I J 59 



(louhle surface ?^ == const., hut r not; fliis, however, is fi-om a 

 geometrical point of view the same case as just mentioned, for 

 z; = const, cuts the double surface now along a curve which is 



double curve tor a definite particular integral. ^ must be zero now, 



it is true, but ' must not, so that the doul)le surface need not belong 

 du 



to E=:0 either, though on the other hand it is quite possible, as 



df , , J 



the case - = is bv no means excluded. 

 du 



A very simple example of a system of surfaces with a double 

 surface not belonging to E=:0, gives the equation 



{.f + if -f u (// -f a) — z'=0; 



the double surface is c =: 0, locus of the secants of the pairs of 

 planes which are found by taking u zero. This plane j == does 

 not satisfy, however, the result of elimination E: y'^ -\-4^:^' =: 0. 



Finally u as well as v may be constant on the double surface; 

 among the particular integrals there will be one in that case, which 

 counts double, and which satisfies E z= or not, according to for 



the particular it and v oi the double surface ' ^^"^^ r being zero 



0)1 or 



in each point of that surface or not; we have apparently to do then 



with the singular plane of contact at a surface (see the conclusion of § 3). 



To wind up with, it is possible that any point of space is node 



to some integral surface or other, this will e.g. be soif each integral 



surface possesses a double curve, and these curves fill the whole 



space; in that case E disappears identically. 



§ 5. In order to illustrate the results of § 4 geometrically we 

 imagine the equation f{;x,y,z,u,c) = in a space of five dimensions, 

 7?5, interpreted as a twisted four-dimensional variety V ^ ; for the 

 sake of distinctness we shall call it T, (/'=0). All points for 

 which u = const, lie in an R^, which is perpendicular to the 

 M-axis, and the same holds good for all points r = const., and these 

 two spaces A\ cut each other along an R^ , which is absolutely 

 normal to the plane iw, and has a point in common with this 

 plane; this R^ cuts V,{f=0) along a surface .t, and if the points 

 of this surface are projected by means of j)lanes parallel to the 

 plane uv on the space A%r of the ,r, // and i:-axis (by which to 

 each point of rr one definite projection is associated), then a surface 

 rt', congruent with rr arises as projection, because the space ofrris 



