AN OBSTACLE ON A TRAIN OF ELECTRIC WAVES. 
311 
which vanishes when A, B, H, &c., vanish, and hence is at most of the order («:p) \ 
where p is the least radius of curvature of the surface at the point Q, it Ijeing assumed 
that the values of ^ 
3 ’ 
, and the similar 
quantities in the immediate 
neighbourhood of the point Q are not of an order higher than Therefore the 
difterence between the components of the actual magnetic and electric forces due to 
the oljstacle at the point P, not near to the surface of the tangent cone Irom the 
point O to the surface of the obstacle, and the components of the magnetic and 
electric forces due to the assumed electric current distribution on the surface of the 
obstacle is at most of the order {kp)~^, where p is the least radius of curvature of the 
surface at the point Q, which is related to the point P in the manner already 
defined. * 
It has been shown above that the principal parts of tlie components of the 
magnetic force due to the assumed electric current distriljution on the surface of tlie 
obstacle are 
’■) {/x2 + 2rt (nmi—U|ai)}/lllli, 
I j-2u (tAj — /r’i)}/RRi, 
iK 
V 
ylK 
{2n {Ip^—mX^)}/liJl-i ; 
in these expressions the quantities /I, jfg are both negative, since A + B and AB— 
are both positive, hence when the obstacle is a convex solid the principal parts of the 
components of the magnetic force in the reflected waves are given at all points P not 
near to the surface of the tangent cone from the point O to the surface of the 
obstacle by the above expressions which are everywhere finite. The points for which 
R —/], R —^2 vanish are situated on the line PQ produced and determine the 
positions of virtual focal lines ; the directions of these focal lines are given by 
and by 
tan 0 — tan xfj sec (p, 
tan 0 = tan xfj cos 
when R = fi, 
when R = f-i- 
If the obstacle is not a convex solid, provided that there is no point P on its 
surface such that a prolate spheroid whose foci are the points O and P can be drawn 
to touch the surface of the obstacle, the same analysis applies, but, P being now a 
point not on the surface of the obstacle, if the surface is concave towards 0 and P at 
* The value of the principal parts of these differences can be obtained bj^ placing on the surface of the 
obstacle an electric current distribution of order {kp)~'^ which will balance the unbalanced, tangential 
electric force of this order. 
