392 Prof. Challis on Integrating Differential 



its equivalent equation (/3), may be considered to be an integral 

 of F = G, although the order of the latter equation is inferior by 



one unit to that of (/S). For since the equation —=0 is a ne- 

 cessary consequence of F = 0, the order of the latter equation 



may betaken to be the same as that of MF + N-^— =0. Also the 



ax 



equation (7) shows that the equation (/3) is not satisfied if either 



df 

 F = 0, or — = 0, but by both equations. 



For another example let 



a ayq 



The integration of this equation by means of the factor p gives 



^=->tl_ + ^ ...... (S) 



vl+f 



b 2 being the arbitrary constant. This result does not admit of an 

 exact second integration ; but it can be proved that the curve it 

 represents is describable by the focus of an hyperbola which is 

 made to roll on a straight line. Following the same course as 

 in the preceding problem, I shall now proceed to find the differ- 

 ential equation of the first order of a curve drawn parallel to a 

 given curve of the kind just mentioned and distant from it by 

 an arbitrary interval. 



Let x Xi y l be the coordinates of any point of the given curve, 

 and x } y the coordinates of the point of the parallel curve cut 

 by the normal at the point x Y y^ or the normal produced. Then, 

 h being the interval between the curves, we have p t =p and 



y=V\- /-, , - > s0 tnat V1-V+ -~Trf=%- Hence, by substi- 



V\+p\ Vi+f 



tution in the equation (8), 



Differentiating to eliminate 5 2 , and putting for shortness' sake 

 P for \/ 1 +jo 2 , the result is 



(.+ |)&>-¥)=<f-p-¥9. 



To eliminate h this equation is first to be solved as a quadratic 

 with respect to h. It will be found that 



/ P 3 \ FP 4 



h*-h(2a+^-yJ>)=^-, 



