296 
ME. A. CATLET’S MEMOIE rPOX CATJSTICS. 
a=-^, the form of the curve is nearly the same as before, only the cusps upon the 
circle through the radiant point lie on the axis of y (see fig. 10). The case ^ 
shown, fig. 11. For «=|, the two asymptotes coincide with the axis oix', one of the 
Fig. 8. a=l. Tig. 9. Fig. 13. a 
Fig. 10. a=^. Fig. 12. a=h 
branches of the curve has wholly disappeared, and the form of the other is modified by the 
coincidence of the asymptotes with the axis of x ; it has in fact acquired a cusp at infinity 
on the axis of x (see fig. 12). When a<\^ the curve consists of a single finite branch, with 
two cusps on the axis of x^ and two cusps at the points of intersection with the cii'cle 
through the radiant point ; one of the last-mentioned cusps mil be outside the refiecting 
circle as long as a>-g-; fig. 13 represents the case «=-§■, for wliich this cusp is upon the 
reflecting circle. For the curve lies wholly within the reflecting circle, one of the 
cusps upon the axis of x being always within, and the other always without the cii'cle 
through the radiant point, and as a approaches O the cuiwe becomes smaller and 
smaller, and ultimately disappears in a point. The case a negative is obAiously included 
in the preceding one. 
Several of the preceding results relating to the caustic by reflexion of a cii’cle were 
obtained, and the curve is traced in a memoir by the Fev. Hamxet Holditch, 
Quarterly Mathematical Journal, t. i. 
