The applications of elliptic functions, by Alfred George Greenhill.

APPENDIX.

I. The Apsidal Angle in the small oscillations of a Top.
The expression given by Bravais in Note VII. of Lagrange's
MJcanique analytique, t. II., p. 352, for the apsidal angle in
the small oscillations of a Spherical Pendulum about its lowest
position is readily extended to the more general case of the
Top or Gyrostat, if we employ the expression on p. 261, ~ 242,
as the basis of our approximation.
We divide the apsidal angle I into two parts, I1 and '2,
such that il == al1- ola,
iP2 = b1-1- b;
and now put a=W3 %-Sw83, b = +qw3,
where q and s are small numbers; so that, expanding by
Taylor's Theorem as far as the first powers of q and s, we may
put ~a '"3 + Sw3 PW3 = a3 + s3e3,
-b ~ r1- qwOt1 = - qw3el;
and now, by means of Legendre's relation of p. 209,
^ (- -03) 1(3 - S- S 3e3) = '^7r - sw3(1 + e3W1),
T2 (Io + q,3)71 - wl(l - qo,3e) = qo3(,1 + elW,).
But, from equation (B), ~ 51,
dn2^(el3 - e =) = 1 - e2 e3
+u-e3 3 —e3
= 1 _ f(~+%) —ea = e —~o^u+r);
e - e3 el1- e3
so that, integrating between the limits 0 and w,?
o
el(w + (W1 + W3) - 3:=(el - e3)2dil 2/(el - e3)udul,
0
0or,l + el= ^/(e -e3)E (Schwarz, ~ 29).
Also (~ 51) (e -e3)w1=./(e-e)K;
so that 1 + e3W1= - V/(e1- e3)(K- E);
and therefore i4' = IT7r + 8W3^/(eI- e(K-E),
i*2 = q(3A(e - e3)E.
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