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A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
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such a point is in fact equivalent to a node and two cusps, and we have thus the 
two circular points at infinity counting together as 2 nodes and 4 cusps; there 
should therefore besides be 2 nodes and 2 cusps, and I proceed to establish the 
existence of these by means of the expressions for (x, y) in terms of 6. 
10. To find the cusps, we have 
^ = — sin 2 6 (3 cos 6 — 2m) = 0, 
= cos 26(3 cos 6 — 2m) = 0, 
which are each of them satisfied if only 3 cos 6 — 2m = 0, or cos 6 = § m; the corre 
sponding values of (x, y) are found to be 
x = m — -fa m 3 , y = + (1 — •§ m 2 )^, 
and we have thus two cusps situate symmetrically in regard to the axis of x; the 
cusps are real if m < f, imaginary if m > f; for m = f, the two cusps unite together 
at the point x = \ on the axis of x, giving rise to a higher singularity, which will be 
further examined, post, No. 12. 
11. The curve is symmetrical in regard to the axis of x, and hence any inter 
section with the axis of x, not being a point where the curve cuts the axis at right 
angles, will be a node. Hence, in order to find the nodes, writing y = 0, this is 
giving sin 6 = 0, that is, 
sin 6 (1 — 2m cos 6 + 2 cos 2 6) = 0, 
6 = 0, x = 2 — m ; 
6 = 7r, x = — 2 — m ; 
but these are each of them ordinary points on the axis of x; or else giving 
that is 
1 — 2m cos 6 + 2 cos 2 6 = 0, 
cos 6 = \ (m + Vm 2 — 2). 
The corresponding values of x are 
x = cos 6 (2 cos 2 6 — 2m cos 6) + m, = m — cos 6, =\ (m + Vm 2 — 2) ; 
each of the points in question, viz. the points 
x = ^ (m + Vm 2 — 2), y —0, 
is a node on the axis of x. 
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