ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
279 
VIII. 
If a ray be reflected at a circle ; we may take a, h as the coordinates of the centre of 
the circle, and supposing as before that n are the coordinates of a point Q in the 
incident ray, «, ^ the coordinates of the point G of incidence, and y the coordinates of 
a point q in the reflected ray, the equation of the reflected ray, treating y as cun-ent 
coordinates, is 
{{h—^){x—a) — {a—a){y—^)}{{a—cc){l-cc)-\-{b—^){ri-^)] 
+ - (a— a)(>?— ^)} =0. 
Write for shortness, 
T,,g=(«— a)(^-a) + (^— /3)(S/— ^), 
and similarly for Nq_ g, &c ; the equation of the reflected ray is 
N„gTg,g+T,gNg,g=0. 
Suppose that the reflected ray meets the circle again in G' and undergoes a second 
reflexion, and let x\ y' be the coordinates of a point (I in the ray thus twice reflected. 
We see first (G being a point in the first reflected ray) that 
^G', gTq, G + Tg/, g^Q, g = 0. 
Again, considering G as a point in the ray by the reflexion of which the second reflected 
ray arises, the equation of the second reflected ray is 
G'^G, G' + T,', G'^g, G'=b ; 
and fi’om the form of the expressions g? T,, g it is clear that 
^G, G'— — Ng'Gi fG, G'— “^Tg/^ G ! 
the equation for the second reflected ray may therefore be written under the form 
^7', G' 'I-G', G G'Ng', G— j 
or reducing by a previous equation, we obtain finally for the equation of the second 
reflected ray, 
G'^^Q, G + fV, G'Nq, G = ii j 
and in like manner the equation for the third reflected ray is 
^7" G"'lg, G"l“ f G"^Q, G— 
and so on, the equation for the last reflected ray containing, it will be observed, the 
coordinates of the radiant point and of the first and last points of incidence (the coordi- 
nates of the last point of incidence can of course only be calculated from those of the 
radiant point and the first point of incidence, through the coordinates of the intermediate 
points of incidence), but not containing explicitly the coordinates of any of the inter- 
mediate points of incidence. The form is somewhat remarkable, but the result is really 
the same with that obtained by simple geometrical considerations, as follows. 
