PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
320 
p ^'-; therefore, when E, is of the same or higher order than (/c>^i)'^3 and is of higher 
order than p [Kp)~"^% K is of lower order than unity. When R is of higher order than 
and is of the same or higher order than p [kp)~‘''\ the quantity K is at most 
of tlie order 
00 
which is at most of the order and therefore, when R is of higher order 
than {piiy'-, and is of the same or higher order than p [Kp)~^^\ K is of lower order 
than unity. Hence, at a point P inside the tangent cone from 0 to the surface of 
the obstacle, the effect of the electric current distribution which balances the 
unbalanced tangential electric force due to the oscillator and the assumed surface 
distribution, and is on the part of the surface inside the tangent cone and remote 
from O, is of lower order than that due to the oscillator, provided that the 
perpendicular distance of the point of contact R of the tangent from O to the 
surface in the plane of incidence is of the same or higher order than p and 
that the distance of the point P from the surface of the obstacle measured along OP 
is of higher order than p {kp)~^^\ or provided that is of higher order than p {Kp)~‘^\ 
and the distance of P from the surface is of the same order as p[Kp)~'^\ In a similar 
way it may be shown that the effect of the electric current distribution on the part of 
the surface of the obstacle next the oscillator which balances the unbalanced tangential 
electric force there, due to the oscillator and the assumed surface distribution, is of 
lower order than that due to the oscillator at a point P satisfying similar conditions 
to those in the previous case. Therefore, at a point P inside the tangent cone from 
0 which satisfies the above conditions the principal parts of the components of the 
magnetic and electric forces are equal to the principal parts of the components of the 
magnetic and electric forces due to the oscillator and the assumed electric current 
<r> 
surface distribution. The same result holds for points P outside the tangent cone 
similarly restricted; the region omitted in the neighbourhood of the tangent cone is 
that lying between the cone through 0 which cuts the surface of the obstacle in a 
curve inside the tangent cone on the side of the surface next O and at a distance 
from the curve of contact of the tangent cone of the order p and a surface 
touching the obstacle along this surface and generated by straight lines in the plane 
of incidence at each point of the curve. 
The principal parts of the magnetic and electric forces at points on the surface of 
the obstacle which are at a distance from the curve of contact of the tangent cone of 
order greater than p{Kp)~'^^ can now be found. Let P be a point on the surface 
inside the tangent cone and on the side of the surface remote from O, and let OP cut 
the surface between O and P in the point Q, then PQ is of higher order than 
p [kp)~^^^ if P is at a distance along the surface from the curve of contact of the 
tangent cone which is of higher order than p [Kp)~''\ and therefore the effects at P of 
the surface distribution on the part of the surface inside the tangent cone and nearest 
