UGO 



parallel with Rxi/i, iind the .t' is nothing but a |)articuhir integral 

 of the given difFerentiai equation. 



Through the straight line l^ at infinity of the plane uv pass x" 

 planes; each of them cuts ^^(/^Oj along a curve, and among 

 them are gd- possessing a node, and whose plane touches therefore 

 the variety. Consequently x" planes of contact pass through l^ 

 touching at F^(/=r:0); the locus of the points of contact P is 

 therefore a surface i2, but in general of course a surface that can 

 only occur in an /?. ; the projection Si' of 52 from l^ on Rxijz is 

 an ordinary surface, which might be called the "apparent contour" 

 of F^(/=0'i on R.niz^ ^"*-^^' tliG lighting axis /^. 



The projection of ii on 74y: > takes })lace exactly through the 

 planes that ha\'e produced il itself, viz. the planes of contact 

 passing through 1^ touching at \\ [f =:. 0) ; now x"* straight lines 

 pass through the point of contact P of such a plane, and conse- 

 quently x^ that touch at V^ (/'^ Ö) in P, and they determine the 

 tangent-/^, in P at V^ {/= 0); as this tangent- /^^ contains all tangents 

 through P at F^ (ƒ=()) it also contains the plane Pl^^; so it is 

 projecting, the consequence of which is that its projection on R^yz , 

 being nothing but its intersection with R,c,jz, 'S only a plane, viz. 

 the i)lane of contact in P' at i2'. 



Out of j)oint P only one perpendicular line can be let down on 

 the plane un and through the foot of this perpendicular passes only 

 one Rs absolutely normal to ///-, from which it ensues that only 

 one surface rr passes through J\ The plane of contact at this surface 

 in P coincides by no means with the one at <2, but does contain 

 tangents of F4 (ƒ=()), as nr too belongs to this variety; the plane 

 of contact in P at jt lies therefore in the tangent-i?^ of i^at F4(/=0) 

 and so projects itself, as the plane of contact at £2, in the plane of 

 contact in P' at £i', from wiiicli it ensues that rr' and i2' touch 

 each other in P' ; ii' is therefore the singular integral of our diffe- 

 rential equation. 



In fact il' is found analytically by making the (fourdimensional) 



df ^ 



first polar space ^ = ot the point at intinity U^ ot the li-axis 



with regard to I"^^ (/ :^ 0), and the tirst polar space ^ = of V^, 



cut each other, in consequence of which the threedimensional 



first polar space of the line U^ V^ = /^^ arises ; the latter cuts 



V^ (/ i=r 0) along the surface i2, and i*! satisfies apparently the 



0/ Ö/' 



result of elimination £'=0 of a and v from/ = 0, " =: 0, — =: 0. 



ou ov 



