NEWSON : REAL COLLINEATIONS. 105 



results when m, n and h are constant, the other when m, n and k are 

 constant. In the first case the group G3(Apl)mnh leaves invariant a 

 net of conies in p and a pencil of cones through A ; in the second case 

 the group G3(Apl)tnnk leaves invariant a pencil of conies in p and a 

 net of cones through A. 



Two-parameter subgroup of GaiApl). — When m, n, h, k are all 

 constant and g and t alone vary, the cc- collineations form a two-para- 

 meter subgroup of GeCApl). Evidently there is a quadruply infinite 

 system of these two-parameter groups. The group G>(Apl)mnhk 

 leaves invariant a pencil of conies in p and a pencil of cones through A. 



Theorem 6. The groujo G«(Apl) contains two singly infinite sys- 

 tems of five-parameter subgroups, three doubly infinite systems of 

 four-parameter subgroups, two triply infinite systems of three-para- 

 meter subgroups, and one quadruply infinite system of two-parameter 

 subgroups. These are characterized by m = c; n=^c; m=c and 

 n = c; n = c and h=c; m = c and k^c; m = c, n=o, h = c; m=c, 

 n=c, k = c; m = c, n = c, h = c, k=c. 



§3. Some Special Subgroups op Ge(Apl). 



Groups of type XIII in GeiApI). — For any constant value of m 

 we have a five-parameter subgroup of GelApl); for the special value 

 m=0 the subgroup requires special attention. Let m=0 in equation 

 IV; it reduces to y(kx — w — Cy)=0, ^. e., the conies in p break up 

 into the invariant line y^O and the pencil of lines kx — w — Cy=0. 

 Hence the coUineation in the plane p is of type V; dualistically the 

 coUineation in the bundle through III is also of type V. Thus it 

 must have a pencil of invariant i^lanes corresponding to the line of in- 

 variant points in p. The collineations in space are therefore of type 

 XIII, and these form a five-parameter group of this type. 



In like manner, when n-==0 the two-dimensional collineations in p 

 and through A are of type V and the three-dimensional collineations 

 are of type XIII. They form a five-parameter subgroup of this type. 



Each of these five-parameter subgroups of type XIII contains a 

 singly infinite system of four-parameter subgroups and a doubly in- 

 finite system of three-parameter subgroups of type XIII. The fun- 

 damental group of XIII is tliree-parametered. The discussion of the 

 details of these groups belongs more properly to the theory of type 

 XIII and will be given in its proper place. 



Subgroup of type XII in Ge{Apl). — When m = and n=^0, the 

 one-dimensional transformations along 1 and in the pencil of planes 

 through 1 are both identical; hence all points on 1 and all planes 

 through 1 are invariant. The collineations in the planes through 1 are 



