314 
PROF. H. M. MACDONALD ON THE EFFECT PRODUCED BY 
where 
Bj = — K (sin^ iff cos^ (f) + cos^ xp), 
Cl = [2 cos (p (C cos^ i// + 3L sin xp cos^ xp cos (p + 3M sin^ xp cos xp cos^ <p + D sin^ xp cos^ (p) 
+ 1(1 /R^ — 1 /Pti^) sin (p cos^ (p cos i/;] [cos^ xp + sin^ xp cos^ 
From these results it follows that the intensity of the reflected waves at a point 
on a caustic is of a higher order than at an ordinary point, the ratio of the two 
intensities depending on where X is the wave-lengtli of the waves. In the above 
investigation it has been assumed that, when R =J'i, Di and Ai are finite, and that, 
when R = J' 2 , Cj and Bj are finite; if, when R =/], Aj is finite and Dj vanishes, 
terms of the fourth order must be retained and the ratio of the intensity at such a 
point on a caustic to the intensity at an ordinary point depends on X~'^^ Similarly, 
if, when R = f. 2 , Bj is finite and Cj vanishes, the ratio of the intensity at the 
corresponding point on the caustic to the intensity at an ordinary point depends 
on X“A Further, at a point on the intersection of the two sheets of the caustic 
Ai and Bi vanish simultaneously and, if Ci and Dj are both finite, the ratio of the 
intensity at this point to the intensity at an ordinary point depends on X“A The 
other cases which depend on the vanishing of more of the constants Ai, Bj, Cj, &c., 
can be similarly treated. 
At a point in the neighbourhood of the caustic either R—or R— is a small 
quantity; if R—/i is small and R— ^2 is not small, Bj is a small quantity and the 
value of I the principal part of the integral 
is given by 
J — CD J — a 
that is 
or 
where 
W = [ cos ^7T dC, and p = 12’ ^Bi^X”''’Di“A 
Jo 
Therefore at a point in the neighbourhood of a Qaustic the principal parts of the 
components of the magnetic force in the reflected waves are given by 
«o = cos 2n (nn.i-n/ri)]/RiR, 
= -2A3V-W-V”^«ArV3DrV3We‘''<^^-''--"^-Tcos(/>[-X2 + 2n(nXi-Ai)]/RiR, 
y„ - cos (/> [2n (//Xi-mXi)]/RiR, 
