) = e T = e 0M * = 1.587245 . . . 62. Derivatives from Bessel's Formula. — Other useful formulae, convenient for the computation of tabular derivatives, are those derived from Bessel's Formula of interpolation (111). The latter may be written in the form F„ = F + na x + Bb + Cc x + Dd + Ee 1 + . . . . (183) where the differences are taken as in the schedule on page 62, o and d being the mean differences defined by (106) ; and where B, G, . . . . have the following values : n(n — 1) m 2 n = 2 = "2 _ 2 _ n(n — l)(n-b) _ w 8 _ ra 2 _re _ 6 ~ 6 4 + 12 (n + l)n(ra — 1)0— 2) _ ?i 4 _ w"_ __ w a ra_ (184) "° ~~ 24 24~l2~24 + 12 (n + l)«.(w-l)(re-2)(w-j) _ ?i 6 w 4 ra 2 m -^ - " 120 = 120 _ 48 + 48 120 Deriving from (184) the values of B', B", . . . . , C", C", . . . . , etc., according to (154), and substituting these in the general form- ulae (159), we obtain 116 THE THEORY AND PRACTICE OP INTERPOLATION, F> (* + «») = ^(a 1 +(rc-i)&+(f-f + T VK + (f 8 -? 2 -T , H &)d + (j? — tV + 55 — T2rr) e i+ • • • ■ J JP» (*+n«>) = 1 (j + (n-i) Cl + (| 2 - 1- T V) d + (S "- f+ AK + • • • • ) 7?'" (*+««,) = I I ^+(»_4.)d+(« f _})e 1 + . . . .) ) (186) F* (< + n«) = 1^+. . . . ) Putting « = in (185) , we get F< (t) = _(o 1 _*&+Tfrc l + 1 i g «Z-. I .i T * l - . . . .) U) ^" (0 = K (»-K-t^+A«i +••••) *""(') = - s ( c l-H+o* + . 0) • • ) ^ lv (0 = -«(«*-K- • • • • ) Fv w = i ( 6l ~ . . ■ . ) 0) (186) Again, putting n = \ in (185), we obtain the following simple formulae : F' (*+*«.) = -(a.-^e. + ^e,- . . . .) (187) F»(t+i») = - t (b-&d + 0) . . . •) 0) ■ ) ^'(H-i-) = ^(d- , . . ■) F- (t+ io.) = I (e, - . . . •) which determine the derivatives of F(T) at points midway between the tabular values of the function. It is important to observe that, *The coefficient of e 1 vanishes. THE THEORY AND PRACTICE OF INTERPOLATION. 117 unless third differences are considerable, a close approximation to F' (t-\-\ to) is given by the simple expression F'(t + lw) = ^ = Fl ~ F ° (187a) which differs from the exact formula only by the omission of the small quantity i(- A j<»+. . . .) The formulae for the derivatives of F(t — nw) are deduced from (Ilia). Let us put, for brevity, b = i(b + b>) , d = l(d +d>) (188) and (Ilia) becomes F_ L = F„ - na' + Bb - Co' + Dd - Ee' + . . . . ( 18 9) o Comparing this expression with the general formula (160), we find that a, fi, y, 8, e, . . . . , in the latter, are replaced by a', b, c, d, e', .... in (189) ; hence, observing these changes, and substituting the above determined values of B', B", . . . . , C, C", . . . . , etc., in the formulae (161), we obtain F< (*_«„) = L (a'-(n-0+ &-% + ^)c>-(f-f-ft+ t \)d _L/7l 4 n s I n l \ „l T^51 — T2+24 TS5,> e — • • ■ F» (t-nm) = ^ (b-(n-i)c> + (| 2 - 1- - ^)d-(f-f+ 5 > 4 )e'+ . . . F'»(t-n») = ^(c'-(n-t)d-+(f-%)e>- . . . .) ) ( 190 ) F* (t-nm) = l(d-(n-i)e<+ . . . .) J-t ( ,_ M(o) = 1(V-. . . .) The values of B', C, U, and JEJ', as computed from the expres- sions B< = n - i , D' ,2 C = £ - s + T V , ^ 1 - » _» _ » J.JL 1 / _ n 4 _ « 8 . » _ i f ( 191 ) — n iitn T2TT ) 118 THE THEOET AND PRACTICE OE INTERPOLATION. are given hi Table VI with the argument n. By means of these co- efficients, values of JF\T) are readily computed from either one of the formulae 1 F'(t+n„>) = -(a 1 + B'b+C'c 1 +D'd+£;ie 1 ) to 1 (192) F> (t-n a' (b) d (d) e' t Jf\ b« ^ cii (b) Cx (d) = -0.3740 C = +0.0282 D' = +0.0692 b = +1.21 e t = -0.30 d = -0.365 a, = +48.95 B'b = - 0.453 C' Cl = _ 0.008 D'd = - 0.025 m F' = +48.464 D = F' = 48 m .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, V = 24J"'(*+i«.) = "^(b-&d+ . = ^(1.33 + ^x0.37) = +0 m .059 ) (>3. Interpolation of Functions by Means 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. 2d 3d t — 2«) t — t + 2w *-* *; F, F, FU FU F' F> F 2 ' a" «' «i a 2 ft y' 7i 122 THE THEOEY AND PRACTICE OP INTERPOLATION. We shall assume that the differences of F(T) beyond J iv may be disregarded; hence the differences of F'(T) beyond y may be neg- lected in the above schedule. Now, by Taylor's Theorem, we have F n = F + n»F t >+ ^ F<> + ^F t '" + ^ *»+.... (194) If. li li Again, since dF< „,„ we obtain, by means of the formulae (175), *•." = -(«-*?) , ^'" = -J . ^o IV = -. (195) (1) 0) 0) in which we have put, for brevity, « = H«'+«i) , y = Hy'+yx) (196) Substituting these expressions for F ", F '", and i^ iv in (194), the latter becomes ., _ . _ , n 2 a> n s a> „ Jl 4 (« J". = F + n»F t >+ — (a-iy) + — ft + — y li li li which may be written i^ = ^ + n» (V '+ | a + I s ft + T \ (I' - 1) y) (19T) By means of this formula we compute F n in terms of the differences of F'(T), instead of the differences of F (T) direct, as in the usual formulae of interpolation. Substituting — n for n in (197), we have F_ n = F - no> (V '-f a+f ft- A (« 2 -l) y ) (198) The values of b = i , r = A(J*-i) (199) THE THEORY AND PRACTICE OE INTERPOLATION. 123 are given in Table VIII with the argument n. By means of these coefficients we readily compute F H = if +nai(.F '+Sa + B/J + ry) F_ n = F -n„ (iP'-| a + Bfl,-r y ) (200) (201) The coefficients in Table VIII are not extended beyond ?i = 0.60, since by this method it is invariably more convenient to proceed from the nearest function F Q . Example. — From the American Ephenwris for 1898 we take the heliocentric longitude of Mercury, together with the daily motion in longitude, for a portion of the month of October. The differences of the daily motion are then taken, as shown below : Date 1S98 Helioc. Long, of Mercury Daily Motion u. P r s O 1 It o / // / II 1 II II It Oct. 11 17(5 51 7.8 4 2 34.3 -14 0.0 12 17.5 10 40.4 9 9.4 - 7 45.8 13 184 41 59.2 3 48 34.3 + 1 42.5 -5.4 6.1 -7.4 15 192 6 33.3 3 36 16.8 1 37.1 -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 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 d 11" 21 m .O. Here we have * = Oct. 15 d T = Oct. 15 d 14" 24 ra .O = Oct. 15 d .60 <« = 2 d na> = T-t = d .60 n = 0.30 Hence, using Table VIII, in connection with (200), we obtain F = 192° 6 '33^3 5 = +0.15 B = +0.0150 T = -0.0239 Whence « = -11 28.95 ft = + 1 37.1 y = _ 5.75 F > = +3 36 16.8 «a = - 1 43.34 Bft = + 1.46 r y = + 0.14 Sum, D = +3 34 35.06 F = F + «« D = 194° 15' 18".3 124 THE THEOEY AND PEAOTIOE OP INTERPOLATION. Differencing the given series of longitudes and applying Bessel's Formula of interpolation, we find F n = 194° 15' 18»2 64. Application of the Preceding Method of Interpolation when the Second Differences of the Series F (T) are Nearly Constant. — When the 3d and 4th differences of F (T) are small enough to be neglected, we may omif the terms containing /3 and y in the formulae (197) and (198) : we therefore obtain F. = *■„+»»«■, (*■„' + *«) (202) F_ n = F 9 -nm(FJ-\a) (203) It will be interesting to determine the error of these approximate formulae as applied when the 3d differences of F(T) are appreciable. For this purpose we write (197) in the form F n = F + no, (F > + | «) + | 3 uj 8 + (| 4 , - T » 5 2 ) . Hence, even when third differences are considerable, these formulae are suf- ficiently accurate for many purposes. THE THEORY AND PRACTICE OF INTERPOLATION. 125 That the formulae (202) and (203) are rigorously true when the 3d differences of F(T) are zero may be clearly shown from geo- metrical considerations, as follows : The 2d differences of F (T) being supposed constant, it follows from Theorem VI that the function is necessarily of the form F(T) = a T 2 + a y T+ a 2 (207) Now, if in the accompanying figure we draw the rectangular co- ordinate axes OT and 01", and plot the curve defined analytically by (207) (regarding y = F(T) as the ordinate corresponding to the abscissa T), it is evident that we obtain a parabola whose axis is parallel to OY. Let us now take OM = t OS = t + w ON = t + wo) Whence MN = nm MP = F(t) = F NQ = F(t+nu>) = F„ M K N Draw the tangents PA, QL ; also, draw PD || QL and PB || MN. dF Then, denoting -5™ by F n ', we have F > = UuAPB FJ = tan DP B Hence we find NA = MP + PB tan APB = F + nu >F > iVX> = MP + PB tan DPB = F n + nwFJ It is therefore evident that to find JSTQ = F n , which lies between N~A and 2TD, we must employ a value of F' somewhere between the values F ' and F n '. Now, let KE be the ordinate erected at the mid- dle point of MJST, and FH the tangent at E. Then, by an elementary 126 THE THEORY AND PRACTICE OF INTERPOLATION. theorem of the parabola, the chord PQ is parallel to EH, and we have, therefore, F NQ = MP + PB tan 0P5 = F + n + | a = 13".732 - 0.217 X 0".147 = 13".700 Whence we obtain 8 = 6° 43' 39".0 + 78 X 13".700 = 7° 1' 27".6 Since the value of n is nearly one-half, we may interpolate back- wards from July 9 d 7 h with equal facility : thus we find n = 0.567 f = 0.283 .. FL? = 13".582 + 0.283 X 0».155 = 13".626 THE THEORY AND PRACTICE OF INTERPOLATION. 127 Whence 8 = 7° 24' 37".4 -102 X 13".626 = 7° V 27".55 which substantially agrees with the above result. Example II. — From the following table of the moon's horizontal parallax (tt), interpolate the value for July 10 a 16" 24 m .O. Date 1898 Moon's Hor. Parallax Diff. for 1 Hour a July 10.0 10.5 11.0 11.5 / // 56 26.1 56 2.5 55 40.7 55 21.1 -2M 1.89 1.73 -1.55 + 0.15 0.16 + 0.18 Here we have T = July 10" 16\40 to = 12 hours n = We therefore obtain t = July 10" 12". 00 4\40 = 0.367 0.183 Fi = -1".89 + 0.183 X 0".16 = -1".86 .-. jt = 56' 2".5 - 4.4 X 1".86 = 55' 54 ".3 Interpolating backwards from July ll d 1 ', we find v = 55' 40".7 + 7.6 X 1".78 = 55' 54".2 65. Choice of Formulae in a Given Case. — When derivatives are required to be computed at or near either the beginning or the end of a tabular series, the formulae derived from Newton's Formula of interpolation must necessarily be employed. In all other cases, the choice lies between Stirling's and Bessel's forms, and should be decided by the value of n. When n = 0, the formulae (175) are un- questionably the best. When n = i, 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 THEORY AND PRACTICE OP INTERPOLATION. EXAMPLES. 1. Given the following table of "Latitude Keduction" 9 — ' corresponding 1 to a change of 10' in , for = 9° 30'; also for F(T) F l F, ^- 2 F. A' 4' 4' A' J" 4," A" 41+ 4/" 4'" Jiv ^ 4L Jv dj Let it be required to find from this table the value of X(4«u F(T)dT (209) 132 THE THEORY AND PEACTIOE OF INTERPOLATION Since dT = oidn T = t + nto ) We have ,7T _ ...^ f ^ 21 °) and therefore Xt-Ha ft F(T)dT = o)j F(t + na>)dn (211) Now, by Newton's Formula, we have F(t + nm) = F + nJ ' + BJ " + CJ "i + DJ* + .... where B, C, D, . . . . denote the binomial coefficients of the nth order. Multiplying by dn, and integrating, we obtain CF(t+no>)dn = C(F + nJ ' + BJ »+CJ '"-± . . . . )dn or CF(t+na>)dn = nF + f +J "CBdn + J >"Ccdn+ . . . . +M (212) where M is the constant of integration. If, for brevity, we put P = Chdn , y = Cbdn , 8 = Cbdn , .... (213) then, from the preceding equation, we derive §F(f+nu)dn = F + i/i ' + pJ " + yJ '" + SJ^ + . . . . (214) Whence we obtain, in succession, CF(t+nu)dn = CF(t + io + no>)dn = F 1 + bJ 1 i + pJ 1 "+ y J 1 '" + CF(t + n+no>)dn= F 2 +iJ 2 '+pJ 2 "+y/l i ll '+ (215) C'F(t +nw)dn = fF(t + i-lo>+ mo) dn = F^+i JU + /3JIU + y^d + . . Summing the integrals expressed in (214) and (215), we find r=i — 1 r=r» — 1 r=i — 1 r=i — 1 J>(<+n») dn=^ F r +i £ A' + P 2 J " + ? 2 J "' + ' ' ' ' ( 216 > r=0 r=0 r=0 r—Q The numerical values of ft, y, 8, . . . . (sometimes called the coefficients of quadrature) must now be determined. These may be THE THEORY AND PRACTICE OE INTERPOLATION. 133 found directly by integrating the expressions for B, C, D, . . . . , as expanded in (163), and then taking the limits of n according to (213). But the following indirect method seems preferable, since it adds a significance to the result. Let us put Q = C(l + y) n dn = C(l+ny + By i + Cif+Dy i + . . . . ) dn (217) where y is supposed constant. Then, if we also put Qi =fjl+yydn we shall have Q< = 1 + \y + &/ 2 + yrf + 8y* + ey* + (y* + . . . . (218) the coefficients being those defined in (213). Again, put that is and we find or (1+y)- = » (219) wlog(l+y) = logs log (1 + y) . dn = — «*» = 1 /f+ x ( 220 ) log (1+y) We therefore obtain /C C dz z (1+ ?/)" (l + y)"dn = I zdn = I , — ... . , = = — ... . . + const. = v J ' + const. V yj •' J\og(l+y) \og(l + yy log(l+y)^ Whence (1 + y)- y , log (1+y) ~ ^ 2 i ~3 4 + 5 ' " - " } y— 5 + s s + ■ • Expanding the last expression by the Binomial Theorem, or by direct division, we obtain Q' = i + iy - &y* + &y* - j\%y* + ?hy 5 - *snx>y s + • • • • (221) Whence, comparing (218) and (221), we find 3 P = "A e = + y = + A £ = -»»ih ( (222) 8 = -^ftf 134 THE THEORY AND PRACTICE OE INTERPOLATION. which are the numerical values of the coefficients of formula (216). It therefore appears that the fundamental coefficients of quadrature are those in the expansion of [log (l-J-2/)] -1 - Let us now regard the functions F , F 1} F % , . . . . F t as first differences of an auxiliary functional series which we shall designate 'F. A schedule containing the new series may be conveniently arranged as follows : r 'F F{T) A' An A'" 'F* t F o >F 1 4/ t + O) Fi 4," ' F 2 A' A'" t + 2u> F s A'' '■*•. 'F. , A'- , AI'' t + (i-l),o 'Ft ■*U A'. A'U t + iia '^m F t The value of 'F is entirely arbitrary. Having assigned a con- venient value to this quantity, the remaining terms in the series are readily formed by successive additions, thus : >F l = 'F + F , >F 2 = 'F 1 + F 1 , , iF i+1 = >F t + F t "We shall now put the second member of (216) under a form more convenient for computation. By Theorem I, we have r = i — 1 J] F r = F^+F.+F, + . . . . +F { _, = iF t - >F r = r = i — 1 ^ 4' = 4/ +4'+4' + • • . . +A>^ = F t - F r = r=i — 1 ^ AJi = A ii+ A ii+ A 'I + . . . . +A'U = A i - A i r=i — 1 2 A'" = A "i+A 1 i'i+A 2 'i'+ .... +A»U = A>>-A " (223) and hence (216) becomes fF(t+ na) )dn = {'Fi-'FJ + i(F i -F ) + ^{Af-A>) + y(A i "-A 'i) + l(A!H-Aii') + t(A?-A\?) + . . (224) THE THEORY AND PRACTICE OE INTERPOLATION. 135 This formula possesses the disadvantage of involving differences 4/, A"> A'"> .... which are not furnished by the foregoing schedule. To obviate this difficulty, we proceed as follows : Put V FA' A" //'" Hence the result for q will immediately follow when the result for 6 has been derived. But we may obtain 6 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+3 = — L- . 1— u whence log (l+i») = _log(l-M) (231) Therefore, by (228), we find 1 1 * 6 = = jy-r—^ = -, — ^ r = w- 1 -£w°+/3«- y w 2 +8M 8 -eM 4 +£M 6 - .... (232) log(l+a;) log(l— u) ' K Accordingly, writing q for 0, 'F^ for u~ x , F t for u°, j'^ for u, etc., as justified by the preceding reasoning, we obtain q = '^ +1 -\F t + M'i-i - r*lU + M'A - <*t* + WU - • ■ • .. (233) Substituting this value of q in (226), and grouping like terms, we get JF(t + na ,)dn = ('F^-'FJ -i(F t +F ) + /3(J'^-J> ) -y(A[L 2 +JJ') + 8(J»i s -J t) <»)-*(A£ i + 4j-) + • • • • (234) Whence, restoring the values of fi, y, S, . . . . , as given in (222), and applying (211), we have F{T)dT = u\F(t + nu)dn = »{ C-P'm- '^o) - i ( F < + K) ~ A (A-i-40 - A (4^+4/0 - t¥* (4^W„"0 - t** (4^ + 4*) - *&St* (AU-*?>- • • • • | (235) THE THEORY AND PRACTICE OP INTERPOLATION. 137 When 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(T)dT = mJ o F(t+no>)dn = »!('*;- 'iQ + i (Ft-F,) - ^ w-jj) + & (4"-4>") - M (4"'-4>"') + T | T (4 iv —V T ) - ^Mf T (4 V -4> T ) + . | (236) We here emphasize the fact that the value of 'F is wholly arbi- trary. 68. As an example in the use of formula (235), let it be required to find* A' = j cos TdT 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 are consistent with a convenient schedule of differences. In all cases the differences should sensibly vanish beyond the third or fourth order. Adopting cu = 4° as a suitable interval in the present instance, we obtain the following table of F(T) = cos T : T '") = - 1.7 A™ + A* = + 48 - dfoW + 4 iv ) = - 0.9 log^ 1 = 0.703392 sum, .£ = + 5.051170 log ft) = 8.843937 o) = 4° — JL "45 logX = 9.547329 .-. X = 0.352638 Since j cos TdT= sin ^F, we find for the true value of the defi- nite integral, X = sin 44° -sin 20° = 0.694658-0.342020 = 0.352638 If it be required to compute X = I cos TdT from the foregoing table, formula (236) at once serves the purpose. Thus we obtain (» = 2) 'F a ->F a = +1.853238 F 2 -F = -56745 + i (F 2 -F ) = - 28372.5 4/ -J ' = _ 8752 - A (4,' - 4' ) = + 729,3 JJi -A " = + 318 + AW- 4") = + 13.3 4"' - 4"' = + 43 -VftW- 4>'") = - ii 2 1 = +1.825607 .-. X = 0.127451 Here the true value evidently is — JT = sin 28° -sin 20° = 0.127451 69. Precepts for Computing the Definite Integral when One or Both Limits Fail to Coincide with some Tabular Value of the Argu- ment T. — Thus far we have considered the limits of the integral F(T) dT to be of the form T' = t + I'd) T" = t + i"a where *" and i" are integers, and hence T' and T" are two particular THE THEORY AND PRACTICE OF INTERPOLATION. 139 values of T for which F (T) has been tabulated. We shall now con- sider the more general problem of finding X when the limits have the form T> = t + n'm , T'l = t + n"o> where n' 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 J" 42° 46' COS J 21° 18' 8' 42" 46' 54" TdT 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 just mentioned, we have computed the values of X from the lower limit T' = 20° to the upper limits T" = 4A° 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 readily 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 J* 21" 18' 87" /•42» 46' 64" cos TdT X" = I cos TdT 20° Jw and therefore J '42° 46' 64" cos TdT = Xii- Xi 21° IS' 87'' We leave the detailed solution of this example to the student as a valuable exercise, exhibiting the spirit of the method employed in problems of this type. The process actually used differs somewhat in 140 THE THEORY AND PRACTICE OE INTERPOLATION. form from the method here explained ; but the principle remains the same. We proceed to develop the general formulae. 70. Let us put li = J F(t + na>)dn (237) and * (i) = 'F t + i F t + £4' + y J t " + 84'" + eJ? + . . . . (238) where i denotes an integer. Then (224) becomes I, = *(i)_*(0) (239) Let us now suppose that (239) has been computed for * = 0, 1, 2, 3, 4, .... , in succession. Then, from the series of values I = *(0)-*(0) I x = *(l)-*(0) J 2 =.*(2)-*(0) (240) thus determined, it is evident that any intermediate value, say /„, can be found by interpolation. To derive a general formula for this pur- pose, we must express the differences of the series (240). Now, by (238), we have * (0) = >F t + iF + pJ > + y4" + 84/" + *4 IT + . . . . * (1) = l F l + %F 1 + /3J/ + y4" + 84'" + £z// T + . . . . * (2) = 'F 2 + $ F 2 + /?4/ + yz/ 2 " + 84'" + e4 iT + . . . . (241) whence, observing the general relation 4&-4M = JM we derive the following schedule of differences : Function 1st Differences 2d Differences 3d Differences J = *(0)-*(0) j x = *m-*(0) Z 2 = *(2)-*(0) /, = *(3)-*(0) F 1 + iJ i ' + pJ 1 " + yJ 1 "' + . . . ^ 2 +i4' + / 84" + y4'" + . . . 4'+ W+W+- • • 4'+£4"+/?4»'+. . . 4'+- 2 -4"+/?4'»+. • • 4"+i4'"+. . . 4"+i4"'+. . . Therefore, applying Newton's Formula of interpolation, we have = T + n (1st Diff.) + B (2d Diff.) + C(3d DifE.) + . = *(0)-*(0)+»(^,+*4/+ j 84/'+y4/ // + . . + 5(4'+i4"+/34'"+ • . ) + (7(4"+£4'"+ ) , ) + J>(4"'H- . . )+ . THE THEORY AND PRACTICE 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 J„ + *(0) = ('F +iF +pJ '+ y .1 "+SJ a '"+ . . . .) +71(^+1 4/+/H"+y4/"+ • • ■ •) +B(J '+tJ "+pj »'+ . . )+C(4/'+Mo'"+ • • )+-Z>W"+ - ■ )+ • • Upon arranging the last expression according to the coefficients 1, -\, /3, y, 8, . . . . , it becomes J„ + *(0) = (>F + nF +B.J '+CJ " + n.J '"+ . + 1(F +u.1 > + BJ o h+CJ >"+ . +P(,l a '+nJ " + BJ '"+ . + y (,l u "+nJ '"+ . + 8(4/"+ . + . Now, it will be observed that the first polynomial in the second member of this equation is simply the expression for 'F n , — the quantity derived from the series 'F a , 'F 1} 'F 2 , .... by interpolation. Simi- larly, the remaining parentheses contain the expressions for F n , j' n , J n ", . . . . , likewise derived by interpolation from their respective series. We therefore have I„ + *(0) = 'F„ + \F n + pj' n + yJ n " + 84,"' + = *(») (242) Whence CF{t+nu>)dn = /„ = *(?i)-*(0) 71. In like manner, if we put V(i) = 'F^-tF. + IU'^-yJlU + MHU-. ■ ■ ■ then, by (234), we have J ( = CF(t + nto)dn = g>(i)-*(0) Therefore, by interpolation (reasoning precisely as above), we obtain (243) (244) C'F{t + nu>)dn =

)dn = *(ni)-*(0)~ , J F '(t+ wo) dn = q> (re") - * (0) the difference of which gives F(t + nw)dn = \(>F n ->F° ) + t(F n -F )-^(J< n -J> )+^ (z/„"-4,'0 - t¥V W'-4,"0 + tI* (^ v -^ v )-^IItt (4T-4T) + • • • • I (247) F(T)dT = o>J.F(i + re)dn t+n'U) */n f = <->K'K» + l-'Fn>)-i(Fn<> + K>)-T\ W^-i-J'J-b (4^- 2 +49 - 1\% WU-^') -t»t (4^-4+4?) - F *i | ff (4V-.-4W - • • • I (249) In these formulae the quantities n, n! and w" are either proper fractions or mixed numbers; while the value of 'F is wholly arbi- trary. It frequently happens that we have to compute X = CF(T)dT for several different values of T; the lower limit remaining fixed and equal to t. In such cases it is convenient to determine the arbitrary quantity 'F Q , in (247) and (248), such that the sum of the terms having the subscript zero will vanish. Accordingly, we may arrange these formulae as follows : THE THEORY AND PRACTICE OF INTERPOLATION. 143 Tate IV — 1 F X 1 A ' 1 //"J- 19 // '" 3 //lv I 863 /f -^0 — — f -^0 + TS^O ^¥^0 TTif!!^ TTTTT ^0 + F7J3S7J ^i Then (a) When the upper limit falls near tlie beginning or middle of the tabular series, find F(T)dT = «>JF(t + no>)dn = fo(!F n +^F K -^J' n +^J n "-^ a J n '"+ T i xs J^- v %ii J[ Jz + . . . .) (b) When the upper limit falls near the end of the series, find Xt+ntii f*n F(T)dT = <»\F(t+nu)dn = »('F*i-t*.-M'^-&J(U-f&4!!U-Thi*l+-- B mxfJl+- ■ • •) (250) Example I. — Let it be required to find X = 10 dT Vt(i— t) ' 0.42787 Here we adopt the interval is avoided. We therefore compute F(T) = 0.02 X 10 0.2 Vr J" jui Jiv v ' \/T(l—T) 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.000000 0.405220 0.808132 1.209418 1.609738 2.009738 2.410058 2.811344 0.405220 0.402912 0.401286 0.400320 0.400000 0.400320 0.401286 -2308 1626 966 - 320 + 320 + 966 + 682 660 646 640 + 646 -22 14 - 6 + 6 + 8 8 + 12 144 THE THEORY AND PRACTICE 01" INTERPOLATION. The computation is now readily effected by formula (249) . Taking t = 0.42, we make 'F = 0, and complete the auxiliary series 'F. For the values of n' and n", we have \ 0.42737-0.42 0.02 0,53054-0.42 M2 = 0.3685 = 5.5270 = 6 - 0.4730 Fn" = + 0.400748 J'«>-1 = + 659 A 11 = + 642 Jill = "Whence, interpolating by Newton's Formula, we obtain iF„ = +0.149636.4 F n+1 = +373075.264 n = 5 -0.20 F„ = +187307.52 - i F n = - 93653.760 n — 1 = 4 -0.20 A' n _ x = + 81139.2 - tV ^'n-i = - 6761.600 n-2 = 3 -0.20 A'^ = + 27648.0 - *»* J'U = - H52.000 n — 3 = 2 -0.20 A'lU = + 5760.0 -ii^^nU = - 152.000 .-. X = +271355.904 which is mathematically exact. 72. Quadrature as Based upon Stirling's Formula of Interpo- lation. — The preceding formulae of quadrature obviously involve the same disadvantages as are inherent in Newton's Formula of interpo- lation. We now proceed to integrate the expression for F(t-\-n -*0 *-* ^o' ^-,1 '*» *\ J!" t + u> ^i j;' 'Ji J'. z/.i" t + 2o> *i 2 4' t + (t-l)o t + iw '^-» ^'w 4' ^ t+ (t+l)tt> t + (t+2)) dn = F + & J " - ,Hi 4 V + iJWn 4 vi (253) (254) 148 THE THEORY AND PRACTICE OF INTERPOLATION. In like manner, we derive fb(t+na)dn = i^+^zV'-^^+inrWin,^ 1 - • • • • Whence, by summation, we obtain J"£*(*+n«,) dn = ^ ^ + ^2 ^" - ****E J ' T + ™W™2 J '' ~ • • • ' (255) ~~ * r=0 r = r=0 r = Upon substituting the relations 2 K = Ji+Ji+Ji + ...+*;= '^ - /i?, -i r = r = i 2/1," = J t » + J 1 " + J t "+ . . . +4"= /*'*,- ^'_» r=0 (256) in formula - (255), the latter becomes jT^*+ «») dn = ('i^ - '*■_,) + a (4' +i -^-0 - ,^ (4+> -^-j) + ^\%W(^ +i -^j)- • • • • (257) Finally, therefore, we obtain F(T)dT = o)| ^(i+nai)^ e-ju) «/— i = „j(/^_'^+A(^Hi-^)-i«^m-^) + 1 riM^(^i-4l i )- ....J (258) When several values of an integral are to be computed from a given series, each having the lower limit t — ^w, it will be more convenient and expeditious to determine the arbitrary quantity 'F-% such that the sum of the terms with subscript — J is equal to zero. The formula (258) may therefore be written as below : IW 1 //' _L 17 //'" 367 /fr _L \ •^-J — IJ ^ T itn d H TFTTCT "4 1 ... j J"(-H(J+iU) /*i+i I F(T)dT = to \F(t+na>)dn (259) (— JU) */— j I = "W +i + A^+i-^^m+i^W^- • • •) ) THE THEORY AND PRACTICE OE INTERPOLATION. 149 As an application of (258), let it be required to find X46° sec 2 TdT Taking w = 3°, t = 31° 30', and 'F-± — 0, we tabulate F(T)= sec 2 T as follows : T —24 and .r=j+(H-i)G) 1 V-/SW= 'F m +&Ji m THE THEORY AND PRACTICE OP INTERPOLATION. 151 the factor w having been previously applied. Whence the expression for I becomes Thus, the value of I for any date T being found by adding the constant l Q to the integral taken from T to T, it is clear that we have merely to increase the above value of 'F_, by the quantity l = 1° 47' 11'. 3 in order to avoid the subsequent addition of this constant to each computed value of the integral. Accordingly, under the heading l -\-'F, . on the line t — \ . The sum l -\- 'F-\- ^j A' is then tabulated in the final column, I, which therefore gives the heliocentric longitude of Mars for the dates indicated in column T. 73. Applications in which the Limits Fall Otherwise than Midway Between Tabular Values of the Argument and Function. — If we put 6(i+i) = >F (+i + ^ J' m - T « T ^+ ^Wsu ^h - • • • • (260) the formula (257) becomes C l F[t + nto)dn = 0(i + i)-6(-i) (261) Whence, if as before n denotes a fractional or mixed number, we derive, by the general method of interpolation employed in §70, CF(t+m*)dn = 8(n)-0(-b) (262) Upon substituting n' and n" successively for n in (262), and taking the difference of the resulting expressions, we get F(t+no>)dn = 0(n")-6(n') (263) 152 THE THEORY AND PEACTICE OF INTERPOLATION. Finally, replacing the functions 0, in (262) and (263), according to the expression (260), we obtain the following formulae : J'V-HKU /*n F{T)dT = <„ \F(t + nu)dn = a \(>F.->F^ + &(J' n -J'^- v H v W'-J'J$+ v fM„(J:-JLd- . . . I (264) F(T)dT = co \F(t+nw)dn = .oK'^-'^ + A^^-^y-^^K'-'-^O + ^VV^K"-^)-- • -i (265) where the quantity 'F-± is wholly arbitrary; and where 'F n , j' nJ /!„', 41, .... (and the similar terms with subscripts ri and n") are to be found by interpolation. "When several values of an integral are to be computed from a given series by (264), the latter may be modified to the more ex- pedient form given below : IT? . 1/f' 4. 17 //'" 387 /p _|_ J-»fr+-nO> /»n F(T)dT = a, j^^+wa))^ ' (266) Example. — Find the value of X = fe s •^0.15 0.48 e being the base of the natural system of logarithms. Taking co = 0.1, t — 0.2, and F(T) = e r , we prepare the fol- lowing table : T 'F F(T)=e T A' A" A 1 " Jiv 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.004840 + 1.216563 2.566422 4.058247 + 5.706968 1.000000 1.105171 1.221403 1.349859 1.491825 1.648721 1.822119 2.013753 + 105171 116232 128456 141966 156896 173398 + 191634 + 11061 12224 13510 14930 16502 + 18236 + 1163 1286 1420 1572 + 1734 + 123 134 152 + 162 Proceeding by formula (266), we find 'F^ = 10- 6 (- Jj X H6232 + ,}j T X 1163) = -0.004840 THE THEOET AND PRACTICE OF INTERPOLATION. 153 whence the column 'F is completed as shown above. Denoting the assigned lower and upper limits by T' and T", respectively, we have T> = 0.15 = 0.20-0.05 = t-iF n = +4.535670.3 J'„ = +0.161674 + sV A' n = + 6736.4 4,"= + 1619 -^khs^'n = ~ 4 -8 Z = +4.542402 .-. X = +0.4542402 The true mathematical value of X is easily found : thus, since Ce T dT = e T we have X = e - 48 - e - 16 = 0.454240159 7i. Quadrature as Based upon Bessel's Formula of Interpo- lation. — Another set of formulae for mechanical quadrature, similar to those already developed, may be derived in the same manner from Bessel's expression for F(t -\-na>). However, since these formulae may be obtained more conveniently by a direct transformation of those developed in the preceding section, we choose the latter course. Putting n" = i, and n' = 0, in formula (263) , we have fF(t + ?io>)dn = 0(t)-0(O) (267) We also have, by (260), 6 (0 = 'F t + 5 > ¥ J', - 5 jj c J' t " + ff ,ftV„ JJ - . . . . (268) 154 THE THEORY AND PRACTICE OE INTERPOLATION. Referring now to the general schedule on page 147, it will be observed that the quantities 'F t , A\, J>", Jj, . . . . are not explicitly given, but must be found by interpolating to halves between 'JFt-i and 'jPVhj A ' i-k an & ^W> etc -> respectively. For this purpose, let us denote the algebraic means of the latter pairs of quantities by ('i^-)j (^'*)> W)> • • • • 5 that is, let us put (J' t ) = i(J' i _ i +^ i+i ) 1 Applying formula (126), we have, therefore, (269) >F t = (>F t ) - ■i(^)+T^(4"')-T^(^)*+ • • • • * t = W)- W) + Tk(^) - • • • • /!>>> = «') - 1(4) + . . . . ^\ = K) - • • • • (270) Upon substituting these values of 'F t , j'i, A'l ', .... in (268), and reducing, we get (*") = W - tV W + tVV (4'") - T m* K) + • • • • (271) Putting i = 0, this becomes 0(0) = ('^ )-A(^») + TVVK")-^lkK y )+ • • • • (272) Whence, from (267), we derive fF(t+ n)dn = 8(i)-6(-±) - '^-» ~ s^'-j + viiv J-i ~ ,W,t 4^ + . . . . (274) In like manner, making if = i-\-\ y and w' = 0, in (263), we obtain f-F^ + raeo)^ = (i + £) - (9 (0) IV 1 \ J ,11 17 /<"' I 3 7 P - C^o) + tV (^'o) - tVV K") + vhih* (40 - • • • • (275) Finally, substituting n" = n and ri = 0, in (263), the latter becomes CF(t + nm)dn = 6(n)—$(0) = 'i^ + & J'„ - t Ht 4." + mftVro ^ - • • • • - C^o) + tV (^'o) - tVV «') + silk (4) - • • . • (276) The equations (273), (274), (275) and (276) give, respectively, the following formulae of quadrature : F(T)dT = j[ ('^)-(^„)] - tV[(4'.)-(^o)] + AY[(4")-(4")] - T *iM(4D-(4D] + • • • • | (277) F(T)dT - a\F(t+no>)dn = •i?^)-AW + AW)-iiib(^)+ • • • • _'JP_j - A ^'-* + *«* ^ ~ WWro Jl» + • • • • \ (278) F(T)dT = uJF(t+nu)dn = »*'*!+* + A j '<-h - *w* -4+j + ^ww 4 +J - • • ■ • - C^o) + tV W - tYY K") + drfk (40 -....} (279) J F(T)dT = a) j F(t+nw) dn = 0*'o)> are means of corresponding tabular quantities, as defined by (269). If, in the formulae (277), (279), and (280), we take (>F,) = & (z?' ) - 7 y ff K") + ***** (A) then the sum of the terms with subscript zero will vanish. But, since (F_, = - i F + T v (z/' ) - tW «') + Mb K) - • • • • \ Cf7t)(IT = ufF(t+nu)dn > (282) = « I ('iQ - T v (z/' ( ) + 7 i 5 v (4") - *m* K)+ • • • • I / /P 1 /// I 1 7 //'" 3G7 A" I \ -^-i — 3T1 •" -1 T 3TB1F -"-J ¥TrTFS0 ^-J T • ■ • ■ j F(T)dT = w \F(t+n) dn \ (284) '*-» = - i^» + A W - AV «') + **Si* (40 - . . . . \ ^(T) dr = c I F(t+nw) dn \ (285) = -> CK + A /*'. - bH_, = J0l = z/Lj = Whence, proceeding by (278), we find (i = 4 ) C*i) = = i('F si +'F lk ) = +1.01022.5 (z/' 4 ) = £(z/' 81 +J' 4J ) = +0.11449.5 - A W = - 954.1 (4") = iW + *») = - 4231 + T y*(^") = - 64.6 (J0 = i(4> +/*!,) = + 852 -*hihVl) = - 2.7 .-. X = +1.00001 Verification : Since Ct sin TdT = sinT— ToosT we have X = sinT- TcosT 7T * = 1 Example II. — Compute the value of (l+o.i ry 158 THE THEORY AND PRACTICE OF INTERPOLATION. Here we take w = 0.1, $ = 0.9, and tabulate F(T)=(l-\-0.1T z )-^, as below : T >F F(T)= (1+0.1 T 2 )-8 A' A" A'" 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 -2664 -180 155 128 104 79 - 57 +25 27 24 25 +22 Proceeding by means of (282), we compute 'F-\ as follows' F = +0.88974 (z/' ) = - 2218 «') = + 26 - i F B = -0.44487 + iV W = - 184-8 -tV*K") = - 0.4 .: F_ k = -0.06819 Whence, completing the series 'F, and observing that the values of 'F„ , J'„ , and _/'," are obtained from their respective series by interpo- lation with the interval 0.15497, we find 'F„ = +1.28846.8 J' n = +0.07754 J'" = + 787 + & J'. = + 323.1 -vHz-l'*" = - ^ .-. A" = +1.29168 Verification : The expression for the indefinite integral is — f "sec 4 TdT = tanr + ^tan 8 T Therefore x = |s- + i(l) 8 l-i 1 + ^ = i- 29167 with which the above result substantially agrees. 160 the theoey and practice of interpolation. Double Integration by Quadratures. 75. Having derived various formulae for the mechanical quadra- ture of single integrals, the corresponding formulae for double integra- tion are now readily deduced. These will serve to compute integrals of the form Y = ( 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-F(T) dT = f(T) + M (286a) where M is the constant of integration. We then have f(T)dT +M(T"-T>) (287) 7" 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 M is generally known from the fact that the first integral has an assigned value (usually zero) corresponding to the lower limit of integration. If we now put T = t + no, , T' = t + n'a> , T" = t + n"*> we have dT 2 = o>W (288) and hence (286) becomes Y =J J F(T) dT 2 = o) 2 J J F(t+no>) dv? (289) upon which relation the subsequent formulae are based. 76. Double Integration as Based upon Newton's Formula of Interpolation. — If we substitute, successively, n' and n" for n in (243), and take the difference of the two results, we obtain J"n" F(t+n)dn = *(ra) or, by (238), fF(t+nu>) dn =fF n dn = 'F n + I F n + $A' n + y J'; + U' n " + (291) the constant of integration being contained in 'F„ , which depends upon the arbitrary quantity 'F . Multiplying this equation by dn, and integrating, we get CCF(t + nw)dn' i = C'F„dn + £ § F n dn + /3 fj'„dn-\- y C.J'„' dn + 8 C.J' u "dn+ . . . (292) Let us now consider a new series, namely — up up up up "F ti the term "F being arbitrary, and the subsequent terms so determined that the quantities F F(T) J' J" J'" Jiv lip 'K t u Fi >F, F - 1 J 'o t + CO "F„ F 1 ■J, 'F. J\ X" t + 2u> "F s 'F* F, -1', K ./;" J t + 3o) "F i F, j.; J? t + (i-2)eo t+ (t-l)<0 up up 'F, F^ F<-, ■i'U ■ c 2 JZ J?-* t + no "F t+1 "F i+2 'F i+X F, i 162 THE THEORY AND PEAOTIOE OF INTERPOLATION. Now, since the differences z/ (r) may be regarded as a series of functions whose 1st, 2d, .... differences are z/ ( ' +1) , z/< r+2) . . . . , it is clear that formula (291) may be applied successively to each of the integrals in the second member of (292). Accordingly, we have f'F.dn = up -*■ n + i'F n + /3F n + yZ/'„ + U 1 : + eZ/- + . . . . *J>. dn = i('K + iF n + /3J' u + yJ!+W+ .... pfj' n dn = /3(F n +iJ' r + pJ! + yJ' i :'+ .... yfd': dn = y(J' % + iJ':+W+ . . . . 8 P^" dn = 8(j':+iJ';'+ .... e C/P; dn = «K"+ .... (293) Summing these expressions, we find, in accordance with (292), CCF(t+n») dn* = »F n + ) dn* = "F n + >F n + T V ^ - ^ J" + ,J ff <" - (294a) the coefficient of z/'„ reducing to zero. We proceed to determine the expansion to which the coefficients of this formula belong. For brevity, let us write (294) in the form J"JV(* + n«>) dn* = »F n + >F n + aF n + bJ' n + cA 1 ; + «X" + Now, from (228), we have logCl + s) = a; - 1 +i* +^+^ 2 + 8 a; 3 +. . . . Also, let us put w = x~* + x- 1 + ax" + bx + ex* + dx s + . . . . (295) (296) (297) THE THEORY AND PRACTICE OF INTERPOLATION. 163 in which the coefficients are taken as in (295). Whence, since the second member of (295) is the combined sum of the second members in (293), it is evident that (297) may be resolved, conversely, as follows : w = a-" 2 + | x- 1 + /3x° + yx + 8x 2 + + i(z- 1 +tx ll +l3x+yx i + + P(x°+ix + f3x*+ + y( X +iX*+ + which may be written w = x- 1 (x- 1 + i x° + fix -f yx 2 + Sx s + + ix"(x- 1 +^x' > + /3x + y x 2 +8a; 8 f + fix (a;- 1 + | x° + fix + yx 2 + Sx s + + yx 2 (ar 1 + £ x" + fix + yx 2 + 8a; s 4- + 8.r 3 (a;- 1 + i x° + fix -\- yx 2 + 8.x 8 + = (x-i+bx»+l3,r + yx 2 + . . . = (ar- 1 + i*»+/ta + y x 2 +&x»+ . Therefore, by (296), we have . )(«-' + i x» + fix + yx 2 + = £log(l+z)£ 2 = (i = x- 2 + ar 1 + ^ .r° - 3 £ T x 2 + ^ cb ; a; 2 a; 8 2+3 ■) 2 a; 4 a; 6 — tt§! sir x + ?TT>f S * (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 + a-)]" 2 , as developed in (298). Whence, in- troducing these values of a, b, r, F n + aF n + l>J' H + c.j;; + d.i;!' + e,J):-+ . . . . = "F R + '^,.+ tV ^ + 0J'„ - ?U X + -*lu-'i" - 55tM -# + and (299) becomes C CF(t + nw)dn 2 = X (w) (300) (301) 164 THE THEORY AND PRACTICE OE INTERPOLATION. Whence, if the integral be taken between the two fractional limits, n and n", we shall have J J F (t + nu) dn 2 = X (n") - X. (n') (302) And if we make the upper limit an integer, say n" = i, we have j'fF(t + n u >)dn 2 = \(i)-\(n>) (303) The last formula involves the disadvantage of employing differ- ences j{, jf, di", .... which are not given when the tabulation of F(T) ends with the quantity F t . To remedy this defect, we pro- ceed as follows : Put v = \(j) = "F ( +'F i + aF ( + bJ' t + cA\' + djl" + eJ? + .... (304) and substitute for "F u 'F t , F t , j t ' t J-', .... the expressions "Fi ="F m -2'F i+1 + F { >F t = >F l+1 - F ( ^ = Fi A> = au + z/;i 2 + j^ s + z/;i 4 + . J'.' = J'.^ + 2z/^ 3 + 3z/;i 4 + . 4'" = A"* + 3Jti + ■ (305) Whence the integral (303) may at once be expressed in terms of the available differences, z/V-i , 41-2 , A'lU , ■ ■ ■ • However, to avoid direct substitution, let us put, as in (229), 1-W (306) and we shall have x- 1 x" = u~ 2 (l-u) 2 = = iir l (i-u) = 2m- 1 + X = u (1-M)" 1 = u + u 2 + V? + M 4 + . . . . X 2 = u%l-u)-* = u 2 +2u s +3u i + . . . . X s = u s (l-u)- s = u s + 3n 4 + . . . . ;> A = w 4 (l-«)-4 = U H- . . . . (307) THE THEORY AND PRACTICE OP INTERPOLATION. 165 Again, from (297), we have w = x~ 2 + ar 1 + ax" + bx + ox 2 + dx* + ex* + .... (308) Now, it is evident that if the expressions (307) be substituted in the second member of (308), the algebraic process will be identi- cal in form with that of substituting the expressions (305) in (304). The w operation involves the quantities w ; x-*, X~\ X°, X, X% X s , ... . ; H-2, u~\ U°, If, it-, M 8 , . . . . ; while the v operation involves, in exactly the same manner, the quan- tities v ; "F-, 'F-. F- . J'.. J" A'" • "V IF 7? /' t" t"< Hence, if we perform the w operation, the result for v is at once known. But the expression which results from substituting (307) in (308) is obtained with greater expedition by the following process : From (298), we have w = {log(l+ic)|- 2 Whence, by (306), we find w = I -log (1 -u) j- 2 = |log(l-w)J- 2 the expansion of which is immediately obtained by writing — a for x in the second member of (297). Thus we find IV = u - m- 1 + au° -bu + cvP- — dip + eu* - . . . . (309) Therefore, according to the preceding reasoning, the expression for v is — v = "F i+2 - >F i+1 + aF t - bJ',_, + c.j;i 2 - rf J- + eJ^ - . . . . Denoting this expression by ir(i), and restoring the numerical values of a, b, c, . . . . from (300), we have v = *■(*) = "F i+2 - >F l+l + aF { - bJ 1 ^ + rj,'\, _ dJ['^ + eJ l Zt - . . . . = "F i+2 - >F i+1 +&F,- ?U J[L, - ,i T J,^ 3 - v nin J\U~ ■ ■ ■ • (310) 166 THE THEORY AND PEACTICE OP INTERPOLATION. Whence, by (304) and (310), X(i) = v = 7T (i) and the formula (303) becomes, therefore, JT>(*+ »»)&»■ = *(t)~X(»0 (311) In the formula just proved the quantity i denotes an integer. JSTow, by the general method of interpolation employed in §70, it is easily shown that (311) is true for non-integral values of i. Thus, writing n" for i, this formula becomes f f F(t + n») dn 2 = tt (»") - 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 "F^ for "F n -\-'F n and for "F n+2 — 'F^ , respec- tively. Thus we obtain the following group : x(») = "F n+1+T ^F n -^^: +^<" - vMU^: J tthU^l - ■ • • »(n) = "F^ + ^T^-^JU-^js^'^-TMhs^U—shU^U- • ■ ■ rr^+wa))^ 2 = x(»'o-x(m') ( iFtf+m^dn 1 = it (n") - X (»') r = | (^(T)^ 2 = o) 2 J^+wo)^ 2 From this group are immediately derived all of the formulae given in the following section. 77. We have already remarked that in the process 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 Q may be taken at pleasure, the integral being independent of its absolute value. Per contra, the double integral will evidently vary with the value assigned to 'F . Hence, unless 'F is fixed by some special consideration, the value of the double integral is indeterminate — a conclusion already derived from (287). (313) THE THEORY AND PRACTICE OF INTERPOLATION. 167 Now, as was previously remarked, the value of the first integral corresponding to the lower limit is usually known in practical appli- cations. We shall therefore denote by IT Q the value of j F (T) dT which results when t is substituted for T. Then, by (291), we have ff = \CF(T)dT =(>F + iF + (3Jl + yJ' ' + U^" + e.l>J+ . . . .) = u>(>F 1 -iF + pJl + yJ' 6 >+8J' " + cJ*+ . . . .) or, upon restoring the numerical values of /3, y, 8, . . . . from (222), and transposing, >F t = ^ + lF +^.l> -& C+ 7 V\F./;"-T»ifC + ir&23.. -JJ- • • • • (314) (0 which determines 'F l , and hence, also, the double integral Y, provided II is known. In practice the value of H Q is frequently zero. Using (311) in conjunction with the relations (313), we obtain the several groups of quadrature formulae given below : / n< _i lpi l // 1 A'< i 19 /l"i 3 /Jiv i 8G3 //v 0) j j F(T)dT* = (o a j J-F(<+wF X = — ° + i F + ^ J' - ^ 4 J;,'f ,VV X'-tIit J i v + *M3o J J- • • ■ • j ji^ZVT 2 = o) 2 J J if* (?+»«) rf!4 2 \ (316) = " 2 i ("^» + i-"^) + a ( F > -K) - a-« (^:-A',') + 5 io (-J:;'-Jn (0 rr^'(r)rfr 2 = o) a jT>(*+Hco)f/rt 2 \ (317) Tw, (^-^ n ) - *i,. (j;'-x) + 2 -u w-j'j') _ 2 2 1 f Itv _ Jiv\ I ]!>_ fjv _jn _ . . . . I '^. = — ° + £ ^ + ^ J' t - Ji X+ VjV -C-xib J i v + u 5 St* J J- ■ • • • (0 rr^('r)dT a = a> % C CF(t+iiio)dn" \ (318) = 1 2 {("^ +1 -"^ +1 )+ a(^-^) -*** t-^— ':.')+ *b w—j*') - zUh (J^-- j?) + *tt* (-^— w - • • • • t 168 THE THEORY AND PRACTICE OE INTERPOLATION. The foregoing formulae are applicable when the upper limit falls near the beginning of the tabular series. When the upper limits falls at or near the end of the given series, the following formulae — like- wise derived from (313) — may be employed : 'F 1 = ^ + iF + T V A\ - J ? < + Jh 4" - rh ^ + Aih *l~ ■ ■ ■ ■ J \F(T)dT* = Vj \F(t+nw)dn* \ (319) = **K"F i+1 -»F\) + T V (F t -FJ - ,i v [d'U- J'o')-*h W-z+J'o") ft) /• /»<+nW /» /*n f J Ji^^tZT 2 = a, 2 ! J i^ + rta/)^ 2 \ (320) = a, 2 { ("F n+1 - »FJ + T V (JP.-Fo) - ,J„ VU-^) -*i„ (*C + O -^(^-^-ilfs^ + A')-. • • -I ZJ~ / I? j_ 1 7T 7 I 1 /// 1 //" X 19 //'" 3 // iv J_ 8 6 3 //v Cf'F(T)dT i = ^C("F(t + no 1 )dn 2 \ ( 32 1) %) */(+7ift) */ «Ai ( -vmvi^-w-iMiVu+JS- ■ ■ ■ ■ \ O) JJFJT)dT* = Jj JF(t+n»)dn* I (322) -MkK'M-^)-4IiW»- s +^)-- • • -I In applications of all the preceding formulae, the value of "i^ (or of "JP when employed) is wholly arbitrary, and therefore may be assigned at pleasure in every case. But when (315), (316), (319) and (320) are applicable, it is frequently convenient to determine "F 1 such that The formulae in question then take the form as follows : THE THEORY AND PRACTICE OP INTERPOLATION. 169 XT JJF(T)dT* = a *j JF(t + na>)dn* = «'("^+i 1 .^-ii,4" + il T 4"- I Hi T 4 T + f Hi4- ■ • • •) ,/J? i = -Ji* , „+vlj><-i l lj i Jo" + v mis4?-vhiv4l+ ■ ■ ■ ■ JJ^F^dT 2 = u>*jJF(t + n A'"— 9 // iv -l- ««» /f v - 1 1 — ^ TT^itls^ii ^+ ^0 T- TTCT^O TBiJ ^0 T ffT)4¥TT /7 — ■ • • • HIT __ 1 B" J. 1 //" 1 /I'" \ 22 1 //iv 19 // v _i_ I \F(T)dT 2 = a, 2 J l.F(* + »ia>)dra 2 «lV"P J. J.P_ 1 //" _ 1 A"' — 221 /fiv 19 ,/v \ /B 1 _. » I IPl 1 /// _ 1 /f"j_ 19 /»'" 3 //■"_!_ SG3 x/v O) II V 1 7? _L J //'' 1 A'" J- 2 2 1" //iv 19 //v I J J F(T)dT 2 = o) 2 j ji^ + W)^ 2 ,.. 2 ('/'P X 1 /V 1 //" 1 4'" 22 1 /I'll 19 ,/v \ (323) (324) (325) (326) The differences which appear in the foregoing formulae, together with the auxiliary functions 'F and "F, are to be taken according to the schedule on page 161. The symbol i denotes a positive integer, while n designates a fractional or mixed number : so that all functions and differences whose subscripts involve n must be derived from their respective series by interpolatioyi. Finally, the quantity H de- notes — as previously defined — the value of J F(T)dT when t is substituted for T : so that we have H ° = [f F ^ dT \^ t < 327 > It may happen occasionally that the value of H is unknown, while the value of CF(T)dT corresponding to T '= t-\-na> is known for a particular value of n. Denoting this quantity by H n , 170 THE THEORY AND PRACTICE OF INTERPOLATION. we may, by any one of the foregoing methods, compute the definite integral F(T) dl = H n - ff and hence find (327a) with which value we proceed as before. Several examples will now be solved as an exercise to illustrate the formulae given above. Example I. — Let it be required to find Si cos TdT 2 on the supposition that jcos TdT=2, when T = 0. We tabulate and difference the following values of ^(^^cos T : T up IF F(T) = cos T A' J" A'" 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 0.86603 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 , * = 0° a = 10° = ^ m = 2 { = 9 Jo Proceeding by (319), the computation of f F x is as follows #„+•«> = + 11.45915.6 F = +1.00000 + * ^ = + 0.50000.0 ^'.= - 1519 + TV ^'o = — 126.6 <= - 2993 _ i /in 5"? n a — + 124.7 J' "= + 139 1 19 /I'" — + 3.7 4 V = + 82 3 //iv TBI! ^0 — — 1.5 .: >F. = + 11.95916 THE THEORY AND PRACTICE OP INTERPOLATION. 171 The column 'F is now completed by successive additions ; hence, also, the column "F, having first assumed "F l = 0. Whence, by (319), the remainder of the computation is as follows : " F w = +136.04398 "F 1 = 0.00000 F 9 = 0.00000 F = +1.00000 A'J = _ 528 JU = - 2993 A"= + 511 4"= + 139 ^ 5 V = + 31 4$ = + 82 log 2 = 2.1334129 log F F(T) = T-z 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. We have, therefore, t = 2.0 <* = 0.1 ff = whence, proceeding by (326), the computation of 'F x and " F x is as follows : + \ F — +0.12500 + T ' 5 J' t = - 193.7 - Jj A 1 ' = - 12.9 - T V J? = -0.02083.3 + ,**< = + 1.3 -*i*4"= + 0.2 .-. '^ = +0.12292 .-. "Z^ = -0.02082 From the completed table we now find n = (2.468-2.0) -H 0.1 = 4.68 = 5-0.32 "F n+1 = +2.36025.6 ^„ = +0.16418 + T V ^„ = + 1368.2 z7;'_,= + 191 -^;_ 2 = - 0.8 ^- 8 = - 36 -?lo^ = + 0-1 2, = +2.37393 .-. Y = +0.0237393 This result is easily verified, for we have j'T-HT = - i+ C io ge r+ cr = -log. 1.234 + 0.468 C also #, = -j,+ C ■ ■ O = £ Hence r= -log. 1.234 + 0.234 = -0.2102609+0.234 = +0.0237391 with which the above result substantially agrees. Example III. — From the table of the preceding example, find the value of THE THEORY AND PRACTICE OF INTERPOLATION. 173 Here we employ formula (324), in which we take 2.15 - 2.0 n = 0.1 = 1.50 = 1 + 4- We therefore obtain 0+1= 2+£) F„ = +0.21633 j;; = + 235 J'" = - 38 »F H+1 = +0.24992.0 + A F n = + 1802.8 1.0 0.2 _ i ,/" 2' = +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 ii F " F t-i '** F^ ■ J '< » J'U -J,-i t + i)dn = 6{n) (328) 174 THE THEOBY AND PRACTICE OE INTERPOLATION. Now, by (260), we have 6 (n) = 'F n + & A' n - ,jJ T A';' + v &J n A" n - . . and hence the preceding equation becomes J>(*+no.)dn = 'F n + A^-iHt^"+ inftVnr ^ - For brevity, let us put « = + A 6 = 1 7 ST¥(S" + 7S7"ffST and (328a) may be written CF(t+nw)dn = fF n dn = 'F n + aJ' n + bJ'J' + cJ' x + (328a) (329) (330) the constant of integration being contained in 'F u . Multiplying this equation by dn, and integrating, we get C CF(t + nm) dn? = C'F n dn + a fj' n dn+ b C/IU'dn + c Cj\ dn + . (331) Applying formula (330) successively to each of the integrals expressed in the second member of (331), we obtain C CF(t+na>)dn 2 = "F n +aF n + bA'i + cA* + . . . . + a(F n +aA'J + bA i :+ . . . .) + b(A'J + aAy-h . . . . ) + c(A™+ . . . . ) = "F n +2aF n +(a?+2b)A , :+2(ab J rc)A*+ .... Whence, restoring the values of a, b, c, . . . . from (329), and reducing, we obtain ffF(t+nF ) = H'F^+'F,) = 'F t -iF which, together with (333), gives 'F, = ^ + i F + T V (J' ) - T W K") + „*!*, (4Q - . . . . (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 : '*» = — ° + *^,+ tV W - M, «') + Mh (4D - • . . . \ C C'F(T) dT 2 = (< + wo)) dn 2 ) ( 336 ) '^ = — ° + i ^„ + tV (A) - tVV (^")+ *ilh (40 - • • • . ) C('F(T)dT 2 = w*CfF(t+no>)dn 2 ) ( 336 ) /^ = — » + * *■„ + a (j' ) - T vv m + *m* wo - • ■ • • \ r (% *r« = ^ r r>(«+«.) «!»• < 337 > j (i?'(r)^r 2 = a) 2 ) I ^(tf+»o)tf» 2 = 2 J ji^ + W)^ 2 ' = ^("ft + ^ft-^h^' + ^Wv^- ■ ■ (339) ) 'ft = -* + i F + tV W - tV* K") + irif h K) - ■ to //p 1 77» j_ 1 A" 3 1 //iv _l_ J |j , (T)^T 2 = 6) 2 I |.F(i + W)dra 2 = a, 2 ("2?. + T V ^ - sh A'i + ^ AW ^ - (340) ) Let us now denote the second member of (332) by y (n) ; that is, let us put y (n) = "F n + ^F n - „i T A'i + ^Ystt ^ - ■ Making n — i-\- j, this becomes y(i+i) = "ft+i+^ft+i-vl^'i+i+^uJl+i (341) (342) It will be observed from the foregoing schedule that "F^, -^Vh> ^"»+i> .... are not explicitly given, but must be derived from their respective series by interpolation to halves. For this purpose, let us put, in analogy with (269), (!'F W ) = i("ft + "F i+1 ) (F i+i ) = i (ft+F i+1 ) (j;; 4 ) = i(4"+^;;o (343) then, after the manner of (270), we shall have "F i+i == ("F i+i )~ Uft +i ) + Thi^+i) - Tlfc «») + (F m ) - I (z/^) + T f* (z/^) - ^ = ^H4 = (344) Upon substituting these expressions in the second member of (342), and reducing, we find y (*■+*) = ("^ +i )-A(^ +J ) + T^T7(^; J )-T^w(^; i )+ • • (345) THE THEORY AND PRACTICE OF INTERPOLATION. 177 Again, by means of (332) and (341), we derive J J F(t + na>) dn 2 = y (n") - y <>') (346) Finally, denoting by H-\ the value of fF(T)dT when T = t — \ a), we shall have, by (328«) , ff_ t = rr>(r)czrl = J CF(t+nw)dn \_y j3'=(-j(i) \y n=-l which gives 'tf-1 = ^*~ A^+iH»^-jMw^+ • • • • (347) By assigning various values to the limits ra' and w" 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 — J, we derive the group of formulae given below : (348) '*» = 5 + * *•, + a w - 7 y* k") + *m* (4 v ) - . . . . JJ J F'(r)rfr 2 = ^{J 2^ +«<■>)*»" (!) "i^ = awy convenient value ; arbitrarily assigned. CCFlffidT* = ) dra 2 \ (352) The last formula may also be written in the following form TT Ijp — I 1 Al j_ 17 /i'H 367 //v I G) "J 1 . = i'^-j+AC^-THTrC^ + TAV^C^)- • • • • jJi? , (r)rfT 2 = o, 2 J Jjp^+Ti^)^ 2 = a, 2 \("F i+i ) - ^ (F j+i ) + T iU (^m) - T^ftW (^Si) +..••} (353) It may be well to again point out the fact that the functions and differences enclosed within parentheses denote the means of corre- sponding tabular quantities, as defined by (269) and (343). Further, that H and H-% denote the values of the first integral of F(T) when for T we substitute t and t — Jw, respectively. Finally, we may add that if in any case H p is given and H q required, it is only necessary to compute F(T)dT = H p -H g (+5(1) and thence find \ ( 354 ) H q = H p — X In the process of double integration by mechanical quadrature it is sometimes convenient to tabulate, not the given function, but o? times that quantity. By this means all differences are multiplied by &> 8 , and thus the final multiplication by that factor is avoided. How- ever, in order that the quantities 'F and "F shall be multiplied by the TT same factor, it is evident that the independent term — (which has the THE THEORY AND PRACTICE OE INTERPOLATION. 179 same fixed value whether we tabulate F(T) or o?F(T)) must like- wise be multiplied by F F(T)B — a' 2 Mcos Tcsc 2 !' J' J" A'" Jlv O 20 24 28 32 36 40 44 48 52 56 60 +0.029974 0.084135 0.133339 0.178619 0.220744 + 0.260304 + 0.060553 0.054161 0.049204 0.045280 0.042125 + 0.039560 -0.017004 0.011689 0.008480 0.006392 0.004957 0.003924 0.003155 0.002565 0.002099 0.001722 -0.001411 + 5315 3209 2088 1435 1033 769 590 466 377 + 311 -2106 1121 653 402 264 179 124 89 - 66 + 985 468 251 138 85 55 35 + 23 -517 217 113 53 30 20 - 12 182 THE THEORY AND PRACTICE OF INTERPOLATION. We proceed by formula (353), taking as our data t = 32° a) = 4° = ,r + 45 i = 4 .Hlj = 2M = 0.868589 Whence, observing that we must now take aH-i instead of the term H-i^-co in (353), the computation of 'F^ is as follows : log uH^ = 8.782752 o)5Li = +0.060639.0 Jlj = + 2088 —h^U = - 87.0 J'_[' h = + 468 + r«*^-i = + I-* .-. 'i^j = +0.060553.4 And for "i^ we find i'-F-j = +0.030276.7 (*•_,) = -0.007436 +^(F-l) = - 309.8 (zT s ) = - 887 -tMtt(^) = + 7.9 (4*) = - 367 +T ^y ffT (j^) = - 0.7 .-.."i? = +0.029974 Upon completing the table as shown above, and continuing the computation by (353), we obtain (i = 4) (»F it ) = +0.240524.0 (*"«) = -0.002332 -AW = + 97.2 (^) 106 + T jfr (^) = - 0.9 .-. F = +0.240620 We easily verify this result analytically as follows : -M cos TdT M I'- ll sin 2 ? ' sin J -McosTdT 2 -, + C n . ?F = Mlog^aniT+CT+C = log 10 tan|T+ CT+ C But 2M .: Y = log 10 tanir + CT r=50° 7=80° H -<- ^+ C = 2M+C : C = ■A' F = log 10 tan f-g" j - log M tan f— THE THEORY AND PRACTICE OE INTERPOLATION. 183 Now we find log tan 25° = 9.668672.5 - 10 log tan 15° = 9.428052.5 - 10 .-. Y = 0.240620 which agrees exactly with the former result. Example IV. — From the table and data of Example III, compute the integral 45° TdT 2 // sin 2 T Y= - "so* Here we employ (351), taking t = 32° as before; we then have for the value of n at the upper limit, n = (45°-32°)-^4° = 3.25 = 3 + 0.25 We therefore obtain "F„ = +0.189420.3 F n = -0.002993 + ^F n = - 249.4 A<: = - 163 - g | g 4',' = + 0-7 .-. Y = +0.189172 Verifying this result as in the last example, we find Y = log 10 tan 22° 30' - log 10 tan 15° = +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 i, 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~ a ; 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 = x, for the times T = 102, 104, 106, 108 and 110 seconds, respectively; assuming that v = 0.6 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 2 x dT* 20000 whence by a single integration + «» and by double integration -IC- 20000dT a + x o (a) 68) We shall first compute the required values of x as given by equa- tion (/3), effecting the double integration by means of (339). The details of the computation are shown in the following table : Table (A). T F(T) = 40000 T- 3 A< J" '¥ "F + ix =a + T5-F=6 ix = a-\-b X 96 98 100 102 104 106 108 110 112 114 0.04521 .04250 .04000 .03769 .03556 .03358 .03175 .03005 .02847 0.02700 -271 250 231 213 198 183 170 158 -147 + 21 19 18 15 15 13 12 + 11 + 0.53730 .57980 .61980 .65749 .69305 .72663 .75838 .78843 + 0.81690 + 3.99667 4.61647 5.27396 5.96701 6.69364 + 7.45202 + 0.00333 314 296 280 265 + 0.00250 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 a 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 &> 2 to w. Accordingly, we tabulate under F (T) the function F(T) = 20000o)T" 3 = 40000I , - S Assume £ = 100, and proceed by (339). To determine 'i^, it must be observed that since F(T), a', d" , .... already contain TT the factor a>, it is here necessary to multiply the independent term — THE THEORY AND PRACTICE OF INTERPOLATION. 185 JT by the same factor : so that, writing v (= H ) for — - in the first equation of (339), and omitting insensible terms, we have '*•, = *. + **•. + * W (y) Hence, substituting v = 0.6, F a = 0.04000, (j' ) = £ (Jl, + Jj) = —0.00240, we find 'F± = +0.61980, and thus complete the series 'F as given above. The second equation of (339) gives simply, "F Q = — T \ F , the term in d" being insensible. But since, by equation (/3), we should afterwards have to add the constant x to each computed value of the double integral taken from T Q to T, it is expedient to tabulate in place of "F the quantity "F + ^ = "F t +ix = -tV^+4.0 = 4.0 -0.00333 = +3.99667 (0 and thence complete the series as given under "F-\- i.r EE a. The reason for this procedure is easily made apparent : for the final equa- tion of (339) gives (since w 2 must now be replaced by co) ffi 20000dT 2 and substituting this expression in equation (/3), we obtain .*". ,T = a ("F t + A F.) + .r = « (!'F t + '-* + A ^.) (S) Therefore, upon forming the column + j\ F'=b, as given above, we have from (S) ix = "Ft+tXt+^Ft = a+b whence the required values of x are derived and tabulated in the final column of Table (A). 186 THE THEOEY AND PRACTICE OE INTERPOLATION. For the computation of the velocity v we employ formula (282), the first equation of which gives or, by adding F to both members, But we shall avoid subsequent additions of the constant v , required by equation (a), if we increase this value of 'F± by the term v = 0.6 ; that is, if we take 'F t = v + iF + T \(Ji ) which is the same as the expression (y), used 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 C-F) (A') -tVW v=(. Completing this column by successive additions of the functions /, we next form the series of mean values tabulated under ('f)-\-x a = c. The columns (8') and — T V (§') =^ are then computed, and finally the column x = 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 - 10000 T- 2 , x = 1.6T+10000Z 1 - 1 - whence, making T = 110, we obtain v = 0.77355 and x = 14.9091 which further verify the results derived by quadratures. 252 188 THE THEORY AND PRACTICE OP INTERPOLATION. 79. It is worth while to inquire what change takes place in the value of the double integral . dT* when, in a particular problem, the quantity H is changed from an assigned value H' to a new value H". This is easily answered. For, if we change H' to H", the value of the first integral — corresponding to any particular value of T — is thereby increased by the quantity H"—H'; or, what amounts to the same thing, the constant of the first integration, M in (286a), is thus increased by H" — H'. There- fore, by (287), it is evident that Y is increased by the quantity {H"—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. Ans. 1.41808. [Note : — Take the eccentric angle E as independent variable, and hence find ■K s = )Vl - e 2 cos 2 .E dE where e is the eccentricity, and s the required length.] 2. Given the equation of a cardioid, r = 1 -f- cos 6 : 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 = a? V2 — sin x , find the area included between the curve, the axis of x, and the two ordinates, x = l and x=}tt. Ans. 0.180518. 4. Compute the value of -# 7T IS dT 1 Vl - 0.82 sin 2 T assuming that the first integral vanishes at the lower limit. Ans. 0.139727. 5. Given a curve in a vertical plane whose points satisfy the relation dhj 4a5 2 - 3 dx* ' ' 5+V^ 190 THE THEORY AND PRACTICE OE INTERPOLATION". — the axis of y being vertical. Find the difference of level between two points whose abscissae are 1.000 and 1.473, respectively ; assum- ing the direction of the curve to be "horizontal at the first point. Ann. 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 V [Note : — This question should be answered mentally.] CHAPTER Y. MISCELLANEOUS PEOBLEMS 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 find S = l*+2*-f3*+ . . . . + r k , where Tc and r are integers. The method of solution is best illustrated by assigning a particu- lar value to I: Thus, let it be required to find S = l i i-2 i + 3 i + . . . . + r 4 We tabulate below and difference the values of T A which corre- spond to T=l, 2, 3, 4, 5 and 6. Thus we find : T >F F(T) = T> J' J" J'" J'v Jv ' F o 1 >F, 1 15 2 'F* 16 65 50 60 3 81 110 24 175 84 4 256 369 194 108 24 5 625 671 302 6 1296 r-1 'F (r-iy r >F r r 4 Now, by Theorem V, the 4th differences of F(T) are constant, and hence the 5th and higher differences all vanish. Whence, if we 192 THE THEORY AND PRACTICE OF INTERPOLATION. consider the auxiliary series 'F — defined as in Chapter IV — we shall have, by the fundamental formula (73), ,F r = ^ e+ , + !^l)(15) + y(y - 1 2 (y - 2) (50) |» V~/ ' |8 r(r-l)(r-2)(r-3) (60) + 9 ' (? - 1) --- (r -V ) = '^ 0+ -(r + l)(2r+l)(3r 2 +3r-l) Therefore, by Theorem I, we have 8 = 'F r -'F t = ^ (r+l)(2r+l)(3r 2 +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_ 2 , F_ t , F , F % , F t , . . . . , and an assigned intermediate value, F n : To find the corresponding interval n. First Solution : The simplest 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 Mercury's distance from the Earth = 9.7968280 : that is, given F n = 9.7968280, to find n. The tabular quantities are here given for every second Greenwich mean noon. Date 189S Log. Dist. of 5 from © A' A" A'n Jiv Jv May 8 10 12 14 16 18 20 9.7560706 9.7652375 9.7768883 9.7905482 9.8057806 9.8221946 9.8394585 + 91669 116508 136599 152324 164140 +172639 + 24839 20091 15725 11816 + 8499 -4748 4366 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) ie somewhat greater than 0.4. Hence we take F = 9.7905482, and interpolate — by Bessel's Formula — the functions corresponding to n = 0.38, 0.41, and 0.44. Thus, computing and dif- ferencing these values, we find n F n J' J" 0.38 0.41 0.44 9.7961736 9.7966267 9.7970810 + 4531 + 4543 + 12 Whence, if we denote by n' 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, »' = 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 n' = 2013 -f 4540 = 0.4434 . . n = 0.41 + 0.4434 X 0.03 = 0.423302 The required time is, therefore, T = May 14 d + 0.423302 X 48 h = May 14 d 20 h 19 m 6'.6 83. Second Solution of Problem II. — Given F„ , to find the value of n. Let m denote an approximate value of n, true to the nearest tenth of a unit, and put n = m + z (356) Then we have F n = ^«+. = F[t+ (»+*)«] = Ftft+nu,) + « w ] 2 2 Since- we have supposed z not to exceed 0.05, it is permissible to neglect z 3 , z*, . . . . in the last expression, which becomes, there- fore, F n = F m +z*F:+i*m'Fi (357) 194 THE THEOEY AND PRACTICE OE INTERPOLATION. To find z from this equation, we first neglect the small term in z 2 , and thus obtain an approximate value which we shall call x. In this manner we find F — F oF' (358) This approximate value of z will now suffice for substitution in the last term of (357). Accordingly, we obtain *■•■*[ 5?) ^' <:i whence, putting y we have s = x — y and equation (356) becomes n = m + x — y (361) Finally, to express F m , i»F' m , and o?F^ in terms of the differences of the given series F, it will be expedient to employ Stirling's Formula of interpolation, together with the expressions for F' m and F'n as developed in §61. The above solution may then be expressed as follows : Determine m = an approximate value of n, true to the nearest tenth of a unit. Thence find F m = F + ma + Bb + Co + Dd + . . . . Z>! = u>F' m = a+mb l> + C'c+ D'd + .... _D 2 = m'F^ = b + me+ . . . . K A > (362) - a and n = m + x — y Here the differences are to be taken according to the schedule on page 62 ; the coefficients B, C, D, . . . . being taken from Table II, and C", B, . . . . from Table V. Finally, Table VII gives the value of y for top argument K and side argument x ; observing that y has the same sign as K. THE THEORY AND PRACTICE OF INTERPOLATION. 195 Example. — Same as in §82. Here we find m = 0.40 ; and hence take from the given table, and from Tables II and V, the quantities *"> = 0.40 a = +144461.5 . . 1 B = +0.080 b = + 15725 . [ = -0.056 = - 4137.5 C" = -0.08667 D = -0.0056 d = + 457 D> = -0.02267 E = +0.01075 e = + 105 ^' = +0.01440 The computation of F m , D 1 and A by (362) is therefore as fol- lows : F = 9.7905482 ma = + 57784.6 a = +144461.5 .BJ = + 1258.0 mb = + 6290.0 b = +15725 Cc = + 231.7 C'e = + 358.6 me = - 1655 Z><2 = - 2.6 Z>'^ = _ 10.4 A T e = + LI E'e = + 1.5 .-. jr. = 9.7964755 .-. A = +151101 .-. D t = +14070 A = 9.7968280 „ F n -F m = + 3525 Whence Z" = A-i-A = +14070^-151101 = +0.0931 » = (A-A)-+A = +3525-^151101 = +0.023329 and we finally obtain m = 0.400000 .r = +0.023329 (Table VII) - y = - 26 .-. « = 0.423303 which agrees within one unit with the former result. 84. Problem III. — To solve any numerical equation whatever involving but one unknown quantity. 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 n 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° where T is expressed in degrees of arc. This equation "may be written F(T) = T - 20° sin T - 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) A 1 A" O 62 63 64 -0^6590 + 0.1799 + 1.0241 +8389 + 8442 + 53 Here we have given F n = 0, to find n. Whence, employing the corrected first difference (§45), we find 1709 T = 63° - !^ X 1° = 62°.7861 8410 85. Problem IV. — Given a series of numerical functions em- bracing a maximum or minimum value : To find the value of the argument which corresponds to the maximum or minimum function. Find by inspection the tabular function which falls nearest the required maximum or minimum value. Call this tabular function F . Then, from the schedule T F(T) A' A" A'" div t (0 - t t + d' we have, by the first of equations (182), = F'(t+nu) (a-l c+ ) + n (b —fe d + . F'(T) 1 )+in>(c-....) + ln°(d ■) + ■ THE THEORY AND PRACTICE OF INTERPOLATION. 197 Therefore, since the condition of maximum or minimum requires that F'(T) = 0, we have, by neglecting 5th differences, (a-i c) +(b -^d )n + i en* + ± d n* = (363) which determines 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 z and n 3 , 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 Mars with respect to the Sun (log r) . Find the time of perihelion passage of the planet. Date 1898 Log r A' 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 b a = +2911 *.-M,= + 2914 e = + 3.5 lc = + 2 d = - 33 M= - 6 and therefore, by (363), or -135 + 2914w + 2» 2 - 6rc 8 = 2914w = 135 - 2to 2 + 6w 8 198 THE THEORY AND PRACTICE OP INTERPOLATION. Neglecting the last two terms of this equation, we have, for an approximate value of n, n = 135-^-2914 = 0.046, nearly; and since for this value of n the small terms sensibly vanish, we obtain as our final value n = 135-^2914 = 0.04633 The date of perihelion passage is, therefore, T = April 30 d + 0.04633 X 8 X 24" = April 30 d 8 h .895 86. Problem Y. — Given a series of numerical values (F_ 3 , F_ 2 , F_ 1} F , F 1: F 2 , . . . .) of any function F(T) which is analytically unknown: To find an approximate algebraic expression for F (T) in terms of the variable argument. Let us put r = T -t (364) and Taylor's Theorem gives F{T) = F(t+r) = F(t) +rF'(f)+£-F»(t) + £-F'»(t)+. . . . (365) Upon substituting in (365) the expressions for F'(t), F"(f), F'"(t), . . . . , as given by (175), we obtain F(T) = F(t) + ~(a-ic+^e- ) r + ~ (b -f s d + ) r 2 + -L(c-ie + . ...) T =+--L («*„-.. . .)T*+_L-(e-. . . .) T 5+. . . . (366) which expresses F(T) as a rational integral function of t, with known numerical coefficients ; r being the value of the variable argument counted from the fixed epoch t, as defined by (364). Example. — From Newcomb's Astronomical Constants we take the following table of the mean obliquity of the ecliptic (e) for every fifth century : THE THEORY AND PRACTICE OF INTERPOLATION. 199 Tear Obliquity A> J" J'" 500 1000 1500 2000 2500 / // 23 41 43.78 37 57.97 34 8.07 30 15.43 26 21.41 23 22 27.37 i a -3 45.81 3 49.90 3 52.64 3 54.02 -3 54.04 // -4.09 2.74 1.38 -0.02 n + 1.35 1.36 + 1.36 Let it be required to express e in terms of r, the latter being counted from the year 1000 in terms of a century as the unit. Since we adopt one century as the unit of time, it is necessary to express cu in the same unit ; therefore we have a, = 5 * = 1000* F(t) = 23° 34' 8".07 a = -3' 51».27 = -231 ".27 K = -2".74 e = +1".355 a-^c = -231 ".496 b = -2".74 co a IL = 50 „»|3. = 750 Whence, by (366), we obtain Coefficient of t = -231.496 -j- 5 = -46.299 2.74 50 = 0.0548 " « t 8 = + 1.355-^750 = + 0.00181 Accordingly, the required expression for the obliquity is — c = 23° 34' 8".07 - 46".299 t - 0".0548 t 2 + 0".00181 t 8 Verification : Putting t = 10 in this formula, we should get the obliquity for 2000. Now we find (For 2000) £ = 23° 34' 8".07 - 462".99 - 5".48 + 1".81 = 23° 26' 21 ".41 which agrees exactly with the tabular value above. It will be observed that the solution given by (366) restricts the epoch, or origin from which r is counted, to some tabular value of the argument, as t. Should the assigned epoch be some intermediate value of T, say T 1} it will only be necessary to write and we have F(T) = F(T;+tJ = F(T 1 ) + t 1 F>(T0+ T 1f"(T 1 )+ .... 200 THE THEORY AND PRACTICE OE INTERPOLATION. Therefore, if we put T x = t + ma> ) we shall have V(T \ - w + p'xI'F'x^f'x C ( 366a ) where t x (= T — T x ) is the value of the variable argument counted from the assigned epoch T x . Accordingly, if -we compute by the usual methods the values of F m , F' m , F'^, F'^, '...., and substi- tute these in (366a), we shall obtain the expression required. As an example, let us express the obliquity (e) as a function of the time (t-j) counted from the epoch 1600.0 in terms of a century as the unit. Reverting to the above table, we take t = 1500* T x = 1600* m = 0.20 Whence we find F m = 23° 29' 28".69 F' m = -46".761 F£ = -0".0443 F'Ji' = +0''.01088 Substituting these values in the formula (366a), we obtain the required expression, namely, e = 23° 29' 28".69 -46".761t 1 -0".0222 Tl 2 +0''.00181t? 87. Geometrical Problem. — A circular well four feet in diameter is centrally intersected by a horizontal cylindrical shaft whose diameter is one foot. Find the volume of the portion of the shaft within the well. Solution : Consider 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 B and r, respectively, we shall have for the length of this rectangular section and for its breadth, or height, h = 2vV 2 -a; a THE THEORY AND PRACTICE OF INTERPOLATION. 201 Therefore, the volume of the differential section is — dV = Ihdx = 4V(£ ! -a; s )(r s -s 2 )^ whence Upon substituting the given values of B and r in this formula, it becomes V = 8^(4 -*«)(* -*«)«** This expression belongs to the class of functions known as ellip- tic integrals, and therefore cannot be integrated directly. Accordingly, we proceed to evaluate V by mechanical quadrature. For this purpose it will be convenient to put whence x = £ sin dx = £ cos Odd and the preceding expression for V becomes V = fcos 2 0Vl6-sin 2 d$ (367) We now tabulate F (6) = wcos 2 >Jl6— sin 2 (where w = 10° = rr -^- 18) as follows : e >F F(6) A' J" J'" Zjiv - 15 - 5 + 5 15 25 35 45 55 65 75 85 95 +105 0.0000 0.6927 1.3427 1.9129 2.3765 2.7201 2.9449 3.0663 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 THEORY A2STD PRACTICE OF INTERPOLATION. Accordingly, we take t = 5° i = 8 t + io> = 85° and proceed by formula (259) : thus, observing that d'^, J^, . . . . and z/j+j, J' t \! h , .... are all zero, and remembering that the factor w has already been introduced, we find and '*■-* = o V = >F i+i = 3.1168 cubic feet 88. Various other problems and applications of a similar nature might be added ; indeed, Astronomy itself presents a large variety of such. But the leading principles of our subject have already 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 INTERPOLATION. 203 EXAMPLES. 1. Derive the expression for the sum of the cubes of the first r integers. Ans. |r 2 (r-|-l) 2 . 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 189S Sun's R.A. a Date 1898 Sun's R.A. a Sept. 13 16 19 22 h m s 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 d 12 h 34 m .8. 3. From the ephemeris of the moon's latitude given below, determine the instant of greatest latitude north. Date 1898 Moon's Latitude Date 1898 Moon's Latitude July 9.0 9.5 10.0 o / // + 5 7 9.3 5 14 28.1 + 5 17 38.3 July 10.5 11.0 11.5 o / // + 5 16 48.7 5 12 9.7 + 5 3 52.8 Ans. July 10 d 3 h 27 m .4. 4. Given the equation sin (z-43°) = 0.92 sm"s to determine the root which falls in the second quadrant. Ans. 101° 17' 43". 5. Given the following table of the longitude of Mercury's as- cending node (6) : 204 THE THEOEY AND PRACTICE OE INTERPOLATION. Year $ / // 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 8 as a function of t ; where r is the elapsed time from 1900, reckoned in terms of one century as the unit. Ans. 6 = 47° 8' 45".40+4266".75T+0".630r a . APPENDIX. ON THE SYMBOLIC METHOD OF DEVELOPMENT. 89. While many of the formulae and results in the foregoing text have been derived by somewhat indirect methods, yet the pro- cesses employed in every case 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 form of development known as the symbolic 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 operation A by the relation AF(T) = F(T+o>)-F(T) (368) from which we formulate the following Definition: The operation of A upon any function of T pro- duces the increment in the function which corresponds to the finite increment a in the variable T. The relation (368) may be more briefly expressed in the form A^„ = F H+1 - F n = J' n (369) where n can have any value. Thus, taking n = 0, and referring to the schedule on page 15, we have A*; = F x - F 9 = JJ (370) Similarly AF, = F 2 -F x = J[ \ AF 3 = F t -F, = 4 (371) A^ s = F s+l - F, = J' s ) 206 APPENDIX. Thus it is evident that the effect of operating with A upon any tabu- lar function is simply to form the first difference of that function and the succeeding tabular value. Whence it is evident that we have AAF = AK) = < \ AA^ = A (JO = 4' (372) AAF, = A(0 = A'.' ) It follows that the operation of AA upon any tabular function produces the second difference bearing the same subscript. But this double operation of A may be conveniently characterized by A 3 ; hence we write A 2 *; = JH , AIF\ = 4[' , , A 2 i? 8 = A 1 : (373) In like manner, i denoting any integer, we have A^ = A(A i - 1 ^ ) = A(J< M >) = A» A'Ji = A(A i - 1 J F 1 ) = A(Ji*- u ) = A i} (374) £^F, = A(A i - , i^ ! ) = A^*'-") = Af and, more generally, n being a non-integer, A^ = (AAA ... . t times) F n = J» (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 a = a, |J = „f; (376) also D*F n = DDF n = "-^("K) = » 2 K (377) WF n = (DDD . . . . t times) F n *=L^\F n = JF™ (378) 92. The fundamental laws or principles governing the combination of symbols of quantity in algebraic operations are the following : APPENDIX. 207 I. The Distributive Law, by virtue of which a (j> + q + r) = ap + aq + ar II. The Commutative Law, expressed by the equation ab = ba III. The Index Law, which asserts the relation a r X a," = a r+ ' ~We proceed to show that the symbols of operation, A and D, when combined each with itself or with symbols of quantity in the manner indicated below, also obey these fundamental laws ; and hence that, wherever found in similar combinations, A and D may be treated alge- braically precisely as if they were themselves mere symbols of quantity. We shall first consider the symbol A. (1). By definition, we have A (F n +f n + ....) = (F n+1 +f n+1 + ....)- (K+A+ • • • • ) = A^„ + A/,+ .... which proves the Distributive Law for the symbol A. (2) The factor a being a constant, we have AaF n = aF n+1 -aF n = a (F n+1 -F„) = aAF n thus showing that A combines with constant quantities in accordance with the Commutative Law. (3) r and s denoting positive integers, the relation (375) gives A'A>J„ = A'CA'-FJ = A'J;;' = -4<: +s) = A*"+*-P; or A'A' = A* 4 "' Therefore, so far as positive integral indices are concerned, the symbol A obeys the Index Law. 93. Eetaining the limitations and the notation used above, similar results are easily obtained for the operator D, as follows : 208 APPENDIX. (1) D(2W„+ ....) = »^ (*.+/,+ • • • • ) dF df = dT + = DF n +Df n + .... (2) DaF n = (o, -^ a,P„ = a. g = aD#„ d r \( _ d° \ „ . J d r \f d s dT r+, These relations prove that — within 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 use of A -1 , A -2 , ...... and of D _1 , D -2 , .... are easily understood : thus, from the foregoing definitions, we have A ('*■„) = F n where 'F n is defined as in the schedule on page 134. Then, in analogy with the usual mode of expressing inverse functions, we may write >F n = a- 1 -?; Whence we have AA-'f, = A('iy = F a (379) which shows (1) that the operation of AA _1 (= A ) leaves the sub- ject function unaltered, and (2) that negative powers of A also obey the Index Law. The relation A -i Fn = i Fn (380) may be taken as the definition of the operator A -1 . Similarly, we have A -2 Fn = u Fn > A -s Fn = m Fn t (381) Again, consider the relation D^ = ->~ = v (382) which, from the point of view above taken, may be written F n = D^v (383) APPENDIX. 209 Then we have DD" 1 ?; = DF = v (384) whence we see that negative powers of D likewise follow the Index- Law. Moreover, from equation (382), we obtain dF n = u-ivdT and therefore F n = oj- 1 CvdT which, with (383), gives D->v = a- 1 CvdT (385) . It follows that the operation of D" 1 is equivalent to an integration. More specifically : Operating upon any function with D _1 integrates that function with respect to T and divides the resulting integral by w. In like manner we have D~ 2 F n = a-tfjFJT* (386) and so on. 95. Having thus defined and explained the use of the symbols of operation, A~ 2 , A" 1 , A , A, A 2 , . . . . , and D~ 2 , D _1 , D°, D, D 2 , . . . . ; and having shown that these symbols may in general be combined algebraically as if they were merely symbols of quantity, we now pro- ceed to derive 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\ r) , of the direct or given series, becomes [Jj r) ] when that series has been inverted. Then, since af, = f 1+1 — f, = j; we have -a^. = f,-f w = [j;j Whence, regarding —A as operator, it follows that (_A) 2 ^ = [J,"], (-A) 3 ^,. = [J,'"], . . . . , (-Ay-F, = [J<'>] and therefore [JJ'>] = (-A)'^,. = (-l)'A^ = (-iyj[ r) (387) which establishes Theorem III. 210 APPENDIX. 97. By definition, we have A^,= F n+1 - F„ hence (1 + A)^„ = F n +AF n = F n+1 = F(t + no> + u>) 2 3 = F+ (»F' A- — F" 4- — F'" A- .... D 2 D 8 „ = j 1 4- D.F -i fa Fa- . . . ■ / D 2 D 3 \ = ( 1 + D + 1 r+ F + ■ ■ ■ -)K= e°F n where e is the base of the natural system of logarithms. "We have, therefore, 1+A = e° (388) which is the fundamental relation between A and D. 98. From (388), we get D 2 D 8 D 4 A = e°-l = D+ E + 1: + K+ .... (389) and hence, by involution, A 2 = D 2 + D 8 + T \D 4 +iD 6 + . . A 3 = D 3 + f D*+ |D 5 +|D 6 +. A' = D> + i D> +1 + ?% (3i + 1) D i+2 + (390) These expressions are equivalent to the formulae (21). Again, from the last of (390), we derive A'F, = (D f + a,D i+1 + a 3 D'+ 2 + . . . . ) F, that is J< 4 ' = tfF^ + a^^Fi'+v + a^Fl'+n + . . . . (391) where for brevity we have written a t , a 2 , . . . . to denote the co- efficients of D*+ x , D i+2 , .... in (390). Whence, if Fi^—aT' -A r pT i - l -\-yT i - 2 + . . . . , we have JJ» = »*jw = ow'^j (2") = ««,*Li which is the algebraic statement of Theorem V- APPENDIX. 211 99. Expressing the relation (388) in logarithmic form, we get D = log.(l + A) = A-^+^-^+ .... (392) whence D 2 = A 2 - A 8 +HA 4 -£A 6 + .... D 8 = A 8 -}A 4 + J A 6 - ... . V (393) From these relations the formulae (45) — or the equivalent group (165) — immediately follow. 100. We next consider the question of reducing the tabular in- terval from a) to mo), as discussed in § 19. Since in the preceding definitions of A and D the magnitude of the interval is arbitrary, we have here only to denote by d and d the corresponding symbols in the reduced series ; evidently the same relations will then exist be- tween d and d as were found above for A and D. Thus we obtain d =m (1 )^= m(u>—j = mD (394) and since, by (388), we have 1 + A = e D we must have also 1 + d = e d = e™ (395) Whence we find m(m — 1) .„ m(m — l)(m — 2) • 1 + d = (l + A)" = l+mA+ V j, ; A 2 + — jir " A + .... and therefore m(m — l) A , m(m — l)(m — 2) . 3 = mA + y ; A 2 +-^ ^ ^A 8 + .... 2 = m 2 A 2 + w 2 (m-l)A 8 + \ (396) d» = m 8 A 8 + which are equivalent to the relations expressed in (64). 101. The equation A^ = F,-F, may be written in the form (l + A)*, = F, (397) (398) 212 APPENDIX. Hence the binomial 1 -|- A may be defined as an operator whose effect is to raise by unity the subscript of the subject function. Whence we have (1 + A) 2 ^ = (1 + A)^ = I\ (1 + A) 3 ^ = (1 + A)-F 2 = F s and generally (1 + A)"i^ = F. (399) We therefore obtain F n = (1 + A)-*. = (l + »A|^AH g(8 -y- 2) A' + )F or *.-*.+ ~*i+ ^F^ 4' + W( "~ 1 ^~ 2) ^" + (400) which is the fundamental formula of interpolation due to Newton. 102. We now find it convenient to introduce a new symbol of operation, which, from its similarity and relation to A, we shall desig- nate V: this operator is defined by the equation \7F { = i^-i^ = A\_, (401) From this relation we at once derive v*F< = vj;_, = A'U V 3 J^. = VJ"„ = J'.". -* '-s ) arm whence it appears that the operation of V upon any tabular function produces the difference of order r 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 downward 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 t _ x (403) also, from (397), we have (1 + A)i^ = F t (404) APPENDIX. 213 Whence we find (1 + A)(1-V)^ = (l + A)^ = F t and therefore 1_ V = (l + A)- 1 (405) which gives log(l-V) = -log (l + A) - (406) Again, combining (388) and (405), we obtain 1 - V = e-° (407) 103. As an immediate application of the preceding relations, let us derive the formula (75). By means of (388), equation (399) becomes F„ = (1 + A)"^ = e«°F whence, changing the sign of n, we find .*■_„ = e-»°F = (e-yF = (1-V)-F Therefore n(n — 1) ,,, n(n — l)(n—2) /m«\ F_ a = F - nJU+ ^ ' '-'- 3 - - V ^ l J- s + • • • • (-108) which is Newton's Formula for backward interpolation, as given by (75). 104. Formula (66) of the text is easily deduced by means of the identity A = (1 + A)-1 Thus we find A ^ = i(i+A)-ir^o = ^(l+Ay-ia+Ay-H^^Ci+A)'- 2 - .... j/<; whence, by (399), we obtain i(i— 1) _, t(t— 1)0'— -) F , (409) j«) = Fi _ iF( _ l + -L—lF l _t--± jj *V* + • • • • ^ * which is the same as equation (66). 214 APPENDIX. 105. "We now pass to the derivation of the fundamental form- ulae of mechanical quadrature. Since D = log (1 -f- A) > we have ' / A 2 A 8 A 4 \ _1 D-^„ = {log(l + A)|- 1 ^ = (^A-^- + ^-^+ . . . .J F n = (A- i +i-^A + ^A 2 - T V 9 TJ A 3 + T | Ty A 4 -^|t T A 5 + . . . .)F n Whence, interpreting the first member according to (385), and the term A" 1 ^ as in (380), we find v-iJfjt = >F n +iF„-^j' n + ^j';-^J':'+ThJ l :-AihJ:+ ■ ■ • (*io) This is the fundamental relation of quadrature, from which the formula (a) of (250) is at once derived. To obtain (b) of (250) in- volving the differences ^4-n ^»-2> ^»-b> • • • • , we have only to employ the relation (406), and the above development becomes D- 1 ^ = \log(l + A)\->F n = {_l„g(l-V)|- l .F. - ^V+^-+- 3 -+- r +- r + ■ . . . ) F„ = (V" 1 - I - tV V- A V 2 - t¥* V s - tIo V 4 - sMb V 6 - • • • • ) -^ the interpretation of which gives «.- 1 /^T='^ 1 -ii^- T V^_ 1 -^;U-T 1 ^^-s-T;l^ , L4- F MI^-5- • • ■ (411) agreeing with formula (5) of (250) . 106. Similarly, we obtain for the second integration D^F n = |log(l + A)|-^„ = (A-^+^-^-f- . . . .y Fn = (A-^A-HtV-^ttAH^ttA^-^I^A^^-^A 6 - . . .)F n Now the first pair of terms in the right-hand member may be written (A- 2 + A- 1 ) F n = A- 2 (l + A)#„ = A" 2 ^ n+1 = "F n+l and therefore the preceding expression becomes ^ff F " dT * = " F »+l + ^ F»- ?h < + ?hj> 4'," - V H b 4.' +^U^n~ ■ . ■ (412) from which (324) immediately follows. APPENDIX. 215 Again, we find D-^„ = \\o S (l + A)\-*F n = j_log(l-V)|-^„ = (V- 2 -V- 1 +i 1 !? —a-oV 3 -aoV 8 -TrgHTrV 4 - i TA^V 6 - • • )F n (413) Transforming the first two terms of the last expression, we ob- tain (V- 2 - V- 1 )^,. = V- 2 (l-V)^„ = V- 2 (l + A)-^„ Now, because the operation of 1 -\- A raises by unity the subscript of the subject function (§101), it follows that the operation of (l-j-A) -1 diminishes that subscript by one unit. Accordingly, we have (v-'-v- 1 )^,. = v-^i + a)- 1 ^; = v- 2 ^ = "F n+l and hence the relation (413) gives »-*ffF n dT* = "F, M +^F,-iioJ:u-?h^-*nh^-vhhJu- ■ ■ ■ ( 414 ) which is equivalent to the formula (326). These expressions complete the fundamental relations of mechanical quadrature. TABLES. 218 Table I. — Newton's Interpolating Coefficients. 23 o fc. 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US © US US "* i# © © © © © us © US © us © 00 US CM b- © © 0 ># tJ< LO ^1 Tj< CO -^ CO CO N CO (M CN # US b- 00 00 CO 00 © © © © © © © © © © © © © © © + + t H CO «D t- CS CO tO CO OS H/ ^# ^* "^ "* CO © ■** b- 00 -* TH -* CO CM i-l CO CO CO CO CO CO CM CM CM CM CM CM ©©©©©© © CO b- © © © © b- © Tfl CO CM CM CM CN CM CM CM CM CM © © © © © 00 US © t-I tH HOStO^IH CM — ' tH t-I t-I CM CM CM CM CM © © © © © © © T-I CO ■<* 00 Tt< t-I b- CO © © © © © NNNrHH © © © © © CM © ■># 00 © © ■* © US © 00 00 00 b- b- HHHrlH © © © © © + + + + + 00 I- ■* iH © CM CO US © © © -^ 00 CM © © © © © tH t-I CM ©©©©©© ©©©©©© © © US CO © ■<* 00 CM © © CM CM CO CO "* © © © © © © © © © © © tH US 00 © CO b- © CO © Tjl -^< US US US © © © © © © © © © © 00 © CO b- © © CM US b- © US © © © b- © © © © © © © © © © t-I © © © T-I CM CO US b- 00 b- b- b- b- b- © © © © © © © © © © + lO lO lO >o lO i^ O O o H « CO ^ lO cot-coos l— I r-t tH i— I i— I rH r-1 >H rH »S kO «5 IO rH CN CO ^ (N « (N « + © us © us © us © © 00 US CM b- 1« tJ tJI T)i ■>* » CM CM CM CM CM CM © US © US © CM US 00 © © CO CM tH © © CM CM CM CM CM us © us © us © 00 US CM b- 00 b- © US CO © US © US © CM US 00 © © CM © 00 © US TH T-I © © © us © us © us © 00 US CM b- CM © 00 © CO © © © © © HHOOO I Interval ©t-ICMCO'^US ©b-00©© i-ICMCO-^US ©b-CO©© USUSUSUSUSUS USUSUSUS© ©©©©© © © © © b- tH CM CO rjl US b- b- b- b- t~ 224 Table IY. — Newton's Coefficients eor F'(T). K o QO H 15 M O fa o O > -i IS 5 Ol t- ^ (N O CO ^ CO (M H O CO 7 "*** ^ CD IO CO C<1 O O CO CO CO 00 t- O IO ■* CO (MOO500 r- C0 CO CO CO CO CO CO ON CN CN t-i^OOOO CO © O H H coiortico CM ^"-JSS 0X|(N(NCM 00 IO IO •<* CM © CM •* CO CM CO -* CO 1- r- CO (ON ODlOH i-l Tfl L-- © CO l-li-l©©© OOOHH ©©©©© ©©©©© + '■'+■ 1 001OCM©00 COIOIOCOOO C0©©i-!"* L-©©©© ©©tH-*© OOOCOIOCO rHi-ICMCMCM cmcocococo ©©©©© ©©©©© 1 1 ' ' 1 > la" 3 o m IN OJ 00 CD © iO it iM h o vd o io us o utt + CO CN t-h CO 00 IO 1Q « (N (N OJ » t- ID "^1 n « H O OS ^■S-*^ ■* Tt< *# T# ^f CO OO OS CO t-t-J-g oo to io-* co c^rlSS COC0C0CO CO CO CO CO « + T -*- + "el 10 o -* © t- oo o IO © CO 00 CO CO CO IO l© ■<* -* CO o o o o o o 1 rjl i-l © CO © N SNHM © © i-H -* © t* © 00 CO CO CM CO CO CO CO © IO © CO t- MNMHH ©©©©© ©©©O© ©©©©© 1 1 ' l' + 1O»O-*C0tH ©oco©t- CM©t-C000 iH-<*©b-© H 1O00N1O SlNWOOrt t-HtHtHCMCM cmcococo-^i ©©©©© ©©©©© + '"""+' '+ n IB 5 O lO X3 ICI lO lO -tf CO o io 00 1^ © lO -«ri CO ia ig> us w ics »o w oot-com i< W N h o + s 1 "a I* 00 CO 00 CO 00 CO m ih t- io co co T* l— © - tO ©10»0-5*l-* ©©©©© ©.©©©© • + + + + + + > IB s cs -* o m O CO © Oi CO CD lO CO OS CO CO CO CO 00 + wt-COCO »C -"-"t-COO CD (MOCSl^ CD lO M N H OJ 00 M C- t- £- t-r-t-J^ CD C005CO-* O t- iff H O oo co m xf co i— i o o oo CO O © CD CD O CD ii3 lO -4* s -4- + 1 O i-l t- IS- CM CO CQ5DHOO COOCOSCO O00 00C1H H»O00HlO 01N(OC»0 NdOOOi oooot-t-co THiHrHrHO OOOOO 1 1 1 1 1 1 n ta' 5 io io io io »o »o c» co i^ cd io ^ OS Oi OS Cft OS c IO IO IO lO IO lO IO IO IO IO coiMt-ho a ooi— cdio -* C3SOS0305 CO 00 00 00 X 00 \a id ia iO id »T5»fl>0»0 M (N H O OS GO I* © lO CO GO CO CO £- ir-lT-t-t- 1 + g 1 "gh CO 00 CO CO CO 00 CO CO IO t- t-I IO CO CO CO CO ■<* "* CO CM t-I © © CO CO CO CO CO CM CM cococoooco cocooocooo Ht-IOCQCC COlOt-i-llO 1O1OCDN00 OOHC0-* t-COlOTHCO CMCMiH©© CMCMCMCMCM CMCMCMCMiH cooocoooco cocooocooo T-lt-lOCOCO CO>Ot-T-llO ©t-©i-ICO lOt-©CM"* OOfc-COCOlO ^CONNH HrirliHH 1-It-iHtHtH + + + + + + n H(N O © 00 t- CO IO IO ^* ^H -* ■* TJH TfHCOCMi-l© ©OOt-COlO ■nH^^-cH-^ COCOCOCOCO "*«Nt-io esoot-coio COCOCOCOCO CMCMCMCMCM o 1 © 1 Interval * O t-I CM CO -# IO o o o © o o © COt-CO©© H N m ^ IO ©©©©t-I HrlHHri COb-00©© i-ICMe0'*>O H rt H rl « CMCMCMCNCM © Table IV- — Newton's Coefficients for F'(T). 225 p- M O t* 03 Eh W O En H o O fe5 s r4 iQ © »0 OS Ift -H rH 1 1 + + + + "*l» 1 115!OHH©t> N CO ■<*•-? 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NO H t- t- © rl CO -* © e i- t-- i- © © © © © + +.07622 .07762 .07894 .08018 .08133 i-H tH CO t * Tf -HI CO H © CI CO -* 10 10 co oo co co co © © © © © + + •^ to' 3 ift >0 W3 *f5 »Hi »0 OS GO t- CD lO T* its *& ia id ira CO IN r* O OS ■^ Tt< ■* -^ CO ia o in i>s »o 00 t- CO lO "^ CO CO CO CO CO O 1ft lO ift ift co cm — © cr. CO CO TO CO C-3 x i- -5 .5 ?1 Cl "M ^-1 1 + s 1 t- CM t~ CM t- CM CD CO ■* CM 00 -^ T-i CD tH CO © US Til tH 1© US CD CD ©©©©©© I t- C; 226 Table V. — Stirling's Coefficients for F'(T). H G H o h H o s o o n 5 -« CO IO £- © t-I O © CO CO t- Jr- 1 rH CO t- OS IM *-£-£-£- GO CO IO t— CO o CO CO CO CO © CM CO »0 t- CO C _. 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Note. — The quantity y has the same sign as argument K. 232 Table VIII. Coefficients for Computing F n = F 4- »* (*■„' + £ « + Bft + Ty). 0.00 .02 .04 .06 .08 .10 .12 .14 .16 .18 .20 .22 .24 .26 .28 .30 .32 .34 .36 .38 .40 .42 .44 .46 .48 .50 .52 .54 .56 .58 0.60 BE 0.0000 + .0001 .0003 .0006 .0011 .0017 + .0024 .0033 .0043 .0054 .0067 + .0081 .0096 .0113 .0131 .0150 + .0171 .0193 .0216 .0241 .0267 + .0294 .0323 .0353 .0384 .0417 + .0451 .0486 .0523 .0561 + 0.0600 Dlff. + 10 11 13 14 15 17 18 19 21 22 23 25 26 27 29 30 31 33 34 35 37 38 +39 '^(t-) 0.0000 - .0017 .0033 .0050 .0066 .0083 - .0099 .0116 .0132 .0148 .0163 - .0179 .0194 .0209 .0224 .0239 - .0253 .0267 .0281 .0294 .0307 - .0319 .0331 .0343 .0354 .0365 - .0375 .0384 .0393 .0402 -0.0410 Diff. -17 16 17 16 17 16 17 16 16 15 16 15 15 15 15 14 14 14 13 13 12 12 12 11 11 10 BIBLIOGRAPHY. List of the Principal Papers, Memoires, etc., upon the Subjects of Interpolation and Mechanical Quadrature. Astrand (J. J.). Vierteljahrsschrift der Astronomischen Gesellsohaft, Vol. X (1875), p. 279. Baillaud (B.). Annales de l'Observatoire de Toulouse, Vol. II, p. B. 1. Bienayme (Jules). Liouville, Journal de Mathematiques, Vol. XVIII (1853), p. 299. Boole (George). A Treatise on the Calculus of Finite Differences, Chapters II and III. Brassinne (E.). Liouville, Journal de Mathematiques, Vol. XI (1846), p. 177. Briinnow (F.). Lehrbuch der Spharischen Astronomie, p. 18. Oauchy (AugUStin). (a) Liouville, Journal de Mathematiques, Vol. II (1837), p. 193. (J) Comptes Eendus, Vol. XI (1840), p. 775. (c) Ibid., Vol. XIX (1844), p. 1183. (d) Counaissance des Temps, 1852 (Additions), p. 129. Chauvenet (Wm.). Spherical and Practical Astronomy, Vol. I, p. 79. Davis (C. H.). The Mathematical Monthly (Cambridge, Mass.), Vol. II (1860), p. 276. Doolittle (0. L.). A Treatise on Practical Astronomy, p. 70. Encke (J. F.). (a) Berliner Astronomisches Jahrbuch, 1830, p. 265. (b) Ibid., 1837, p. 251. (c) Ibid., 1852, p. 330. (d) Ibid., 1862, p. 313. Ferrel (William). The Mathematical Monthly (Cambridge, Mass.), Vol. Ill (1861), p. 377. Gauss (Carl P.). Werke, Vol. Ill, p. 265. Grunert (J. A.), (a) Archiv der Mathematik und Physik, Vol. XIV (1850), p. 225. (b) Ibid., Vol. XX (1853), p. 361. Hansen (P. A.), (a) Abhandlungen der Koniglich Sachsischen Gesellsohaft der Wis- senschaften (Leipzig), Vol. XI (1865), p. 505. (b) Tables de la Lune, p. 68. Herr und Tinter. Lehrbuch der Spharischen Astronomie, Chapter II. Jacobi (O. G. J.), (a) Crelle's Journal, Vol. I (1826), p. 301. (6) Ibid., Vol. XXX (1846), p. 127. Klinkerfues (W.). Theoretische Astronomie (2d edition, 1899), pp. 67 and 490. 233 234 BIBLIOGRAPHY. Kliigel's Mathematisch.es Worterbuch. See article "Einschalten," which includes a brief history of the subject. Lagrange (J. L.). (a) (Euvres, Vol. V, p. 663. (6) Ibid., Vol. VII, p. 535. Laplace (P. S.). Mecanique Celeste, Vol. IV, pp. 204-207. LeVerrier (U. J.). Annales de l'Observatoire de Paris (Me"moires), Vol.1, pp.121, 129, 151, 154. Loomis (Elias). Practical Astronomy, p. 202. Maurice (Fred.). Connaissance des Temps, 1847 (Additions), p. 181. Merrifleld (C. W.). British Association Report, Vol. L (1880), p. 321. Newcomb (Simon). Logarithmic and Other Mathematical Tables, p. 56. Newton (Isaac). Principia, Book III, Lemma V. Olivier (Louis). Crelle's Journal, Vol. II (1827), p. 252. Oppolzer (T. R.). Lehrbuch zur Bahnbestimmung, Vol. II, pp. 1 and 596. Radau (Rodolphe). (a) Liouville, Journal de Mathematiques, 3d Series, Vol. VI (1880), p. 283. (b). Bulletin Astronomique, Vol. VIII (1891), pp. 273, 325, 376, 425. Rees's Cyclopedia. See article " Interpolation." Sawitsch (A.). Abriss der Practischen Astronomie, Vol.11, p. 416; or see the one volume edition, p. 818. Tisserand (F.). (a) Comptes Bendus, Vol. LXVIII (1869), p. 1101. (b) Ibid., Vol. LXX (1870), p. 678. (c) Traite de Mecanique Celeste, Vol. IV, Chapters X and XI. Valentiner (W.). Handworterbuch der Astronomie, Vol. II, pp. 41 and 618. Watson (James 0.). Theoretical Astronomy, pp. 112, 335, 435. Weddle (Thomas). Cambridge and Dublin Mathematical Journal, Vol. IX (1854), p. 79.