SUPPLEMENTARY CURVES. 



187 



Hence we see that each of the supplementary curves (4) 

 and (5) can he considered as the locus of a point P, such 

 that 



- is to 



/ PQ..paA ' 



VaQ. AQ-/ 



AR,. AR,. AR3. AR, \AQ,. AQ. 



in a fixed ratio ; and thus, in a geometrical point of view, the 

 curves (4) and (5) can be considered as two branches of the 

 same curve. 



IX. 



If the equation to a given curve be of the form 

 y^—2t/.cf>x + {^l^xy=z o, 



where and ^ denote any given functions, it is required to 

 find the supplementary curve in relation to a system of 

 straight lines parallel to the axis of y. 



By solving the equation as a quadratic, we obtain 



y = ^x±x/ {{ci>xy~-ifxy} (1), 



from which it is evident (VI, b) that the equation of the 

 required curve will be 



2/ = </,a;± y {{fa;y — {<t»xy} (2). 



(a) It is evident from the last two equations that the curve 

 whose equation is 



y = «^^ (3) 



bisects all chords of the curves (I) and (2) which are parallel 

 to the axis of y. Hence the curvilinear diameter which bisects 

 chords parallel to the axis ofyis common to the supplementary 

 curves (1) arid (2). 



(6) By clearing the equations (1) and (2) of radicals, we 

 obtain 



(^ — ^xY = {<},xy—{ylrxY (4), 



(^ — >i>xy = {^xy—{<t>xY (5); 



