24 Mr. A. Cayley on the Purism of the 



which gives the values of A, B, C ; or again in the form 

 2 (Ac + crt + ah) {bcv- + caf + ahz^) 

 - (be + ca + ab)^{x^ + y^ + z^) 



+ 4abc{aa;^ + bi/ + ez^) = ; 



where it should be observed that bcx^ + caij^ -\- abz'^ =^0 is the 

 equation of the conic which is the polar of ax'^ + by^ + cz'^=-0 

 with respect to *^ + y^ + 5^- = 0. It is very easy from the last 

 form to deduce the equation of the auxiliary conic, when the 

 conies ax^-\-bi/ + cz'^^(), .r^ + //-4-2'^=0 are replaced by conies 

 represented by perfectly general equations. 



The condition A-f B + C=-0 gives, substituting the values of 

 A, B, C, 



/» V + cV + a%'^ - 2o-be{a -\-b + c)=0; 



or in a more convenient form, 



{be + ca + ab)^—4abc{a + b + c)=0, 



as tlie condition in order that it may he possible to inscribe in the 

 conic .v' + i/'^ + z'^ — O an infinity of triangles, the sides of which 

 touch the conic ax^ + bi/",+ cz^ =^0 : this agrees perfectly with the 

 general theorem. 



It is convenient to add (as a somewhat more general form of 

 the equation A + B + C = 0), that the condition in order that it 

 may be possible in the conic Aa,^ + B?/^-|-Cy^ = to inscribe an 

 infinity of triangles the angles of which are conjugate points with 

 respect to the conic AjA'^ + BiJ/^ + CiC^ = 0, is 



A B C; ^ 



But the problem to find the condition in order that it maybe 

 possible in the conic a^ -\- y"^ + z'^ =^Q to inscribe an infinity of 

 triangles the sides of which touch the conic ax'^-\-by'^-\-cz'^=.0, 

 may, by the assistance of the geometrical theorem to be presently 

 mentioned, be at once reduced to the problem, — 



To find the condition in order that it may be possible in the 

 conic x^ + y^ + z^ — to inscribe an infinity of triangles the sides 

 of which are conjugate points with respect to a conic 



The theorem referred to is as follows : — 



Theorem. If the chord PP' of a conic S envelope a conic a, 

 the points P, P' are harmonics with respect to a conic T which 

 has with S, cr, a system of common conjugate points. 



Take for the equation of S, 



