1153 



with the general mluiion /{,v,y,ii) = (we represent, witli a view 

 to the geometrical interpretation that is to follow, the arbitrary 

 constant by u), then the result of elimination E=0 of 



Ou 



i-epresents the locus of tho.w points in the .?7/-pIane, for which the 

 equation /{,T,i/,u) = 0, considered as an equation in u, possesses a 

 double or multiple root, hence the locus of those points, through 

 which one particular integral less passes than through an arbitraty 

 point (if we restrict ourselves to a double root); it ïs obvious now 

 to surmise that a point which is node of a definite particular integral 

 will belong to this locus, because the curve with the node passes 

 twice through (hat point, but in general this is incorrect, as may 

 appear from very simple instances. The equation 



has as general solution 



in which a represents the arbitrary constant, consequently a system 

 of equilateral hyperbolae. The result of elimination E of a from 

 this last equation and its partial derivative with regard to a : gives, 



Ez=.x'-2y'-2E' = 0, 

 and this is really the envelope of the equilateral hyperbolae. 



Let now ~ = 2w-2a = 0, and ^ = ~ 2if = 0, then we find that 



the point ,y = a, y = is a node for the particular integral which 

 is^ determined by giving to a e.g. the value R, thus for the pair 

 of straight lines 



(w-Ry-y^^o-, 

 but the node .r = /?, y = does apparently not lie on E. And in 

 fact through the point x = R, y=0 passes not only the particular 

 integral a = R, but also a = (,v'~y' = R'^), viz. two, just as 

 through an arbitrary point. 



What consequences has this for the differential equation? 



By solving it with regard to /> we find for each point the tangents 

 of the integral curves passing through tiiat point, so in our case 2 ; 

 but in the point ,v =: R, ?/ = we must find 3 now, viz. the two 

 45"-Iines, and the line parallel to the //-axis; hence the differential 

 equation must disappear identically in this case, which occurs at 

 once through substitution. 



75* 



