108 
DR. G. E. GOLDSBROUGH ON THE INFLUENCE OF 
The equations ( 2 ) may be replaced, under the suppositions made, by equations ( 3 ). 
Equations ( 2 ) form a system of n pairs of linear equations of the second order. The 
complete integral will therefore involve 4 n arbitrary constants. The system of 
equations (3) will give the same result, for the solution of (3) will be a function of s 
involving four arbitrary constants. By giving s its n values, 0 , 1 , 2 , ..., (n— l), we 
arrive at the complete integral involving 4 n arbitrary constants. 
Now it has been shown by Tisserand # that for large values of n, whatever the 
value of s may be, the limiting value of L s is 0'0194n 3 , N s = 2 L S and M s = 0. 
These values largely simplify the discussion of the stability of the system. 
Lastly the equations (3) may be written, for convenience, in the form 
p" — 'Ikct' + (Oj,o + 0i,i COS 0 + ... + 0],COS V 0 ...) p 
+ (0 2 ,i sin 0 + 0 22 sin 20+... +0 2>r sin r<p +...) o- 
= 0 3 , o + 0 3 i COS 0 + ... + 0 3j r COS V(p + ... 
it -f- 2 Kp T ( 0 i_ i sm (p T 0 j o sin 20 + ... 4 ~ 0 ^ r sm vcp - 1 - ...) p 
+ (05,0+0.5.1 COS 0+ +05,r COS ?'0 + •••) O' 
= 0 fi , 1 sin 0 + ... 4- 0 6 , r sin r<p + ... 
(4) 
The values of the quantities 0 are : 
0 
1, 0 
-527 
q 2 1/2/200,) 2T 
— OK — WV K K — —5 + VK 1 j s 
pa“ 
©l,r = 
02, r = 
0 
3,0 
02 .! = 
e 3 , r = 
04. r = 
© 5,0 - 
05,, = 
06, r = 
06,1 = 
/ 2 /2 e b r 
■v K K 
ca" 
/ 2 / 4 / 3l , 
— V K K V - - 
da 
1 / 2 /*l 3 OOp 2 TV 
2 ,V K K — - VK IV 
ca 
/ 2 r*i 3 d&i / 2 /*/, 
V K K 77- 1 —VKK 13 
da 
/ 2 /->/ 3 OO 
V K K -- 
36, 
ca 
ttVAi,; 
0 a 
— w 2 N s ; 
AYV6 r ; 
/ 2 /' 2 / 3 ,,7 
v k k V 0 r ; 
/ 2 /*/ 3 7 / 2 /«/, 
VK K V 1 — VK K 3 
(r *0) 
(r s* 0 , l) 
(r * 0) 
(7-^1) 
(5) 
* ‘ Mec. Celeste,’ vol. ii., p. 184. 
