304 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
Now, ri + /' is stationary at the point Q for which 
therefore 
whence 
^1 1 ^ _ A 
R “ 
^2 = 0; 
that is 
Ri -R 
In this result the upper sign corresponds to the case where the point P is external 
to the tangent cone from the point O to the surface, or inside the tangent cone and 
on the same side of the surface as the point 0 ; the lower sign corresponds to the 
case where the point P is inside the tangent cone and on the side of the surface 
remote from the point O, and in this case the point Q lies on the straight line OP. 
Considering first the case where OQP is a straight line, writing 
= — R] sin (f), Zi = —Ri cos </>, 
it follows from the relations 
that 
hence 
and therefore 
Zi/Ri + ^s/R = 0, a^i/Ri + Xg/R = 0, 
x’a = R sin (f), ^2 z= 0, ^3 = R cos (f) ; 
= Ri^+2fRi sin ^ + 2^Ri cos (f> +^^ + r)^+ 
= R^—2^R sin (^ —2^Ri cos (^ + ^^ + 17^+^^, 
7’i = Ri + I sin (f) + i, cos (f) + ^ (f^ + 77^)/Ri—sin^ (^/Ri+ ..., 
r = R —I'sin (f)—C cos <^ + |- </>/R+ 
whence 
n + r = Ri + R + I- cos^ 4 > + V^) (Rr^ + R“^)+ •••, 
the remaining terms involving higher powers of f and 77. 
Now 
= Ife-'--' ..^11 + (IIJ+ dJ} didr,. 
where the integral is taken throughout the area bounded by the projection on the 
tangent plane at Q to the conducting surface of the curve of contact of the tangent 
cone from the point O to the conducting surface. Hence the value of the principal 
part of 
