MR. W. M. HICKS ON TOROIDAL FUNCTIONS. 
Gil 
nexion with the following pages consists chiefly in the fact that the potential is 
expressed in a series of the form 
G k (v) 
and that the orthogonal co-ordinates employed are closely allied. 
I. 
GENERAL THEORY OF CONJUGATE CURVILINEAR CO-ORDINATES IN THREE 
DIMENSIONS. SPECIAL CASE. 
1. It is well known that if Laplace’s equation be referred to a system of orthogonal 
co-ordinates u, v, w it takes the form 
where 
b_( u M , i/JLM . 6 (VLW \ =0 
bu \VW burbv\UW bv)^bw \UV bw) ' 
Let us now take u, v to be any conjugate functions of p, 2 ; p, z being the cylindrical 
co-ordinates of a point. Also take w to represent a series of planes through the axis 
of z, so that 10 = tan -1 yjx. 
Then u, v, w are orthogonal surfaces and 
So that equation (1) becomes 
b_ 
bu 
bw 2 
In this write rf)=xfjl x /p, then 
0 
b^t 1 1 J b p : \ h £'\ 
bu~ tr 3 ~ r 2 P 2 \bu j 
■xPp/bpp brp\ 
2 p [bu^bv*) 
4 K 2 
b-yjr 
~buf~' 
0 
