ttofeERTS — On the Equation of the Tangent. 



63 



where 



A a a^ 2 + 2a,xy + a 2 y 2 = ife « 5 



Q = q x 2 + 2q x xy + q 2 y- a y 2 e q 2 , 



and 



B = bj? + Zb x x 2 y + Shxif + h?f m y 3 e b s , 



B = B 2 - A*Q, 

 thus expressing X, Y, and ^ in terms of a parameter x/y, which we may 

 call 6; it being understood that § is an operation which, when applied to 

 any one of the ten quantities which enter into the above qualities, converts 

 it into one of lower weight, so that $c r = rc r _, ; c,. being typical of any one of 

 these quantities whose weight is r; while §# = - y and Sy = 0- 



The coordinates of the corresponding point are expressed as follows : — 



X' = Ax 



y - Ay • . (3) 



Z' = D- y/B 



We now proceed to find the equation of the tangent at a point on the 

 curve determined by the equation (1), and we arrive at the following result 

 without much difficulty, X, Y, and Z being current coordinates : — 



X j yj - AB X + — ^ [y (2BJ - A*K) - AB : ] } 



( 2 <y b ' 



- Y \xJ -i AB. + ~— [x(2BJ - A 2 K) + AB,] j 

 I 2 V 'B I 



+ 3A*Z=Q; (4) 



the equation of the tangent at the corresponding point being 



X j yJ- AB t - - 1 — [y(2BJ - A*K) - AB>] j 



where 



and 



Y \xJ+ AB,- 



B> 



--— [x (2BJ - A-K) + AB,] 



yB ) 



(5) 



+ 3A 2 Z = 



dB 



~dx~ 



dy 



J*J(A,B), K-J{A,Q). 



[11*] 



