Determination of the Coefficients of the Series R. 452. In order to complete the developement of R, the coefficients A and B, and their differences, must be determined. Let (a2aa' cos ẞ+a3) A4, +4, cos B + A. cos 2ẞ + &c. The differential of which is A-12saa' sin ẞ= 4, sin ẞ + 24, sin 28 +34, sin 3ẞ+ &c. multiplying both sides of this equation by A, and substituting for A-, it becomes 0 2saa' sin BA, + A, cos B + 4, cos 2ẞ + &c.} If it be observed that cos B sin cos 28, &c. when the multiplication is accomplished, and the sines and cosines of the multiple arcs put for the products of the sines and cosines, the comparison of the coefficients of like cosines gives in which may be any whole number positive or negative, with the exception of 0 and 1. Hence 4, will be known, if A, A, can be found. Let A¬~1 = B。 + B1 cos ẞ + B, cos 28 + &c. multiplying this by (a2 2aa' cos B + a'2), and substituting the value of A-' in series A+ A, cos ß + A, cos 2ß + &c. * =(a2 - 2aa' cos ẞ + a2) ( B, + B1 cos ß + B2 cos 2ß + &c.) the comparison of the coefficients of like cosines gives A1 = (a2 + a'2). B. - aa'. B-1)-aa' B(+1)• But as relations must exist among the coefficients B-1), B, B(+1) similar to those existing among A(1), A;, A(+1), the equation (119) gives, when s + 1 and i + 1 are put for s and i, If this quantity be put in the preceding value of A, it becomes (120) (121) (122) whence may be obtained, by the substitution of the preceding value of B(+1) A(i+1)= ̧s(i+s). aa'(a2+a'2) B (i-1) + s { 2(i − s) a2a12 — i(a2 +a'°)°}B; − s + 1). aa' (i − s) (i If B be eliminated between this equation and (121), there will B(i-1) result, B = 2 1 (i + s) (a2 + a'2) A; − 2 (i − s + 1) . aa' . A (i+1) or substituting for A(+1) its value given by equation (119), 2 8 (a12 - a2)2 If to abridge =a, the two last equations, as well as equa a tion (119), when both the numerators and the denominators of their several members are divided by a2, take the form A;= (i − 1) (1 + a2) A (i−1) − (i + 8 − 2) . œ . A-2), (123) (is) a s α. (1 2 1 (8 − i ) (1 + a2) . 4, + 2 (i + s − 1) • a' . A (-1) All the coefficients A2, A3, &c., B。, B1, &c., will be obtained from equations (123) and (125), when A., A, are known; it only remains, therefore, to determine these two quantities. c being the number whose hyperbolic logarithm is unity; therefore a12 - 2aa' cos ẞ + a2 = {a' - ac2N=T}. {a' ac-9N=1} consequently, but A~'= {a' — ac2Ñ−1}~'. {a' — ac−®N=1}—'. (a' — ac®N=1)~ = _—_{1+ s ac®N=1+ $(8 + 1) 2 22=1 + &c. }, 2 whence it appears that c, and e-I have always the same. coefficients; and as CNI+CN1 2 cos iß, it is easy to see that this series is the same with 2aa' cos Ba2)~' = ↓ A。 + A, cos B + &c. A= · { 1 + s2x2 + 128 These series do not converge when s; but they converge rapidly when s; then, however, 4, and 4, become the first and second coefficients of the development of (a12 2aa' cos ß + a2)3. Let S and S' be the values of these two coefficients in this case, and as the values of A。, A, may be obtained in functions of S and S', the two last series form the basis of the whole computation. Because A, A, become S and S' when s, and that B becomes A.; ifs, and i = 0, equation (124) gives and ifs, and i = 1, equation (125) gives i=1, and substituting the preceding values of 4, and 4,, it becomes. 454. It now remains to determine the differences of A, and B, with regard to a. A = Resume and take its differential with regard to a, observing that gives A = a12 2aa'. cos ẞ + a3 or, substituting the values of A A+ A, cos B+ A cos 0 a a dA Ꭶ da Ꭶ da cos 28 and A-1 in series 28 + &c. + (a2 - a12) × B. + B, cos B + B2 cos 28 + &c.} = and the comparison of like cosines gives the general expression, or, substituting for B, its value in (124), it becomes (126) If the differentials of this equation be taken with regard to a, and if, in the resulting equations, substitution be made for from the preceding formula, the successive differences of A, functions of A(+1), A(+2), will be obtained. Coefficients of the series R. 455. If be put for s in the preceding equation, and in equation (123), and if it be observed that in the series R, article 446,. dA da is always multiplied by a, de A da2 by a2, and so on; then where i is successively made equal to 0, 1, 2, 3, &c. the coefficients and their |