358 PKOCEEDINGS OF THE AMERICAN ACADEMY. 



1=1, .. . 4. 





i = i, 



Expanding these equations and making use of (12), this condition 

 reduces to 



2 U' 



(13) 





dUfc = 0, i = 1, 



dUk = 0, i =1, 



For these equations to be compatible requires five conditions, but as 

 we saw a tangent S^ depends upon six constants (six ^'s are arbitrary) ; 

 hence there are gc^ solutions of (13), that is, there are oo^ >SVs which 

 have two-point contact with V^. 



Then in St There are oo^ C^s tangent to V^ in a line p which are 

 also tangent to V^ in a line infinitely near to p. 



The consecutive tangent tti's to V^^ (in >S9) along one of the directions 

 above have a line in common. This line of intersection and the line 

 joining the point of contact are in a sense conjugate directions. 



III. Three-Parameter Families T^s* 



11. The tangent space TTg to Y^ at a point x is of three dimensions ; 

 hence there are '^^ aSVs passing through vrg (and consequently tangent 

 to Fs). Then in >S^4 There are oo^ Cg's tangent to a three-parameter 

 family of lines Vz in a given line. 



The osculating >SVs which have a fixed osculating S^ in common 

 generate an S^. Then in /S4 The ruled surfaces Cx, which have in com- 

 mon four infinitely near fixed lines of V^ generate a C^. 



12. The osculating planes which pass through a given direction 

 generate an St, and if the direction is varied these aS's will generate 

 a cone Fg* having ttq for vertex. The S^ is determined by the points 



f=0, f = 0, f = 0, A = 0, l.f^„du4u^ = 0. 



Then if dui is varied (that is, if the tangent line is varied) the cone is 

 generated by the *SVs which project v^ from the points of the surface 

 generated by 



^fikduiduk = 0, 



and hence is a cone of order four and dimensions six. Then 



