An Introductory Treatise on Dynamical Astronomy (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1918 Excerpt: ...mutual distance. It is therefore important to consider the nature of this expansion, or rather of A_M in general, where s is half an odd integer. For this more general form will give the, derivatives of A-1, A3 being a rational quantity, and these will naturally occur when A-1 is expanded in terms of any contained parameter. It is convenient to consider first the case of two circular, coplanar orbits. Then, if H is the difference of longitude in the plane, A2 = a,2 + a, ?--2-, as cos H a, Oj being the radii of the orbits. Let a, a2, a = Oj/oj, iH = log z, t2 =--1 and therefore '-2-a A2 = 1 + a2-2a cos H = (1-az) (1-azl). Hence the function to be examined is, F = (1-az)- (1-az-1)-= f b/z'-00 = (1 + a2-2a cos H)' = %bs + 2 bj cos iH. I Since the function is unaltered when z and zx are interchanged, bs'i = bgi, and i may be treated as positive. The coefficients be are called Laplace's coefficients. By Fourier's theorem, 1 f W =--(1-az)' (1-az-1)-' z-1 dz 7TI J Cf. H. v. Zeipel, Encykl. dtr Math. Witt., vi, 2, pp. 560-665. The first (complex) integral is due to Cauchy; the path of integration is taken round a circle of unit radius. By introducing the Weicrstrassian elliptic function. f (-) =-Ha + --") Cauchy's integral clearly becomes an elliptic function, and Poiucare has shown how this function can be reduced to a calculable form. But another method will be followed here. The coefficients bj are easily developed as power series in a." For, with the use of gamma functions, v 'v pr(-)r(P+i), r(-)r(?+i) and therefore, when p = q + i, ' 7r()32r(7 + i + i)r(g + i) C"+3 = - - + (j+V) a'-i- (J+-)TS (9) whence cn-(n = 3, 5, ...) are found in succession. It only remains to form c'i = pni-...