1889-90.] Lord McLaren on Reflexion-Caustics of Curves. 295 
epicycloid which is the caustic of the circle of curvature at the 
given point ; and the caustic of any reflecting curve is evidently the 
envelope of all the epicycloidal caustics of the circles of curvature. 
5. Equivalents in Cartesian Coordinates of the Polar Curves whose 
Caustics have been found. 
For polar curves having integral indices and coefficients n and 
m, the equivalents in rectangular coordinates are easily found. 
Cos mO, when expanded in terms of cos #, is always a homogeneous 
expression in sin # and cos #, provided that m is an integer, and it 
is easily proved by the inductive method that, if the expansion he 
homogeneous for any one value of m, it is also homogeneous for 
m+ 1, and therefore for any integral value of m. The expan- 
sions of cos mO for even numbers from m = 2 to ra=10 are as 
follows : — 
Cos 2# = cos 2 0 - sin 2 # ; 
Cos 4 0 = cos 4 # - 6 cos 2 # • sin 2 # + sin 4 # ; 
Cos 60 = cos 6 # - 15 cos 4 # • sin 2 # +15 cos 2 # • sin 4 # - sin 6 # ; 
Cos 8# = cos 8 # - 28 cos 6 # • sin 2 # + 70 cos 4 # • sin 4 # - 28 cos 2 # • sin 6 # 
+ sin 8 # ; 
Cos 10# = cos 10 #- 45 cos 8 # • sin 2 # + 210 cos 6 # • sin 4 # -210 cos 4 # • 
sin 6 # + 45 cos 2 # • sin 8 # - sin 10 #. 
The law of the expansion is this : — It contains the even terms of 
the binomial theorem with the sign of every alternate term changed. 
The expansion of sin mO contains the odd terms of the binomial 
theorem, with the signs of alternate terms changed. 
From the values of cos mO above given the equivalents of polar 
equations for even values of n and m may be written out as 
follows : — 
For equations of the form r n = a n sec (mO), or a n = r n cos (m#), 
suppose n= 10 ; 
a io_ r io cos (io#). 
— a? 10 — 45# 8 i/ 2 + 210d? 6 y 4 — 210a? 4 ?/ 6 + 45a? 2 g/ 8 - 2/ 10 ; . . . (1) 
a 10 = r 10 • cos (8#) = (x 2 + y 2 ) • r 8 • cos (8#), 
= ( x 2 + y 2 ){x 8 - 28 x 6 y 2 + 7 Oxhfl - 28 x 2 y Q + y 8 • } ; . . . (2) 
