312 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
the point Q (fo, rjo, ^o) determined as before, the points Pj, P2 on the straight line QP 
for which R— f\ or R—^2 vanish are both external to the surface of the obstacle if 
A + B cos^ (f) + cos <^/Ri 
is negative, for then and y’g are both positive. The principal parts of the magnetic 
force in the reflected waves given by the above expressions tend towards infinite 
values as either of the points Pj, P2 is approached, and the points P^, P2 now 
determine the positions of actual focal lines whose directions are given by 
tan 9 = tan xj/ sec It = J\, 
tan 9 = tan ifj cos </>, 11 = / 2- 
The principal parts of the components of the magnetic force were evaluated above 
on the assumption that /’I'b 11“^ —were both finite ; if either of these 
quantities is very small, some of the terms neglected in that evaluation are of the 
same order as those retained, and it is necessary to calculate the principal part of the 
magnetic force taking account of these terms. In the equation of the surface in the 
neighbourhood of the point Q (^0, Vo, L) the terms of the third order must be retained 
and the ec^uation to the surface with Q as origin of co-ordinates and the same axes of 
reference as formerly is now 
^ = i(Ae+2Hir) + Br)^) + i(Ci^+3Lev + SMirj^ + I)rj^}. 
The value of ri -f- r is now given by 
r, + r = B,-f-E,+i(A'e+2H'^77 + BV)-fi(C'f+ 3 L'fp-i- 3 M'^ry^+DV) 
where 
A'= cos^ 2A cos (^, B'= (Ri“^ + B“^) + 2B cos (^, H'=2Hcos(^, 
C' = 2C cos (f) + 3 A sin (f) cos (f) (R“^—Ri'‘^) + 3 sin (f) cos^ (f) (l/R^— l/R/), 
L' = 2L cos <^-t- 2H sin ^ cos ^ (R“^ —Ri“^), 
M' = 2M cos (f) + B sin ^ cos (IR^ — RR^) + sin (j) [I /IP — 1 /Ri^), 
D' = 2D cos (f). 
At the point Pj, for which R = fi, these relations become 
A' = K cos^ (f) cos^ xjj, B' = K sin^ xfj, H' = K sin ifj cos xfj cos (f), &c., 
where 
/r-fr - K, 
and, turning the axes of £ and y through the angle 9 given by tan 9 = tan ip sec (f>, 
7 \ + r is given by 
r, + r = lb-tR 4 -i(A,f+B,p^) + i(C,P-i- 3 RP>? + 3M,er?^+Di7^=^), 
