1889 - 90 .] Lord McLaren on Reflexion-Caustics of Curves. 291 
values of lines and angles from which points on the caustic may be 
computed. 
These are, in the first instance, obtained in terms of the coordi- 
nates of a fundamental curve, r n = a n cos ( nQ ), whose caustic has 
already been found. The fundamental curve coordinates r, 6, <o, p, 
&c., are distinguished by the suffix (1). The coordinates of the 
given curve, or primary, and those of its pedal, negative-pedal, and 
caustic, are distinguished by the suffix (2). These are — 
i p2i w 2 i V 2 x l / 2 ’> ^2 
(1) I consider those to be corresponding points of the fundamental 
curve, and the given primary curve, <£ 2 ( r 2 A)> which 
r x = r 2 ; accordingly the suffixes for r are omitted in the proof. 
Hence for such corresponding points, 
2 . ___ 1 <y\j 
cos” ttfl^cos” m0 2 ; 0 2 = — '@i> • • • (1) 
(2) For the perpendicular on tangent (or radius- vector of the 
pedal), we have p 2 = r cos v 2 ; 
o d$ 2 , ,\m/ n „ d0 x 
. \ p 9 ~ r z • 7 " = r * d — 7 - = — r 2 • ; 
A as as m as 
n 
*P Q — P\ ) 
/2 m 1 
• • (2) 
also, 
p 9 np , n 
cos i/ 9 = —= 1 = — • COS V , . . . . 
z r mr m 
• • (3) 
also, 
n 
w 2 = v 2 + 62 = v 2 + — d 1 . . . . 
. . (3a) 
(3) For the radius- vector of the negative-pedal — observing that, 
by the general law given above (5), between p and r = ^i between 
r and v = v — we have by similar right-angled triangles, 
C2= k = ^- secm02= ^ } ' seOB0i; (Byl) - 
mv, 
v 9 = — 4 j 
2 n 
w 
For the angular coefficient, 
'l' 2 = V2±6 2 = V2±—01 ( 4 «) 
