BREWSTER: COLLINEATIONS OF SPACE. 301 



contain all of the continuous subgroups included under the head of 

 the continuous groups of like symbol. Hence, in writing out these 

 groups, we shall not express in full detail the structure of the con- 

 tinuous groups, leaving that to be understood from previous work. 

 m y G 2 (lmpF)== 9 G 2 (lmpF)+oc 2 12 T(p7F) k = _j. 



§ 3. Type VI. 



40. The continuous group G Gi(ll'mm'nF) may be combined with 

 oo 1 involutoric collineations of type XII and the invariant figure as a 

 whole be left unchanged ; i. e., there will exist one involutoric °T(PsF), 

 which will interchange 1 and m, 1' and m' for each plane passing through 

 the two invariant points, the intersection of 1 and m, 1' and m'. These 

 combine to form a one-parameter mixed group m e Gi(ll'mm'n'F). 

 There can exist no m ,5 G 2 (H'uiF), since, if we combine with this a 

 transformation which interchanges the systems of generators, 1' will be 

 thrown into a position m', thus giving a new invariant figure and de- 

 stroying one of the essentials of the group property. 



The group 6 G3(lmF) may be combined with oo 2 of these involutoric 

 collineations, exactly as in the case of the group !) G 2 (linpF), and there 

 exists the three-parameter mixed group m 6 G3(lmF). 



There can exist no other mixed group of type VI, since the com- 

 bination of 6 T(PsF) with 12 T(PsF) does not alter any part of the in- 

 variant figure. 



Theorem. There are two mixed groups of transformations of type 

 VI, each composed of a continuous group of the same symbol and in- 

 volutoric transformations of type XII. Expressed symbolically, they 

 are : 



m 6 (*i(ll'mm'rcF)== c Gi(ll'mm'raF) + ex 1 12 T(p7F) k = _! 

 m ,i G3(]m^F) = <! G 3 (lm^F)+ ex 2 12 T(p7F) k= _ 1 . 



§4. Type III. 



41. First class. — It is readily seen that there can exist but one 

 mixed group of this type and class, there being but one corresponding 

 continuous group having similar invariant parts on the two systems 

 of generators. This is the continuous group 3 Ga(lmF), which, if com- 

 bined with oo 2 12 T(P6-F)k = _i, gives us the mixed group 



m 3 G 3 (lmF)^ 3 G a (lmF)+ ex 2 ,2 T(P7F) k = -i. 



42. Second class. — There exists in this class a continuous group 

 3 G 2 (lmF) a , which if combined with oo 2 12 T(PsF) k = _i, results in the 

 mixed group m 3 G 2 (lmF) a . 



m 3 G 2 (lmF)a= 3 G 2 (lmF)a+ oo 2 12 T(p7F) k= _ 1 . 



