196 



MR. ROBERT FINLAY ON " 



XIII. 



The results which have been obtained in the last two 

 numbers can be extended to a certain class of algebraic 

 curves of every degree. Thus, if a curve of the nth degree 

 have a multiple point of the order n — 2, and if this point 

 be taken as the origin of rectangular co-ordinates, the polar 

 equation of the curve will be of the form 



A,r" + 2B,i^-^ + C,r^-^ = o, 



where A^, Bi, C,, are algebraic functions of sin d and cos 6 of 

 the degrees n, n — 1, and 7i — 2 respectively. Rejecting the 

 common factor r"~^, and solving this equation as a quadratic, 

 we obtain 



A.r = - B. ± /.(B."- A, CJ (1); 



and by changing the algebraic signs of the quantities under 

 the radical sign we get 



A.r = -B,± v/(A,C,-B;) .(2), 



which is the polar equation of the curve supplementary to 

 the given one in relation to the multiple point which we have 

 taken as the origin of co-ordinates. Since these equations 

 are of the same form as those obtained in the last two 

 numbers, the same reasoning will lead to results in every 

 respect similar to those which have been there developed. 



XIV. 



The reader will not fail to perceive that the theory which 

 I have endeavoured to explain can be extended to surfaces 

 with the utmost facility. In the case of supplementary 

 surfaces, the general problem to be solved is as follows : — 



j4tii/ surface being given, to find the locus of the imaginary 

 points in which it is cut by a system of straight lines, drawn 

 so as to succeed one another according to any given law. 



