530 
Proceedings of the Royal Society 
and the first of the second set of three equations becomes 
or 
Thus 
and 
u*(X" + Y" + 2uX' 2 + 2uY' 2 ) = u\X! 2 + Y' 2 - Z' 2 ) , 
X" + Y" = - u(X’ 2 + Y’ 2 + 71 2 ) . 
X" = Y" = Z" = C , 
1 = - 1 [(* - af + (y - by + (z - c)s] + C, 
or, as we may take the origin where we please, 
u oc 
1 
CTp 2 + D ’ 
This is, therefore, the only value of u which satisfies the conditions 
of the problem, and the last equation in § 4 above shows that 
either C or D must vanish. If C vanish, u and q are both 
constant. 
(6.) If D vanish, we have by § 3 above 
d q . <r' = Y.dp I? = - y = - dUo 
This gives 
q — aJJp 
where a is any constant versor. 
Also 
d<r — 
c 2 
fp 2 
all pdp(V p)~'a~ 
so that cr is the Electric Image of p rotated through any angle 
about any axis through the centre of the reflecting sphere. (H § 12.) 
(7.) If the equations of any three systems of orthogonal sur- 
faces be 
^\ = C 2 , F 2 = C 2 , F 3 = C 3 , 
we may obviously write for the flux of heat through each the ex- 
pressions 
VF VF 2 = u 2 qiq ~ l , VF 3 = u^qkq~ x ; 
so that we have three equations of the form 
V (ufliq- 1 ) = a Y , 
