104 
DR. G. R. GOLDSBROUGH ON THE INFLUENCE OF 
d 2 d 2 a /3F\ , „ / 3 2 F 
_ Pa _ 2 «„ - = (g jV J o (^T /o 
/ ^2T1 \ 
3 F 
M 00>- 
/ x K/1 0 
3 2 F 
n d 2 rr A 9 clp± /_1 0F\ „ 
at at \r K dOj o ** r^dr^d 0a -o 
+ 
M 
TxdO^ddJ o J 
L 
• (2) 
In order to determine the derivatives, we write in the formula for A a , 0 A = 0 '- 0 A, 
and a = T\/r'. Then 
A, v 1 = {l + a 2 —2a cos 0x} 'F/ 
= {^b 0 + 3 X cos 0 + ... + b l cos i<p + ...} -f-r' 
by Fourier’s series. 
This series will be taken as absolutely and uniformly convergent. 
We find then 
oF M + to a to x 36 0 06 t - • 1 to' 
— =-v- + 7 . 7 ' i + ...+ „ 1 cos ub + ... f- - cos 0 
0r A ?V r 0a 0a r j r 2 
- y. 
[r x -r M cos (0 M —0 A ) . cos(0 M —0 X ) 
1 
TO 3 
J-^A/x 
*. 2 
rF 
"/V o „ ^ , 
,x cr K or 
0 M + to x , to' 0 2 / 17 , 7 • \ 
• I—u (2^0 + + ^i cos «0 ...) 
W r' 3 0a 2 
-TO,, 
1 3 (n-?v cos(0^-Oa)) 2 1 
M id 3 
M L-L/ X/i 
I) 5 
-L'A/x 
PA 
+ ~«i u 
cos (0 M — 0a) + 3 {n-?y cos( 0 M - 0 x )} {r >t -rACOs( 0 ^-0 A )} 
D a 
T) 5 
J-'Ajx 
+ 
2 cos (O m -0a) 
2 0 l 'F 
/ M 00,0Ca 
'TO 0 /7 • .7 * \ TO « 
^ — (67 sm 0 + ... + ?6 4 - sin ?0 +...) — ^ sin 0 
o-a 
V 
TO,, 
vy sin (0 m -0a) _ 3 {r A —^ cos (0 m -0a)} r x 7y sin (0 M -0 X ) 
sin ( 0 M — 0 X ) 
T) 3 
J-^/xA 
T) 5 
J-^A/x 
r 2 
(<r M —cr A ), 
1 3F to' / - 7 \ to' • 
^3ft'7V l ( " ,+!S ' Sln¥+ '"^? 5Sm ^' 
-TO,, 
r - sin - F sin (0,-0 t ) 
I) 3 
