SGPPLEMENTAEY OUEVES. 



189 



From the last two equations it is evident that each of the 

 supplementary curves can be considered as the locus of a 

 point P such that 



PR. .PR. PR.n . / PQ. .PQ, PQ »y 



AR,. AR, AR.« ** ^ VAQ, .AQ, A qJ 



in a fixed ratio ; and thus we see that the two curves (4) and 

 (5) can be considered as two branches of a geometrical locus, 

 which is determined by a unique geometrical condition. 



X. 



In some cases the equation to the supplementary curve can 

 be obtained with great facility. Thus, the equation to the 

 Conchoid of Nicomedes being 



m^x^ =^{p — x)' (.T« + /) (1), 



it is evident that the equation of the curve which is supple- 

 mentary to it in relation to the axis of y will be 



Wi»a?» = {p — xf [x' — y^ (2), 



and the supplementary curves (1) and (2) will possess pro- 

 perties analogous to those which have been noticed in the 

 preceding examples. 



(a) It is evident from the form of these equations, that the 

 origin is a double point on each of the supplemementary 

 curves (1) and (2). For the curve (1) the tangents at this point 

 are represented by the equation 



P'f + {p'-~ni-)x'-o (3), 



and for the curve {2) they are represented by the equation 



— p^y^ + ip^ — m') .r' = o (4). 



(&) When m>p, the tangents (3) are real and the tangents 

 (4) are imaginary, so that the origin is a double point of the 

 curve (1) and a conjugate point of the curve [2); but, when 

 m < p, the tangents (4) are real, and the tangents (3) are 



