1154 

 § 2. A point satisfying the three equations 



ow Olf 



is a node for a definite integral curve; we ask when this point 

 satisfies moreover 



Ou 



consequently belongs to E=0. 



If we differentiate ƒ =0 partially with regard to .t' and //, we find: 



0/ dfdu 



— + ~ — =0 



dx dii d,v 



^ + — — =z .) 

 dy du dy ' 



I, 



Ö/* df df du 



Now, if -^ = — =0,-- need of course not be zero, lor — and 

 d.v dy ou dx 



du 



-- may be zero, and the latter is the normal case (cf. the geome- 

 trical explanation in § 3). Let us suppose viz. that in a particular 



df 

 case -- is 0, then we can easily determine another system of curves 

 du 



where this is not the case; we have only to replace the equation 



f{x,y,u) = 0, by 



if) {w,y,u) =f{x,y,n) -f r/(?<) = 0, 



in which g{u) represents a function which is zero itself for that 



particular value of u which produces the nodal curve in the system 



/ = 0, while its derivative (j'{i() is not zero for the same value. It 



is evident that the svstem of curves ff=0 has with /=:0 the nodal 



, dy Ö/' 

 curve in common, because for the ?< of this point is a; = /, -—=:—-, 



^ ' da-, dx 



drp df d(f df df d(p 



^"^^; on the contrary v~ = t^ — \- q'iu), and if ^ = 0, t- == ^• 

 dy dy du du "^ ^ du du 



We arrive, however, at quite different results if the system /=0 



contains a nodal locus. On this locus y and u are functions of .r, 



because not only one or more points of this locus must be determined 



by x, but at the same time the values of u, by which the integral 



curves are indicated for which those points are nodes. And as the 



values of .v, y, and u, which ai'e associated to each other by the 



locus, satisfy at the same time ƒ =0, we can say that for each 



point of that locus 



