370 
A MEMOIR UPON CAUSTICS. 
[145 
XXXI. 
The theorem, “ If a variable circle have its centre upon a circle S, and its radius 
proportional to the tangential distance of the centre from a circle G, the envelope is 
a Cartesian,” 
is at once deducible from the theorem— 
“ If a variable circle have its centre upon a circle S and its radius proportional 
to the distance of the centre from a point C', the locus is a Cartesian,” 
which last theorem was in effect given in discussing the theory of the secondary 
caustic. In fact, the locus of a point P such that its tangential distances from the 
circles C, G' are in a constant ratio, is a circle 8. Conversely, if there be a circle C, 
and the locus of P be a circle S, then the circle C' may be found such that the 
tangential distances of P from the two circles are in a constant ratio, and the circle 
G' may be taken to be a point, i.e. if there be a circle G and the locus of P be 
a circle S, then a point G' may be found such that the tangential distance of P 
from the circle G is in a constant ratio to the distance from the point C'. 
Hence treating P as the centre of the variable circle, it is clear that the variable 
circle is determined in ihe two cases by equivalent constructions, and the envelope is 
therefore the same in both cases. 
XXXII. 
The equation of the secondary caustic developed and reduced is 
(P + iff - 2/j? (a 2 + (jjl 2 + 1) c 2 ) (P + if) + 8 c 2 /jb 2 ax + a 4 - 2a. 2 c 2 (f + 1) + (/a 2 - l) 2 c 4 = 0, 
or, what is the same thing, 
{f (P + if) — (a 2 + (f + 1) c 2 )} 2 + 8 c 2 /jb 2 ax — 4P (c 2 /j, 2 + (f +1) a 2 ') = 0, 
which may also be written 
(*’ + S'* - ($ + ( 4 + c! )] + I* ^ "7? ( c! + (1 + ï ù “ 2 ) = °- 
which is of the form 
(P + y 2 — a) 2 + 16 A (x — m) = 0 ; 
and the values of the coefficients are 
a = - 2 cl + (1 + -) c 2 , 
f fJL~