G20 
MR. W. M. HICKS OK TOROIDAL FUNCTION'S. 
Hence 
From this 
Cl, {pn —3 CL<- 2 .r-i + c «_3* 3.>— id~ • • • — 1 .*'—l) 
tt «.i——(C„_ 2 + C «_3“t” • • • + c l) 
tt„. 3 = {c w _ 2 (c ; ,_ 1 .+ . . . +c 1 ) + c„_3(c„_5+ . . -+Ci)+ • • • HCgCj 
= Sum of products two and two together, with the exception of all products where 
the subscript numbers are successive. 
Assume that ( — sum of products of the c m up to c#_ 2 , r together with the 
exception of any in which successive subscripts occur. Then 
a n.r+i= (—) r+1 {c„_ 2 (prod. up to c*_ 4 , r together, &c. . . .) 
- i - C„_g( 55 5 5 j • • • ) 
+ *.. } 
= ( —) ?+1 {Prod. (r+1) together up to c„_ 3 without successive subscripts.} 
Whence by induction the assumption is seen to be universally true. It may be 
thus stated, a n%r is the sum r together of the terms 
5 2 T~ 
(2«-3) 2 
2.4’ 4.G’ 6.8’ ' ‘ *(2w-2)(2w-4)’ 
all products being thrown out in which, regarding the numbers in the denominators as 
undecomposable, a square occurs in the denominator. 
We have 
a 
'//■O' 
1 
(4?i- 3)(n- 2) 
Cl„. i = — 
4(?i — 1) 
This result is of very little use for application. If the co-efficients (a„) are needed 
for particular values of x they can be very rapidly calculated by means of equation (15), 
while if their general values are to be tabulated, equation (16) will serve to calculate 
them in succession. 
Further, 2(3„ is the same kind of function as a lly in every way except that it does not 
contain c x ; in fact 2fi u is the same function of c 3 , c 3 , . . c„_ 2 , that a n _ x is of c v c. 2 , . . . 
c „_ 3 ; calling this a , „_ 1 we can then write 
