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Full text of "Scientific Papers - Vi"

1916]                         APPROXIMATE  SPHERES   AND  CYLINDERS                           389
C2 = -i-e-, Q being the eccentricity. It must be remembered that a in (30) is not the semi-axis major, but the spherical radius of equal volume. In terms of the semi-axis major (a), the accurate value of the capacity is ae/sin"1 e.
We may now proceed to include the terms of the next order in C.    The extension of (25) is
tfiPi 4- . . . + GPPP]
...+(q + 1) CqPq],    ...... (33)
where in the small term the approximate value of Hn from (25) has been substituted.    We set
where. Jn is of order C2 and depends upon definite integrals of the form
rPnPpPqdfi, .............................. (35)
n, p, g being positive integers.
In like manner the extension of (26) is
+ BCiP9+...+te(p + I)CpPp}. (36)
Here, again, the definite integrals required are of the form (35).
These definite integrals have been evaluated by Ferrers* and Adamsf. In Adams' notation n + p + q = 2s, and
/q^_     2     A(s~n).A(s-p).A(s~q}                  ^
w __._._                 __^_             ..... ,    ......... (6i)
n                                 ...     1 .3. 5 ... (2n  1)                          /oo\+
where                          4 (n)  ____L__J ...................... (38)J
In order that the integral may be finite, no one of the quantities n, p, q must be greater than the sum of the other two, and n + p -t- q must be an even integer. The condition in order that the integral may be finite is less severe than we found before in the two dimensional problem, and this, in general, entails a greater complication.
But the case of a single term in Bu, say GPPP (p), remains simple. In (36) Jn occurs only when multiplied by On, so that only Jp appears, and
* Spherical Harmonics, London, 1877, p. 166.
t Proc. Roy. Soc. Vol. xxvn. p. 63 (1878).
 [Following Adams, A (o) must be taken as equal to unity.    W. F. S.] unexpectedly, that there is a linear acceleration amounting to