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A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
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so that the equations 8 = 0, T =0, or, what is the same thing, S' = 0, T = 0 give 
and therefore 
( æ ~ m ) 4 m2 {27 (« - m) - 8m 3 } = 0, 
the second factor of which gives x = m — f Y m 3 , and thence « 2 + y 2 — m 2 — 1 = — f m 2 , that 
is, x- + y 2 = 1 — | m 2 , and therefore y- = (1 — £ m 2 ) — (m — m 3 ) 2 , = (1 — | m 2 ) 3 , that is, we 
have 
x=m — 27 m 3 , y 2 = (1 — ■§ m 2 ) 3 , 
which, as appears above, gives the two cusps. 
14. Similarly, in the equation 16 F = S 3 — T- = 0, the intersections of the curves 
>8 = 0, T = 0 must include the cusps; the curves in question are the two circles 
3 (x- + y 2 ) + to 2 — 3 = 0, 
18??i (x 2 + y 2 ) — 27a? — 2m 3 + 9m = 0, 
meeting in the circular points at infinity, and in the two cusps. It is to be added 
that the tangent at the cusps coincides with the tangent of the last-mentioned circle, 
18m (x 2 -f y 2 ) — 27x — 2m 3 4- 9m = 0, 
or, as this may also be written, 
15. The axis of x meets the secondary caustic in the two nodes counting as 4 
intersections, and besides in 2 points, viz. the points x = 2 — m, x = — 2 — m ; these 
correspond to the values 0 = 0 and 6 = tt respectively. But to verify them by means 
of the equation 
16F=$ 3 — T 2 = 0 
of the curve, it may be remarked that for y = 0 we have 
>8=4 (3# 2 + m 2 — 3), T= 4 (18m« 2 — 27« — 2m 3 + 9m); 
and writing herein « = + 2 — m, we find 
>8 = 4 (2m + 3) 2 , T= 8 (2m + 3) 3 , 
values which satisfy the equation S 3 — T i = 0.