AN OBSTACLE ON A TRAIN OF ELECTRIC WAVES. 
309 
When the point P is on the surface of the conductor, 
P* —• 0, 9’ Pl5 ^ ^1? ■“ P'15 ^ “ ^l5 
and therefore the principal parts of the components of the electric force at a point on 
the surface are given by 
X - - 2 iKH (z/i cos + 
Y = —(rj cos (^ + w) 
Z = —^LK^n (z^i cos (i> + n) 
hence the principal part of the electric force at a point on the surface is normal to the 
surface, and therefore the principal part of the electric force satisfies the condition 
that the electric force tangential to the surface vanishes. It follows from this that 
the principal parts of the components of the magnetic and electric forces at points not 
near to the boundary of the geometrical shadow are those given above. 
To find the region within which the point P lies when the above values cease to 
represent the principal parts of the components of the magnetic and electric forces 
due to the obstacle, it is necessary to find the order of the terms neglected by taking 
the limits of the integral representing the principal part of the integral 
jg-c.(r. + r)^^g 
to be infinite. In the evaluation of I this integral was replaced by an integral taken 
throughout the area enclosed by the curve which is the projection on the tangent 
plane at the point Q of the curve of contact of the tangent cone from the point O to 
the surface of the obstacle with this surface. The actual limits of the integral 
j j e- T. drj' 
are quantities of the order Kd or quantities of higher order, where d is the least 
distance of the point Q from the boundary of the curve throughout whose area the 
integration is taken ; hence the part of the integral neglected by taking the limits to 
be infinite is at most of the order (/ccZyPi +/ccZYR)^'''" compared with the part retained, 
when the point P is inside the tangent cone from the point 0 to the surface of the 
obstacle and on the side of the surface remote from the point O. Again, when the 
point P is outside the tangent cone from the point O to the surface, the actual limits 
of the corresponding integral are also quantities of the order kcZ or quantities of a 
higher order, and therefore, since A + B and AB —are booh positive, the part of 
the integral neglected by taking the limits to be infinite is at most of the order 
{Kd^fJii + Kd^jR)~^^^ compared with the part retained. The region within which the 
point P lies when the values obtained above for the principal parts of the magnetic 
