306 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
lienee, to the order of approximation adopted, the components of the total magnetic 
force at the point P are given by 
LK 
a 
y LK^ j! e 
cK(Vi-Ki-R) 
V Ki + II Ki + II V El + II P, + Pv, 
0 , 
r 
LK^ X K] R) ^^2 Hi K) 
V V p7+p 'Pi+P^ 
- 0 , 
= 0 
that is, the principal part of the magnetic force vanishes at all points inside the 
tangent cone from the point 0 to the surface which are on the side of the surface 
remote from 0 and not near to the boundary of the tangent cone. The principal part 
of the electric force also vanishes for these points, and therefore the surface condition, 
that the electric force tangential to the surface vanishes, is satisfied for points on the 
portion of the conducting surface remote from O inside the tangent cone and not near 
to the curve of contact of the tano-ent cone with the surface. 
o 
When the point P is external to the tangent cone or inside the tangent cone on the 
same side of the conducting surface as the point 0, the relations to be satisfied are 
xJR, + x,/R - 0, z,IR,-z,IR = 0, 
it follows that 
and 
and, as above, writiii 
a'l = —Pi sin (f), Zi == —Pi cos cf), 
X2 — P sin (f), y2 = 0, 22 = — P cos (j), 
= Pi + ^ sin (f> + C cos + cos“ (f) -!-p^)/Pi + .. •, 
whence 
that is 
r = P—^sin <j) + C cos + cos“ <^ + p^)/Pv;..., 
i\ + r — Pi-f P + |-(^^ cos^ + (Pi“^ + P~^) + 2^cos 
^’i + r — Pj + p +{(P^ ^ + Pt cos^ (f) + 2 A cos (f)} + 2 H cos 
+ ^ 7 ]“^ {Pi~^ + P~^ + 2B cos <j)} + — 
Writino' 
^ 11 cos d—171 sin 6 , 77 = sin d + 771 cos 6 , 
and choosing 6 so tliat the coefficient of ^1, vanishes, 6 is given liy 
cot 2d = {2 (A—P) cos (^ —(Pj-^ + P”^) siiP (^}/ 4 H cos 
and 
= Pi + P + ^ (Ai^i^ + Bi')7i^)+..., 
where 
Ai + Bj = (Pi ^-+-11 ^) (1 + cos^ (^) + 2 (A + B) cos 
A^Bi (Pi“^ + P~^)^ cos^ ^ + 2 (Pa^-^ P“^) (A+ B cos^ </>) cos ^+4 (AB— H^) cos^ 
