630 
MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
If we try </> = AS, we shall find that both the equations 
f" — (2 n +1) /+ 2 n~ AC=0 
2/'-2wAS=0 
can be satisfied simultaneously if f—nX.C. 
Hence, whatever \ be, the equation 
x/c+i=A^rCi//,,+S-J^ 
holds. 
Again, we may also determine f </>, so that 
In this case the equations for/and </> are 
4 >" — (2 m — 
S du\Sj 
= 0 
> 
= 0 
J 
which are satisfied by (f>-— BS,/= — ??BC. 
Hence, v// / ,_ 1 = B(— 
The toroidal functions themselves are i/z/^/S, and the two particular integrals are 
represented by P m n , Q„ i w . For these functions the above equations become 
P»Z.« +1 = A„ { 2S P 'M.V + (2 » +1) CP OT .„} 
P w =B»{ 2SPh,, - (2n -1 )CP„, ? ,} 
and similar equations for the Q. 
Since the solutions P„ Ui of the differential equation are multiplied by an arbitrary 
constant, we may, when we confine ourselves to one of the above equations, put A or 
B=l, and after solving the equation of mixed differences multiply the result by an 
arbitrary constant. But if we wish to combine both formulae so as to eliminate the 
differential co-efficient in them, then the P in both must be the same, and a relation 
will hold between A and B. This we proceed to find. Dropping the ( m) as unneces¬ 
sary, write 
P„=A*., {2SP'„_ 1 +(2» -1 )CP,_ 1 } 
and substitute therein 
P,- 1 =B42SP'.-(2n-l)CP,} 
Whence 
