50 
MR. F. W. DYSOX OY THE POTENTIAL OP AN ANCHOR RINO. 
3. To obtain tbe expansion of j 
li 
of—, tlie correspondino’ expansion of -. „ 
c ^ to 1 Jji ^/[r- + c" — 2cr sm 9 cos 0] 
tiated with respect to c, R and ^ being considered as functions of r 
Now 
Pt cos X — ^ 0 — c ~ V7 
Pt sill X = >' cos 6 — z 
Therefore 
cos X s -1^ «■>'X a, = 1 I 
'Nti 0 ^ 
Oil , I 
Sin X 0,, + fv. COS X 0^, = 0 J 
Solving these equations 
or: 
= cos X 
d<p 
0 \/{r^ + c- — 2cr sin 9 cos (f)] 
(1(f) 
in 
R 
dc 
dc 
sill 
X 
Again 
and 
0 V 
cos X - R sin X ^ 
ail 
dz 
ail ay 
sin X 0^' + R cos X w = 1 
a? 
1*' 
I 
J 
Solving these equations 
ail . ay 
-0, = sin X ; R 0r = cos X . 
From these formulae it follows tliat 
and 
and that 
and 
^ (R" cos j/x) iRR' ^ cos [n 
(I /coswyX _ 
dc \ R" / 
cos (a + 1) y 
h« + i 
d 
(h 
d [cos ?/ •’ 
dz \ IP 
X\ _ 
n sin (a 4 1) y 
H« + ‘ 
)X 
I 
— (IP cos ny) = ~ iiR" ^ sin (n — l) X 
J 
^1 - 
• 0 \/ C 
4- c- — 2cr sin 9 cos (f>) 
ascending powers 
must he differen- 
6, and c. 
Let us write 
