298 KANSAS UNIVERSITY SCIENCE BULLETIN. 



reciprocal polars with regard to the quadric surface which they leave 

 invariant. 



In each 1 Gi(ll'mm'F) r , where r is a rational number, there ex- 

 ists one of these collineations of type X, which we shall therefore 

 term a singular transformation. The value of k and the position of 

 the axes in each case depends upon the value of r — i. e., whether it 

 be odd over odd, even over odd, odd over even. If we combine oc 1 of 

 the one-parameter groups of type I, where r is odd over odd, we ob- 

 tain no group of type X ; nor, if we combine each of the singular 

 transformations with the transformations of type I of the group, or 

 with those of type XI occurring in this group, do we obtain any new 

 singular transformations. However, if r be even over odd, we see 

 ( art. 35 ) that the group of type XI and the singular transformation 

 of type X have their invariant systems lying in opposite directions. 

 But a combination of a collineation of type X and one of type XI in 

 this position we have shown ( art. 23 ) to be of type III. Each 

 singular transformation of type X combined with each of the group of 

 coliineations of type XI gives such a resultant, These resultants 

 do not form a continuous group, but there exists a discrete set of 

 them, and they must be classed as singular transformations of type 

 III. In the third case, r being odd over even, the axes of invariant 

 points are reciprocal polars with regard to the given quadric, and the 

 combination of this transformation with those of types I, IX, XI, 

 gives only type I. 



For singular transformations in type I, we may then state in sym- 

 bolic language : 



J Gi( ll'mm'F) r= ° Z U 'S. T.hnm A F) k = _ , 



o 



1 G2(ll'mmF) r= «Zcx 1 10 S.T.(^ 77/ AF) k = _, 



o 



1 G 3 (lmF) r= «Zoo 1 10 S.T.(mm'AF) k = _ 1 



1 Gi(ll'mm'F) r = ^Z K, S.T.(«VF) k = _ 1 



o 



1 G 2 (U'mF) r= «Zoo 1 10 S.T.(«>F) k = _H 3 S.T.(ll'mF). 



o 



1 G 3 (lmF) r= «Zoo 1 ^S.T.tZT^F^-i+oc 1 »S.T.(lTmF) 



o 



1 Gi(ll'mm'F) r== iiZ 10 S.T.(ll'mm'w;i'F) k . M 



e 



1 G 2 (ll'raF) r= oZcc 1 10 S.T.(ll'mm'7rVF) k = +1 



e 



>G3(ll'F) r= ° Za 2 10 S.T.(lT'm m'77VF) k= +1 . 



e 



It should be carefully noted that these transformations of type X 

 cannot combine to form a group, since the resultant of two of them, 



