MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
57 
§ 5. The integral 
sin- \[rd\lr 
lo GOS^p■ + «-} 
may be reduced to elliptic functions. 
Let it be called I. Then 
= -f 
sill -v/r d (cos \lr) 
- 0 \/ cos + a~} 
— [ cos — 2ali cos ifj + crj difj 
fill j 0 
I (R- + cd) cos -ifr — 2cR (1 — sin- -v/r) , 
«R Jo — 2f(Picos-v/r + a^] 
Therefore 
3P 
= 2f 
dy^r 
^{R^ — 2aR cos + «-} 
cos yfrd-^ 
^{R- — 2ffR cos ij!r + «-} 
Writing a for a/R, 
2 
3R Jo 
dyjr 
y/{1 — 2c( cos yjf + «-} 
^ / I cos yfr dyfr 
.3R W «/ J 0 
These integrals may be transformed by putting 
This gives 
sin — xjj) = a sin (f). 
cos (j) ] 
dip = d(f) 11 
= d(f) \ l 
y/ cos {(p — l/f) i 
« cos (p 
\/( I — «- sin- (p) j 
Also 
Therefore 
or 
Therefore 
or 
sin (p (cos xp — a) = cos (p sin xp. 
sin^ (p (cos xp — a)^ = cos- (p sin^ xp, 
cos^ xp — 2 a cos xp = cot® (p — cot® (p cos® xji. 
cos® xp cosec® (p — 2a cos xp rx^ = cot® (p, 
(cos xp — a sin® ^)® = cos® (p — a® sin® (j) + a® sin^ (p 
= cos® cp — a® sin® (p cos® (p. 
* This method of reduction was given by one of the Referees. 
MDCCCXCIII.—A. I 
