SATELLITES UPON THE FORM OF SATURN’S RING. 
109 
The best methods of determining the values of b r , and its brst and second 
derivatives for known values of u, or aja', are given by TisserandA The complete 
evaluation of a number of these quantities for various ratios, applicable to the solar 
system, is given by Ponteooulant.I For the purpose of estimating the order of the 
numerical values of the quantities 0 rs , we may take the highest ratio a likely to 
occur as that of the outer edge of the ring to the mean distance of Mimas. This 
ratio is 07461 (see Appendix for data). Pontecoulant gives the values for 
rx = 072333, which we may use to avoid laborious calculation. If we take 7 = 7. 10~ 8 , 
the value for Mimas, we find 
0 3iO = -20-2590 + 67530^; 
0i,i = — 1 ’35 . 1CU 6 , 0i, 2 = -P53 . 10“ 6 , 
0 21 = -4M7.1CT 7 , 0 2 , 2 = —7'39 . Itr 7 , 
0 3 ,o = -F46 . 10~ 7 —179.10- 7 K, 0 ; . 2 = 
0 5 , o = — 13'5060rL s . 
It is clear that, compared with 0, 0 , all products and squares of the remaining 0’s 
may be neglected. 
0 1)3 = —1'63 . 10~ B ... ; 
0 2 , 3 = -9-34. 10“ 6 ...; 
4'07. 10- 7 , 0 3 , 2 = 3'69 . 10- 7 ... 
§ 3. Solution of the Equations. 
{a) The complementary function. 
The equations 
p" — 2/co-' + p20i, r cos rcj) + <t 202 . r sin r<p = 0, 
rr" + 2 Kp + pA0 4 , r sin r<f> + cr20 5 , r cos r<f> — 0 . 
belong to the class of homogeneous linear differential equations with periodic 
coefficients. The integral is known to be the sum of the forms e' : f (</>), where J (</>) 
is a periodic function of cf> with the same period as the coefficients in the equations (6). 
Equations of this form in one dependent variable have been discussed by Whittaker,J 
Young, § Ince,|| and BakerA The present solution is a simple extension of the work 
of these writers. 
Let 
P = cAA, 
t = e rA X , 
* ‘ Mec. Celeste,’ vol. i., p. 270, et scq. 
t ‘ Systeme du Monde,’ vol. 3, pp. 353-376. 
\ ‘ Pioc Inter. Congress Math.’ vol. 1, 1912; ‘Proc. Edin. Math. Soc. xxxii., p. 76. 
§ ‘Proc. Edin. Math. Soc.’ xxxii., p. 81. 
|| ‘ Monthly Notices R.A.S.’ lxxv., 5, p. 436. 
1 H. F. Baker, ‘Phil. Trans.’ A., vol. 216, p. 129. 
R 
VOL. CCXXII.—A 
