188 



MB. ROBEBT FINLAY ON 



from which it is evident (III) that each of the supplementary 

 curves touches the system of straight lines represented by the 

 equation 



{<f> XY — {yjr xY ~ O (6) 



at the points in which these lines are cut by the diametral 

 curve (3). 



(c) By taking the sum of equations (4) and (5) we obtain 



{y — <f,xy= O, 



from which we see (III) that the supplementary curves (4) 

 and (5) touch each other at each of the points in which they 

 are cut by the curve (3). 



(^ Let a straight line be drawn from the origin A to any 

 point P in the curve (4), meeting the curve (3) in the points 

 Q.J, Qj, 0,3, &c., and meeting the system of straight lines (4) 

 in the points R,, R^, R^, &c.; then if ^ar be an algebraic 

 function of x of the wth degree, and ^jrx one of the mth 

 degree, where m is not greater than n, we shall have (V) 



PQ, .PQ, .PQ, PQn ^ I y--cf>x 



AQ, .AQ, .AQ3 AQ„ — M ' 



PR, .PR,.PR3 PR,„ ^ i^xY — ifxY 



AR. .AR,.AR AR,« T 



where M and T are the absolute terms of the functions <f)X 

 and (<^ xY — {-^ xY respectively. Hence, by substituting these 

 geometrical expressions in equation (4), it becomes 



/ PQ..PQ,...PQ. v _ PR,.PR....PR.. 

 ^ VAQ,.AQ,...AQ„/~ ^AR,.AR,...AR,, ^ ^' 



Similarly, if the same straight line meet the curve (5) in any 

 point P, we shall have 



/ PQ,.PQ....PQ. x _ _ PR,.PR....PR.. 

 VaQ. .AQ,...AQ«/ ^AR,.AR,...AR,/*"^°^' 



