J 158 



r/{2i,i') mav be determined, which become zero for llie said vabies of 



dq dd 



u and V, while ~ , -^ do not become zero for the .same values. 

 On ov 



If we consider now the sjstem of surfaces : 



<f{a;,y,z,u,v) = f{x,y,z,u,v) + g(ii,v) = , 



where y represents one of that infinite number of functions then 



this new system has the same locus of nodes as the old one, while 



for those nodes ^-- and — - are certainly not zero. 

 ou Ov 



Let there be a nodal locus, formed l)y a surface; then .c and // 



may both be chosen as independent variables, while z, u, v become 



functions of them, and as the equation ./*=:0 must always be 



satisfied, we find by differentiation : 



Ö/' , ^/ , 0/ Ö// , d/dr 



.-+ T- /' + ;r ;»"'+" ^ A" = ^ 

 ax Oz ou ox Or ox 



bf df df du df dv 



dy dz du d>i ' dv di/ 



which crpiations reduce themselves into the last two terms, as iii 



df df Ö/' , , , 



each point of the double surface ^ =.— = ." =0. It the determinant 

 ' Ox Oy oz 



I du dv ' 



dx dx 



du dv 



dy dy 



=1=0' 



then ^ = -.- := 0, and the double surface consequently satisfies 

 du dv 



E=z{), in this case there exists no functional connection between 



u and r, i.e. the curves ?<=: const., ?; ^ const, cut each other on 



the double surface only in a limited number of points, or in other 



words, each particular integral possesses a finite number of nodes. 



Is on the contrary the determinant really zero, then v is a function 



of u, so that on the double surface the curves u = const, and v = 



const., coincide: there are now only x' particular integrals possessing 



nodes, but each of them possesses in that case a double curve and 



its locus is the same surface as just mentioned, which need not, 



however, belong to E=:0. because ^ and -"- need not be zero. 

 ^ du dv 



They may be zero, though, and in that case the double surface 



does belong to E=0. 



There is still another possibility. It may be that for the whole 



