MR. F. W. DYSOiST OX THP] POTENTIAL OF AN ANCHOR RING. 
51 
Ij h;is been expanded as a function of II, and c. 
The expansion of divide may therefore be found by difierentiating this by the rule 
d d , c sill y 3 
- = ^i-COSXgJT--j^3-- 
It is rather simpler to use the formulas for djdc (R" cos «y) proved above. 
Take, for example, the term 
1 9/ + 3 3 
c 64 
S'^ cos y. 
This 
9/ 4 3 IG 
61 
4 R cos y. 
d /9l + 3 IG 
dc \ 64 c 
I R cosy = 
(9/ + 3) K2 (I 
" f C' 
x) + R 00,s X p 
d [(9/4-3)1R 
dc 
- _p 
64c‘ 
64c* 31 
IH cos y 3 /dl + 3 
64 3c V c 
In performing the partial dilferentiations with respect to R and c, it must be 
remembered that I = log 8c/R — 2, and is a function of R and c. 
Let 
Ii = 
1+2 , / 4- 1 Lcos y , f(2^ + I) IR (31 + 2) R^ cos 2y 
^ + 4 +1 16^5 + 
, , (9/+ 3)R-5 cos y , (15/ + 7) IH cos 3y'| , 
c/li_ cos y I + cos 2y I [ (8/ — 3) R cos y oR cos 3y 
dc 
cR 
4c- 
+ 
32c^ 
323* 
1(18/- 3) R2 (18/-8) R2 cos 2y BIG cos 4y; 
' 1 I28c* ' r d" • • • 
I28c* 
128c* 
, /_+ 1 , (3/ + 2) R cos y , f(9/ 4- 3) Rd , (15/ 4- 7) Id cos 2y-, , 
+ 2cd- + -+ -R— + dTN ^ + 
64c* 
64c* 
/ + 1 (2/ + 1) R cos y [(6/ 4- 1) R" (9/ + 3) Pd cos 2y', 
'^c2 23* 1 16c* ■*" 16c*ri--- 
cosy 
cR 
2/4-3 I cos 2y 
^ 4c2 + 4c' 
(20/ + 11) R cos y 3R cos 3y 
323* 
323*, 
1(12/ 4- 5) Pd (12/ 4- 9) Id cos 2y 5Pd cos 4y 
t 128c* 64c* 
*d 
T28c* 
— &c. 
H 
