176 



MR. ROBERT FINLAI ON 



u==l+f,{x,y)+f^{x,y)+f,{x,y)...+f„(xy) = o (2), 



where fm {nc, y) denotes a complete homogeneous function of 

 X and y of the wth degree, such as 



A a;"" + B x"^^y + C x'^-^y^ + Sec, ; 

 then if p denote the radius vector of the curve w, d and t^ the 

 cosines of the angles which p makes with the axes of x and y, 

 we shall have 



x = p6, y = p(p', 

 and by substituting these values of x and y in equation (2) 

 we obtain 



'^ + r,f,{e,<f>) + r\f^{e,<l>)+r\f,{e,^)...+r'^,fn{d,<}>)=o, 

 which is the polar equation to the given curve. Now since 



Pi> P»> Ps> P« ^^^ ^^6 roots of this equation, we obtain, 



by the theory of equations, 



J_ + J_ + i_ + &c. = — /, {3, cp), 



Pi P. Ps 



PiP, Pi Pa Pzp3 



hence, by substituting these expressions in equation (1), 



we get 



(Pa„) :(AQ„):=l+r./.(^, </.) + /■*./, (5, <^)...H-r«./„(^,<^); 



and since rd =^ x' and rcj) = y', this equation can be written 



in the form 



(PQ„) : (AaO = 1 +/, {x\ y') +/, {x', y') ... +fn {x', y') = u\ 



where u' denotes the same function of x' and y' as u is of 



X and y. 



When n is odd it may be demonstrated in the same manner 

 that (PQ») : (AQ.„) = — m'; hence, generally, we have 



PQ,.PQ,.PQ, PQ. 



AQ,.AQ,.AQ3 AQ„ - ^ ^' 



where the upper or lower sign must be taken according as n 

 is even or odd. 



(a.) When the proposed equation is of the first degree, the 

 demonstration still holds good, and equation (3) becomes 

 PQ^ : AQ,== — m'. 



