In-and-circumsa-ibed Triangle. 23 



of the line joining the other two points, i. e. that the tliree points 

 are a system of conjugate jjoints with respect to the last-men- 

 tioned conic. The problem is thus reduced to the following one: — 

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

 conic 



to inscribe an infinity of triangles such that the angles are a 

 system of conjugate points with respect to the conic 

 x^-^if + z'^^O. 

 Before going further it is proper to remark that if, instead of 

 assuming ul'l" + ^m'iu" + 'yn'a" = 0, we had assumed 

 l'li' + m'i)i" + n'n" = 0, 



this, combined with the equations 



p + „,'2 + n'^ = 0, /"2 + m"2 + m"2 = 0, 

 would have given l':m': n' = l" : m" : n", i. e. two of the angles 

 of the triangle would have been coincident : this obviously does 

 not give rise to any proper solution. Returning now to the 

 system of equations in/, c/, h, &c., since the equations give only 

 the ratios f:9:h;f:g':h';f:(j": h<*, we may if we please 

 assume 



J"2_f.y'2 + ^//2^1^ 



which, combined with the second system of equations, gives 



We have, consequently, 



A + B + C = A(/^ +/-^ +n + B(/ +g'' +g"') + C(A^ + l^'^+in 



= (Ar + B/ + CA2) + (A/^ + By2 + CA'2)+(A/"^+By'^-f C^, 



i.e. A + B + C = 0, 



for the condition that it may be possible in the conic 



AA-2-fB?/^ + Cr^ = 

 to describe an infinity of triangles the angles of which are con- 

 iueate i)oiuts with respect to the conic w'^ + y- + z"- = Q. 



The equation of the conic Koc^ + ^if-\-Gz'' = Q may be written 

 in the form 



^ («2/,2 _ ^2^2 _ c2«2 ^ 2abc^)Z^ = 0, 



