32 Proceedings of the Royal Society of Edinburgh. [Sess. 
ie i 
*ji 
where F is an arbitrary function, the most important case being 
'0A dV 
lav (V^a) * 
When (23') is used, the quadratic (11) becomes 
'0A\ 2 dV 2 
(24) 
+ 2(V -- a)dyl . 
v0V/ 2 (V-a) 
We shall now assume that Y x = Y — a = constant is one of a triply 
orthogonal system of surfaces. Making a slight change in notation, we 
denote the corresponding curvilinear coordinates by y 1 = Y v y 2 , y Zi and 
set the square of the element of distance in the space x v x 2 , x 3 , equal to 
Hicfa/5 + H 2 efa/! -\-H 3 dy% so that (24) becomes 
(25) . . . (Hi-^-)^|h^ + H 3 */1+2V 1 ^ > 
where A x = 0A /dy v or say 
( 25 ') .... a 1 dyl + a 2 dy% + a 3 dyl + a^dyl. 
As a further restriction on Y, we now assume that it, and therefore also 
Y 1 , satisfies the Laplace equation which is covariant relatively to (25), 
namely 
/0fi , AW _ la^aw i v ito ( sw\ . 
' ^ ci/i 2a? dy t dt / ( 2«,2 a . dy. di/j) 
or, if W is a function of y 1 only, 
■ 3 2 W j 0 log (a 2 a 3 a 4 /a 1 ) 0W 
5- + 
d Vi 
= 0. 
d Vi 
In particular, if W = Y 1 =y 1 , 
0 f a^a Q a 
dy 
where / is an arbitrary function of y 2 and y 3 , y± being absent since the a’s 
are independent of y tl . Inserting the values of the a’s from (25), we get 
;(“) = 0, or f { y»« s ), 
(27) 
HoHsV, „ 
“ A 2 %)> 
H 1 -2V, 
so that H 2 H 3 is the product of a function of V x alone and a function of y 2 
and y z alone. 
If now Vj_ is a function of r= Jx\ + x\ + xl, we may use spherical polar 
co-ordinates with y 2 = S and y s = </>, so that 1^ = 1/ 
H 3 = r 2 sin 2 0. (27) then becomes 
= /dViX 
dr 7 ’ 
H 0 = r 2 and 
TT JL( d A\ 2 
2Y 1 \0Y 1 / 
y r i V v (y constant) 
