106 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
wtr„ 
' a a F 
00 M 0?v<j 
m! 3 
(0i sin 0 + ... + 10 ,-sin r % 2 sin 0 
n/ 2 0 
a 
a 
<T\ 
+ zm u 
S cos 1 
!.p— a) 
1 7 r/ll 
8 a 2 sin : 
3 1 
(,u-A) 
1 TTj 
hi 
+ 
n 
a~ 
[cr IJL — <T K ), 
1 0F 
■l\ 00A/O 
= (... + ib i sin id>...) 
a a 
m • 
— 77 . sm 0 
a 
+ 
cos(m-A)W» —t sin fju—X) 2~/ 
_4« 2 sin 2 (m—A) ~jn cd 
n 
:2 F 
nv 
7\0?V00/*/ 0 
m' _0_ 
a r2 a da 
J 
(0j sin 0 + ... + ?'0, sin 70...) 
— 4- 2 (0j sin 0-j- ... T ?0, sin i<j >...) — 
f 3 cos ( 
yU— A) 
7r/n 
l8« 3 sin 2 
(p —A 
) "! n 
Pa 
+ S?? 1 M 
0 cos (/U—A) irln 2 - / . Xo / 
- Q , • -w - TV / + —3 sm U - A) 2 7 T n 
L 8 a sm (/*—A) 7 r/n a 
Pfo 
0 2 F 
-(X,, 
\r K dduddJ 0 
n' 7 7 -97 \ m! 
— (0j cos 0+ ... + % b l cos ?0 ...J -I—^ cos 0 
/ ^ / cl 
m 
a!a 
Ta 
+ 
cos (fx — X) ‘hrjn _ 3 sin 2 (yu—A) 271-/71 
_ 8 a 2 sin 2 (yu — A) tt/ii 32a 2 sin 5 (// — A) 7 r/n 
— -^2 COS (/X — a) 27r/n (cr^ — cr A ). 
a _ 
I 11 the summations of the right-hand members, 77 takes all integral values from 
1 to n, except yu = A. 
Next assume that all the small particles forming the ring are equal to one another. 
That is m A = m. 
Further, let / 3 A+1 = /3/> A , for all values of A. 
Then 
PA PA + n /d pA> 
whence 
18 * = 1 ; 
or 
(3 — cos + / sin , where 1 = y/( — 1), 
71 7i 
and s takes all integral values from 0 to n— 1 . 
The quantities appearing under the signs of summation are then : 
m 
n CL 
m 
2 9 COS (ya-A) 277/71 = —^; 
; CL 
- — sin (// — A) 27r/n = 0 ; 
/x CL 
. m 
- =“K; 
4a 2 sin (,u —A) n-jn a 2 
