26 Mr. A. Cayley on the Porism of the 



are conj ugate points with respect to tlie couie oux^ + ^tf -f 7/^ = 0, is 



-+a +-=0. 

 a P 7 



Or writing this equation under the form /Sy + ya + «/3 = 0, and 



substituting for a, /3, 7 their values, we have the equation ah-eady 



found, as the condition in order that it may be possible in the 



conic .r^+i/^ + ^^ = to inscribe an infinity of triangles the sides 



of which touch aw^ + by'^ + cz'^ = 0. 



Theorem. Let 



ax^ + bf + cz^=0 



he the equation of a spherical conic, and let {^ -v '• 0>^ point on 

 the conic, be the pole of a great circle cutting the conic in two 

 points ; the conic intersects upon the great circle an arc given 



by the equation 



cosS= {a + b + c)^/^+v^ + ^ 



\/{a + b + c)2(P + 7/2 + ^) -4{bc^ + c«7?2 + ab^) 



Hence if a + b + c = 0, S=90°; or there may be inscribed in the 

 conic an infinity of triangles having each of their sides equal to 90°. 

 It is worth while, in connexion witji the subject, and for the 

 sake of a remark to which they give rise, to reproduce in a short 

 compass some results long ago obtained by Jacobi and Richelot. 

 The following are Jacobins formulae for the chords of a circle, 

 subjected to the condition of touching another circle ; viz. if in 

 the figure we put 



CP = R 



cp=:r 

 Cc=a 

 /. ACP = 2<^ 

 /. A'CP=2<^'. 



Then it is clear from geometrical considerations that 

 d<^ _ d(f>' 



MA~ MA'' 

 We have 



MA'^ = cX^ -7M' = a" + R2 + 2«R cos 20 - r^ 

 = (« + Rf-r2-4«Esiu2 



= {(ff + R)2_r2}(l-Fsin20), 



