SUPPIiEMENTAUI CUBVES. 



179 



and in virtue of these equation (11) becomes 



+ 2 A, A,-2 A,A, + A;-2 A,A. + A/, 



and this may evidently be written in the form 



M^ = (A,- A3 + AJ^ + (A,- A, + A.r (12). 



In the general case we should evidently arrive at a similar 

 result, and hence equation (9) gives 



«'=MxPM,.PM,.PM, PM« (13), 



whereM'^i^CA— A3+A— &c.)'+(A,— A^+A— &c.)'...(l4). 



{d.) The formulas demonstrated in this number are of great 

 importance in reducing geometrical expressions to algebraic 

 form, and the converse. Let us consider, for instance, the 

 equation 



Ui = u + kvv'= o, (II, 4.) 



Let A P be a straight line drawn from the origin A to 

 any point P in the curve u^, and let this line meet the conic 

 section u in the points Q,, and Q^, and the straight lines 

 V and «' in R^ and R, respectively; then, by equations (3) 

 and (4), 



^ PQ,.PQ, ^_PR, ,^ PR.. 

 "" AQ^.AQ,' ^ AR/ ^ AR,' 



' * *"' AQ^.Aa. AR^.AR, 



Hence we see that the curve u^ may be considered as the 

 locus of a point P, such that if a straight line be drawn from 

 it to a fixed point A, meeting the curve u in Q,^ and Q^, and 

 the straight lines v and ©' in R^ and R», the compound ratio 

 PQ.. PQ, : AQ, . AQ, shall have to PR.. PR, : AR., AR. 

 a fixed ratio — k. 



