§ fi. Amorig the oo^ spaces R^, which are absolutel_y normal to 

 tlie plane u. d, and according to what precedes, cnt the snrfaces rr ont 

 of V^i/= 0), there are oo^ for wliich that snrface of intersection jt 

 possesses a node, and which tonch V^ accordingly in that point ; 

 this node of rr will in general, how^ever, be only a simple point of 

 V4., i.e. the plane passing throngli /^, and that point will in general 

 cut V^ along a curve that possesses no node in that point; and from 

 this it ensues that the projection of that point on Rx,/z will in 

 general not belong to A'^0. With this we have proved that the 

 nodal curve, which in general is jiresent in on i- system of surfaces rr', 

 will as a rule not belong to JiJ^=i). 



Conversely the possibility is, however, not excluded that V^{/z=0) 

 contains a double curve; now the plane passing through /^ and a 

 point F of this curve cuts F, along a curve which does have a 

 node in F, and the consequence of this is that the projection of the 

 double curve does belong this time to E=zO; and finally the 

 case is not excluded that both phenomena occur at a time, and 

 consequently the system of surfaces jt' contains two different 

 double curves, of which one does belong to E ^ 0, the other 

 does not. 



Let ]\{/=0) contain not a curve but a surface of nodes; as 

 any plane passing through /^ and a point F of this double surface 

 cuts }\{f=:0) along a curve with a node in P, the projection of 

 the double surface will belong to E ^= \ and as an R^±iiv cuts 

 the double surface in general in a finite number of points we have 

 here to do with the case that each surface of the oo".t' possesses a 

 finite number of nodes (see § 4j. 



A double surface in the system n', may, however, have a ([uite 

 different origin. Among the spaces R^ i. m^ there may be some that 

 touch Vi{f=0) not in one but in an infinite number of points, 

 so that the associated surface rr possesses a double curve, which, 

 however, is not at the same time a double curve of T^ ; in that 

 case there are surfaces rr' with a double curve not belonging to 

 E = 0. And if all R^^iiv, which touch V ^, have this property, 

 then we find in the system rt' a double surface, locus of double 

 curves of oc^ surfaces ci' , which does not belong to E^=0. The 

 case is even not excluded that a certain R, ± iiv touches V^ ( /—O) 

 in all the points of a surface, the analogon of the singular planes 

 of contact of a ring, then the difierential equation jjossesses a par- 

 ticular integral counting double, not belonging to A' = 0. If, however, 

 that integral counts double l)ecause the associated surface rr in A*, is 

 a real double surface of V^i/^^O), which happens to lie in an 



