In-and-circumscribed Triangle. 515 



lx + my + n2=0 in the points (x X) y x , z x ) and (x v y<» * a ), then 

 *8 '• Vi '• ^=^z x {ym 9 + un^) 

 :*A(«* + 0P) 

 :x x y x (fip+um 2 ). 

 We have, in fact, identically 



ly i z 1 (/3m 2 + y;n a ) + mr,;*, (y/ 2 + an 2 ) + rc^y, (am 2 + /3/ 2 ) 

 = (amncr j + finly x + y/m.?,) (/#, + my x + n*,) 

 — lmn(ax x * + /3y, 2 + yr, 2 ) 

 «yi**i W + 7™ 2 ) 2 +#?, V(yZ 2 + «" 2 ) 9 + y#, V(«»* 9 +#*)*= 

 «/3y { - ^i 3 — w 3 y ! 3 - w 3 ^! 3 + my x -\-nz x Px* + nz x + lx x m*y , 2 

 + £r, + my i n 2 y x 2 —2lmnx 1 y i z x } (fe, 4- my, + n*,) 

 - (/ 4 /3yx, 2 + iM 4 y«y i 9 + ***#?,*) (a*, 2 + /3y, 2 + y?, 2 ) = j 

 which show that if £r, + my, + nz, = and ax, 2 -f /%, 2 + y,r 2 = 0, 

 then also lx 2 + my 2 + nz^ = and a# 2 2 + /3y 2 2 + y? 2 2 ==0 : this is, 

 of course, as it should be. 



I shall now consider I, m, n an given functions of x u y,, z x 

 satisfying identically the equations 



~lx x + my x + nz x = 

 -l%c + ?n 2 ca + n 2 ab = 0, 



equations which express that lx + my + nz = is the tangent 

 from the point x x , y x , z x to the conic tf# 2 4-6y 2 + c2 2 =0. And I 

 shall take for a, (3, y the following values, viz. 



a = ax x * + bij* + cz*-a{x* + y* + z x 2 ) 

 (3 = ax x s + by x * + cz x *-b(x x * + y* + z x S) 

 y^axf + byZ + cz*— c{x x q + y x * + z x *) ; 



bo that x v y x , z x continuing absolutely indeterminate, we have 

 identically ax, 2 + /3y 1 2 + y^ 1 2 = 0. Also taking © as a function 

 of x x , y x , *„ the value of which will be subsequently given, I 

 write 



x 2 = ®y x z x (/9n 8 + ym 2 ) 



y q =®z x x x {yl* + an 2 ) 



z 2 = ®x x y x (am 2 + /3Z 2 ) ; 



so that x x , y x , .*,, are arbitrary, and # 2 , y 2 , s 2 arc taken to be 

 determinate functions of x x , y,, z x . The point {x v y 2 , s 2 ) is 

 geometrically connected with the point (x t) y,, s,) as follows, viz. 

 (x 2 , 7/ 2 , z 2 ) is the point in which the tangent through {x i ,y i) .; l ) 

 to the conic ax 2 + by- + cz 2 = meets the conic passing through 

 the point {x x , y v z x ) and the points of intersection of the conies 

 2 L2 



V 

 u 



