AN OBSTACLE ON A TRAIN OF ELECTRIC WAVES. 
315 
where W has the value given above. When Ji— f 2 is small and 11—/’i is not small, 
the principal parts of the components of the magnetic force in the reflected waves are 
given by the corresponding expressions in which Ai is replaced by Bi, Bj is replaced 
by Aj, and Dj is replaced by Ci. At a point in the neighbourhood of the intersection 
of the two sheets of the caustic, Ai and Bi are both small, and the expressions for the 
principal parts of the components of the magnetic force in the reflected waves will 
contain the product of two integrals which are values of W corresponding to two 
values of p. 
When the obstacle is of any form, if P is a point not inside the boundary of the 
geometrical shadow, there may be more than one point Q on the surface of the 
obstacle such that OQ + QP is stationary, or there may be points Q, Qi, Qg, ... Q„ 
such that OQ + QQ 1 + Q 1 Q 2 +...Q„P is stationary; in the first of these cases the 
principal parts of the components of the magnetic force in the reflected waves is the 
same for all the different points Q ; in the second case the principal parts of the 
magnetic force in the reflected waves at P is obtained by placing an electric current 
distribution on the surface in the neighbourhood of Q double that due to the 
tangential electric force from O, an electric current distribution in the neighbourhood 
of Qi double that due to the tangential electric force arising from the current 
distribution in the neighbourhood of Q and so on. 
The magnetic and electric forces due to the assumed distribution over the surface 
of the obstacle have still to be obtained at points which are near to the boundary of 
the geometrical shadow. Taking first the case where the point P is inside the 
tangent cone from the point O to the surface of the obstacle, and on the side of the 
obstacle remote from the point 0, the principal parts of a, /3, y, where 
a. = 
I f 
27rYJJ 
ill 
(■( 
y = 
27rVjJ 
' y-y] + n {zy]-yC) 
02 
rrj J dr rr^ 
02 giK(V«-r,-r) 
T rrx J drdi\ ri\ 
K [^n{xy} — y^) 0^ giK(Vf-ri-r) 
dS, 
dS, 
rri 0r dr^ ri\ 
dS, 
have to be calculated when the limits can no longer be taken infinite for the purpose. 
As before, the principal parts of a, /S, y are given by 
27rV B, ’ 27rV R ’ 
where I is the principal part of the integral 
If 
R RjR 
2 S 2 
(7S. 
