1155 



Ö.C dydx dud.i; 



df df dy 



If now - and :;^ are both zero, and it' ■ ^poc, which can occur 



ö.y oy dx 



only in a few points, and niaj' even tliere be avoided by rotation 

 of the axes of coordinates, then we must have 



Ö/' du 

 du dx 



Now T^ =: will be the normal case, viz. the nodal locus will 



du 

 as a rule belong to E=0; for if this is not so must be = 0, 



dx 



and u consequently constant along the locus, which means that the 



said locus is a particular integral to be reckoned twice. It is not 



df du 



excluded, however, that ^ and — are both zero; in that case too 



On dx 



there is a particular integral to be reckoned twice, which now does 



belong to E =z 0, and the difference between this case and the 



preceding one where there was also question of a particular integral 



to be reckoned twice is not particularly striking. The geometrical 



explanation in the following § will cast sufficient light on this case. 



§ 3. If we consider the parameter u as a third coordinate, our 

 equation f[x,y,u) = represents a surface; the sections ?ii=n constant 

 produce, if projected on the try-plane, the various integral curves. 

 The curve E=0 lying in the .ry-plane contains apparently all the 

 points of the property that the straight lines, passing through those 

 points parallel to the ?^axis, cut the surface in tw^o coinciding points; 

 hence it contains : 



l^*^^ . the apparent contour of the surface for the point at infinity 

 of the ?<-axis as lighting point, and this is apparently the singular 

 integral ; 



2"*^. the projection of an eventual double curve or cuspidal curve 

 of the surface, situated in such a way that any plane //- = constant 

 cuts it in a certain number of points; the system of curves f=0 

 then contains a locus of nodes or cusps in such a way that each 

 integral curve contains one or more of those points ; 



3^'^. the projection of an eventual double or cuspidal curve which 

 is lying in a plane w = constant ; in that case one integral curve 



