1 1 5G 

 counts double, so that - = (see § 2), while at the same time 



4^1'. the projections of eventual conical points of the surface; in 



Ö/' 0/ a/' 



this case the two equations I of § 2 are satistied bv s~=^=r^=^^' 

 ' Ox Oy Ok 



— and — =1= 0, and the points under discussion are nodes of E. 

 dx dy 



Generally speaking there are, however, a number of tangent planes 



to be constructed to the surface perpendicularly to the u-axis ; they 



cut the surface each along a curve with a node in the point of 



contact, but as this 'node is not at the same time a node of the 



surface itself, its projection will in general not lie on E ; so we 



have now integral curves with nodes, not belonging to E, in the 



df df du du Ö;" 



equations I of § 2 is —=:--= - = — = 0, but :^ =|= 0. And 

 • Ox Oil Ox d// On 



if such a node does belong to E after all, it is because the projecting 



straight line of the point of contact on the surface touches that 



surface e.g. in another place, or happens to cut it m a point of a 



double or cuspidal curve ; the node of the integral curve is then, 



however, a simple point of E. 



Finally something else is possible. A tangent plane perpendicular 



to the ?ó-axis may, after the fashion of the two singular tangent 



planes of a ring, have an in unite number of points of contact ; in 



that case a certain integral curve counts double, however without 



there being the slightest cause for belonging to E ; for the points of 



(lit 

 contact on the surface are simple points. We have then - = 0, 



ax 



0/ 

 but not as under 3), at the same time ^ = Ü. 



Ou 



§ 4. Passing on to the partial differential equations we represent 

 the complete integral of F{x, y, z, p, q) = by ƒ (.r, ?/, z, u ,?;) = : 

 it determines a system of oo^ integral surfaces. 



Elimination of u and v from the four equations 



/z=:0 , f = , 1 = , f- = 

 Ox Oy Oz 



gives two relations between x, y, z, and so a twisted curve, locus of 

 the nodes of oo^ surfaces out of the complete system ; this twisted 

 curve does not in general lie on the result of the elimination ^= 



