640 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
Now it is easily shown that 
F, +1 +F_ ] -(SP.+2CP / ,) = 0 
with a similar formula for Q,„ we may hence write the above equation 
(A. ra+ 2 A«)P (A« A^_^)P /(—j—— Q »+i Q«_! 
with initial equation 
(A 3 -A 1 )F a -A 1 F 0 =Q / s -Q' 0 
To get a first integral of this write the successive equations in order, multiply those 
containing Pd +1 , Pd_i by Pd and add, we get 
(A /<+1 —A /( )Pd +1 Pd—A 1 P / 1 Pd=PdQd +1 —P / 1 Qd+-r 1 (PdQd +1 —Pd +1 Qd) 
= PdQd +i —P'oQ 1+9 -o _1 (- r + 1 ) 
Put 
then since 
Hence 
and 
A _ A — AA+ii 
CL/( +l ■ CX/t TV 
Qb +1 , (AjP'j —Q'^P'o . 77 n~ 
P' 
^+1 
PbP' 
2 P' Y 
— U L il 
+1 
(A 1 P' 1 -Q' 1 )P' 0 =L„ £=*. 
2 = fijd+l ' 11+1 F"+1 Qd) 
A /i+ i A„—C(+i+ 
— 2;t+ i{( n + liJ T a ) X *+l~ (tt S + a)E A } 
A __ A _ (^ + !) 8 + « _1_9N-'—- + -A y —-tfiy 
n )l+ i -a-!— 2n+l ' l +l^—' 2 4 r 2 _i‘L- 3 -A 
. pi + l) 2 4- a _ 4“ a 
A ' t+1 = 2 vt + 1 Xn+l + 2il 
A„ is undetermined to the extent of a; but since the velocity potential must be 
finite everywhere, a must be chosen so that the series £A, t P (l (to) shall be convergent. 
It will first be necessary to prove that A n is finite when n is large ; a must then be 
chosen so that A n vanishes for n infinite, and lastly, it will remain to show that with 
