194 



MR, ROBERT FINLAY ON 



to the given ones in relation to a point of intersection of 

 conjugate common secants of the two given curves. 



XII. 



To find the curve which is supplementary to a given curve 

 of the fourth degree, in relation to a double point A on the 

 given curve. 



If the point A be taken as the origin of rectangular 

 co-ordinates, the equation to the given curve will be of 

 the form 



Ay* + B^-jj + Cy^x^ + J)yx' + Ea;* 



+ 2{A.y + Byx + C'yx^ + D'x^) 



+ K'y+ Wxy + C'V= o (1). 



Let r and 6 be the polar co-ordinates of the point x, y; 

 then, the preceding equation becomes 



A,r* + gB^r'-H C,r*= o (2), 



where, for the sake of brevity, we assume 



Aj = A sin* ^ + B sin" ^ cos ^ + &c. \ 



B, = A'sin'^+ B' sin' ^ cos 5 + &c. I (3). 



C, = A" sin* 6 + B"sin ^ cos ^ + C'cos' 6. 



Now, by rejecting the common factor r% and solving 

 equation (2) as a quadratic, we get 



A,r = - B. ± 1/ (B.'- A, CJ (4), 



which may be considered as the polar equation of the given 

 curve ; and by changing the algebraic signs of the quantities 

 under the radical sign, we obtain 



A,r = - B, + / (A, C, - B,0 (5), 



which is the polar equation of the required curve. It is 

 evident from the values of A,, B„ C, given above, that the 



