118 Proceedings of Royal Society of Edinburgh. [sess. 
Let P be any external point, and take T on PS (produced) so 
at T for unit surface-density ; so that the sum of the values of 
the first side of the equation is inversely as SP ; and the pro- 
position is proved. 
Although no mention has been made of Electric Images, in the 
above investigation, it is obvious that nearly all their chief 
elementary properties have been proved, almost intuitively, in 
the course of these three or four lines. The others are obtained 
at once by applying the same method to the case in which P is 
inside the spherical shell, and T outside : — remembering that the 
potential at T is now inversely as the distance of T from the 
centre, 0, of the sphere ; and referring the potential of E to a 
point S 1 on OS produced till OS- OS 1 is the square of the radius 
of the shell. 
that 
PS- ST = KS-SE = b 2 . 
Then we have obviously, from similar triangles, 
SK-EP = SP-KT. 
E 
Thus 
