276 
IklE. A. CAYLEY’S I^IEMOIE UPON CAUSTICS. 
then the equation 
where x, y are to be considered as current coordinates, and which inYolves of com'se the 
arbitrary parameter, is the equation of the circle, and the envelope is obtained in the 
usual manner. This is the well-known theory of Gekgoxxe and Quetelet. 
III. 
There is however a simpler construction of the secondary caustic in the case of the 
reflexion of rays proceeding from a point. Suppose, as before, that Q is the radiant point, 
and let G be the point of incidence. On the tangent at G to the reflecting curve, let 
fall a perpendicular from Q, and produce it to an equal distance on the other side of the 
tangent ; then if y be the extremity of the line so produced, it is clear that ^ is a point 
on the reflected ray Gq, and it is easy to see that the locus of q is the secondary 
caustic. Produce now QG to a point Q' such that GQ'=QG, it is clear that the locus 
of Q' will be a curve similar to and similarly situated with and twice the magnitude of 
the reflecting curve, and that the two curves have the point Q for a centre of similitude. 
And the tangent at Q' passes through the point q, i. e. q is the foot of the perpendicular 
let fall from Q upon the tangent at Q' ; we have therefore the theorem due to 
Dandelix, viz. 
If rays proceeding from a point Q are reflected at a curve, then the secondary caustic is 
the locus of the feet of the perpendiculars let fall from the point Q upon the tangents 
of a cuiwe similar to and similarly situated with and twice the magnitude of the reflecting 
curve, and such that the two curves have the point Q for a centre of similitude. 
IV. 
If rays proceeding from a point Q are reflected at a line, the reflected rays will pro- 
ceed from a point q situate on the perpendicular let fall from Q, and at an equal 
distance on the other side of the reflecting line. The point q may be spoken of as the 
image of Q ; it is clear that if Q be considered as a variable point, then the locus of the 
image q will be a curve equal and similar but oppositely situated to the cuiwe, the locus 
of Q, and which may be spoken of as the image of such curve. Hence it at once follows, 
that if the incidental rays are tangent, or normal, or indeed in any other manner related 
to a curve, then the reflected rays will be tangent or normal, or related in a corresponding 
mqnner to a curve the image of the flrst-mentioned curve. The theory of the combined 
reflexions and refractions of a pencil of rays transmitted through a plate or prism, is, by 
the property in question, rendered very simple. Suppose, for instance, that a pencil of 
rays is refracted at the first surface of a plate or prism, and after undergoing any number 
of internal reflexions, finally emerges after a second refraction at the first or second 
surface ; in order to find the caustic enveloped by the rays after the fii’st refraction, it is 
only necessary to form the successive images of this caustic corresponding to the difierent 
reflexions, and finally to determine the caustic for refraction in the case where the 
