MR. W. M. HICKS OX TOROIDAL FUNCTIONS. 
647 
this value of a the series SA„ P„(^) is convergent, from which the convergence - of </> 
will flow at once. Now 
_Q 'n _ Qh-i - CQ„ 
X n p- rip _ J) 
n n L n —i 
Qh-i—CQ „ 
^ C< o TA 
S 2 ^ 1 lh- a 
CH» Q«_ 
CO co _J_ ^ 
Both the series on the right are finite, hence so also are —% ] x r and — jx r , and 
A n tends to a hnite limit with increasing n. It is therefore possible to give « a value 
which shall make this limit zero. It is given by 
whence 
o , 
co 7*" -f ci 
2S 1 i^rr r '-“*«= 0 
O . o . 
. n l + a n , 00 v •“ A « 
A *= 2 V-i x *- 2 S -sizr x ' 
(43) 
Lastly it remains to consider the conve-rgency of the series 2A„P„(n). When n is 
which is 
very large — A i!+1 tends to the limit — 
>1 + 1 
(n + l) 2 + « /Q»_ 1 _p Q« \ 
< (2»+l)S»^ 1 P»-J 
Also since u<u 0 P,,(w) < P„. Hence the series under consideration is 
1 ^ v~ J Q,i_ 2 P M p Q„_ 1 P» ] 
<^( 2 ^- 1 ) llbl iaT J 
The sum of the first set of terms is < 
(C “ Q and of the second set is 
S" An — 1 
<— g 2 ; both of these are finite. Hence the sum 2A„P„ is finite anrl d 
fortiori the sum 2A.,P„(w) sin nr. 
Finally then the velocity potential for fluid motion due to a tore moving parallel to 
itself through a fluid at rest at infinity is 
ff> =—v/ 2 V f C ■ 
IT 
■ c% ( A ,,P„sin nv 
where A.„ is given by (43) and 
