Scientific Papers Volume 3 (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. 1910 Excerpt: ...are as specified above in the equation for a. By forming continued products of ratios of successive q's, we can find all the q's as multiples of qs, and q, = I. In the cases EEC, OEC, OOS, EOS, these are the required coefficients for $. s + 2?i In the cases EES, OES, OOC, EOC we put qf8 =----j, a- and thus s find the coefficients for Ps. 2n) i, s + 2n-2j... i, s+2j When s= 0, 72//0 is equal to that which would be given by the general formula for JH& when we put in it n--1, s = 0. Hence it follows that the qs for s = 0 have double the values given by the general formula. If we change the sign of s, the two continued fractions in the equation for o-are simply interchanged. Hence cr is unchanged when s changes sign. Also, since i, t is equal to--i--1, t, a is unchanged when--i--1 is written for i. A consideration of the forms of the q's and p's shows that-g+atP""1"2 7---s t is equal to. '-qs_cPs'ik, and therefore + s ffli" _ _+__ jr6 Ip;-s JPrs # tiPU Pf IP'-i-i 7. Rigorous determination of the Functions of the second order. If a numerical value be attributed to /3 it is obviously possible to obtain the rigorous expressions for the several functions. Thus, if /3 were we could determine the harmonics of the ellipsoids of the class c2 = (a2 + 62). In order to show how our formulae lead to the required result I will determine the five functions corresponding to i = 2. The case of i = 3 will be considered in Paper 12. Then B-l and CDaa = _, j--+ cos29 (24) s = 2, sine; EES type. Both fractions disappear and a vanishes, but is not needed for determining the functions. Noting that (/ ' = 0, and /2' = 1, P. 2 = n(?2'P1,2 = 3 ("'-rO'2-1) (25) &22 = sin20 (26) We can write down the functions of / by sy...