Smyly — A Note on Certain Curves, etc. 



371 



i + A, 



This equation remains unchanged if we substitute c - 2X, a + \ 

 af 



for c, a, b, f, g, respectively ; hence the bicircular 



« + A' b + \ 



quartic may be generated in four ways as the envelope of a circle, 

 whose centre moves on the quadric 



a + X 

 and which cuts the circle 



b+k 



-1=0, 



X' + i/'^ + 2 



+ 2 



iff 



a + X ' b + X 

 orthogonally ; A being zero on a root of (i.). 

 The curve 



^ dS 

 dx 



+ c - 2A = 0, 



dy 



dF dF 

 dx dy 



0, or 



« + ^ y +f 



= 



(ii.) 



is a rectangular hyperbola, and passes through the centres of the circle 

 and conic, and also through the vertices of their common self -conjugate 

 triangle ; it also passes through 16 other points closely connected with 

 the quartic : — 



At any point 1' of i^ draw the tangent TP; from C, the centre of S, 

 let fall the perpendicular CP on TP; on this line take the limiting 

 points t, t' ; then t, t' are points on the quartic, and tT, t'T are normals 

 to the quartic at i and f. If T and P coincide, then tf is a double 

 normal, and Tis its middle point; in this case, CT is normal to the 

 conic. Hence four double normals can be di-awn through each centre 

 of inversion. 



The coordinates of any centre of inversion are 



a+ X' b + X' 

 and the equation of the corresponding focal conic is 

 x^ y^ 



+ 



a + X ' b + X 

 The equation of the normal to this conic at x'y' is 



^-^' ^ y-y' ^ Q. 

 x' y' 



a + X J + A 



1. [A = 0, or a root of (i.)J 



