ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
803 
consists always of two ovals ; the shape uf the exterior oval is best perceived from the 
figui’es. An examination of the figures will also show how the same curves may originate 
from a different refracting circle and radiant point. 
XXXI. 
The theorem, “ If a variable circle have its centre upon a circle S, and its radius pro- 
portional to the tangential distance of the centre from a circle C, 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, C' are 
in a constant ratio, is a circle S. Conversely, if there be a circle C, and the locus of P 
be a circle S, then the cmcle C' may be found such that the tangential distances of P 
from the two circles are in a constant ratio, and the circle C' may be taken to be a 
point, i. e. if there be a circle C and the locus of P be a circle S, then a point C' may 
be found such that the tangential distance of P from the circle C 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 the 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 
~ -f + 1 (.3?^ + -f- Sc^iJt/as 
+a^-2«V(/A^+I)+(jtA^-I)V=0, 
or, what is the same thing, 
which may also be written 
which is of the form 
and the values of the coefficients are 
, c^a 
m 
