MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
629 
To complete the general expression for i ft we must therefore add a term 
. . Ucosli u + cos v —sinh u cos6 
A sm v I 
J i 
d<p 
sin 2 v + sinli 2 u . sin 2 </> -y/cosh u — sinh u cos cf> 
■ ■ (25) 
We shall denote this by the letter AO, so that the solid angle varies as H\/C — c. 
III. 
SECTORIAL AND TESSERAL FUNCTIONS. 
12. The differential equation which has to be considered in the general case is 
rfh/r 9) 4m 3 —1 
(0 
which in the case of sectorial functions becomes 
dHr 4m 2 -1 
civ? 4 sinh he 
xp=0 
In the rest of this paper we shall call n the order of the function and m the rank. 
Calling the solution of (7) xp m , nJ we proceed to show how r fj nul can be expressed in 
terms of xp mi q, xp M ^. 
Dropping the m for the time, assume 
^+i=^ -/( w )+^7^( w ) 
Then writing (4m 2 —l)/4 = \, and substituting in the equation which xp, l+1 satisfies, 
making use of the equation for \Ij„ to express and , in terms of xk, and we 
du~ civ* cm 
shall get 
A 
— Vc | — {% n + 1 )J +/" + 2 n 2 (f>'—— j + 'I'*' {2/'+ cf>" — 2 n + 1 \4> 
Now choose f, j>, so that 
r-(2»+l)/+2»V'-fK| 
(p' — (2n+ !)</>+2/ 1 —0 
= 0 
