44 KANSAS UNIVERSITY QUARTERLY. 



For r= — 1, 2, and | the conies have double contact at A and C, B 

 and C, and A and B, respectively. 



The case where r= — 1 is essentially different from the other two 

 cases where r=2 and r=|. In the first case the conies in all the 

 planes through AC have double contact at A and C ; in the second 

 case the conies in all these planes have only one point in common, C 

 when r=2 and A when r=^. The invariant surfaces in the two cases 

 are very different and worthy of attention. Let 8 = 1 and r^ — 1 in 

 equations (I), (II), (III) ; we thus get 



(I), xz=Cyw; (II), xz = Cyw, (III), x=Cz. 



In this case we have a family of invariant quadrics. Again, let 

 8 = 1 and r=2 in the same equations, and we have 



(I),x2y = Cw-^z; (II), xw2=Cyz-^-, (III), x = Cz. 



Let s = l and r=|, and we get 



(I),xy^=Cwz^; (II), x% = Cy^z; (III), x=Cz. 



In both these latter cases the invariant surfaces are ruled surfaces 

 of the third order. 



Instead of taking the edge BD for the line of invariant points any 

 other one of the six edges may be made the line of invariant points ; 

 hence there are six such cases as the above to be considered. 



Theorem 12. There are eighteen one-parameter subgroups of hGs 

 ( ABCD ) for which the path curves are conies ; six of these groups 

 leave invariant a family of quadrics and twelve of them leave invari- 

 ant families of cubic surfaces. The first six are given by the follow- 

 ing values of r and s: (—1, 1), (1/2, 0), (2, 1/2), (1, —1), (0, oo, rs 

 ■=^ — 1), ( oc, 2). The remaining twelve result from the following val- 

 uesofrand s: (2, 1), (1/2, 1), (2, 0), (-1, 2), (1 2, -1), (1, 2), (1, 

 l/2),(0, 00, rs=2), (0, 00, r8=l/2), ( ex, -1), ( ex, 1/2), (-1, 2). 



Subgroups whose path cn7'ves are hoisted cuhics. — If we give to r 

 and s such values that two of the families of invariant cones (IV) 

 . . . (VII) are quadric cones so situated that all the cones of both 

 families have one edge of the tetrahedron ( ABCD ) in common, the 

 path curves of the resulting one-parameter groups will be twisted cu- 

 bics ; for the intersection of two quadric cones having one element in 

 common is a twisted cubic passing through the vertices of the two 

 cones and having the common element for a secant line. 



For example, let r== — lands = — 1 ; equations (IV) and (VII) re- 

 duce respectively to 



xz=Cy''^ and yw=Cx^. 



The first eqviation represents a family of quadric cones having dou- 

 ble contact along AB and AD ; the second represents a family of cones 



