302 KANSAS UNIVERSITY SCIENCE BULLETIN. 



§5. Type I. 



43. First class. — There are three continuous groups of type I, 

 first class, having corresponding parts on each set of generators. 

 Each of these may be combined with 1 -'T(PsF) k __ 1 , giving us a cor- 

 responding mixed group. In the first of these groups there are 

 necessarily two invariant points, and therefore but cc 1 12 T(PsF) k = -i. 

 This gives us the mixed group n^G^ll'mm'F). In the second case, 

 there is but one invariant point; hence, cc 2 12 T(PsF) k = _i, and the 

 third case drops to a continuous group. 



In symbolic language : 



m 1 G 2 (ll'mm'F) = 1 G 2 (irmm'F)+ oo 1 12 T(P7F) k = _! 

 m 1 G4(lmF) = 1 G4(lmF)+oc 2 12 T(P.7.F.)k = -i. 



44. Second class. — The second-class groups are derived from those 

 of the first class by making the parameter r any constant value, and 

 we have : 



m 1 Gi(U'mni'F)a= 1 Gi(ll'mm'F)a+ oc 1 12 T(p7F) k = _! 



m 1 G3(lmF)a = 1 G 3 (lmF),+ oc 2 u T(p7F) k — i- 



F.— Table of Groups of Collineations of Space Leaving 

 Invariant a Quadric Surface. 



Type XI. 



Symbol. Additional invariant figure. Lie.* 



(1) u Gi(lfiF). All generators of one system, and 



one generator of other system. (VI. A) 



Type X. 



(2) 10 Gi(/r nF). All generators of one system, and 



two generators of second sys- 

 tem. (VI. D) 



(3) 10 G2(//a F ). All generators of one system, and 



one generator of other system. (V. A) 



(4) 10 G 3 (/iF). All generators of one system. (IV. A) 



Type IX. 



(5) ,J Gi(lmpi!F). One generator of each system, 



their common tangent plane, a 

 line of invariant points in that 

 plane tangent to the quadric. (VI. B) 

 (()) :, G2(lmpF) . One generator of each system and 



their common tangent plane. (V. E) 



*Theorie tier Transformationsgroppen, vol. Ill, pp. 251-2'>4. 



