292 KANSAS UNIVERSITY SCIENCE BULLETIN. 



surface the resultant transformation evidently leaves invariant the 

 figure (l 2 l'mm' 2 ). In the plane of I'm we have the resultant of an 

 elation and perspective collineation, the axis of the elation passing 

 through the vertex of the prospective collineation. This is a per- 

 spective collineation having the same vertex and axis as the first. 

 The complete invariant figure of the space collineation is (lil'mim'^F), 

 which is that of type VI. Taken in any other position, the two com- 

 ponent transformations result in one of type I. 



§ 5. Type III. 



26. The combinations of type III, first class, are comparatively 

 simple. But one thing is to be noted : the resultant of a transforma- 

 tion with one of any other type is, in general, never anything other 

 than type II or type I ; but type III may be combined with any other 

 type to produce a transformation of type I. 



3 T(ll'mF) + 3 Ti(ll'mF) == 3 T 2 (U'mF) 

 3 T(ll'mF) + 3 Ti(ll'imF) = 3 T 2 (ll'2mF) 

 3 T(ll'mF) + 3 Ti(lil'imF) = 3 T 2 (l 2 r 2 mF) 

 3 T(ll'mF) + »Ti(hl'imiF) = = 1 T(M'>mjm'sF) 

 3 T(ll'mF) + 3 Ti(H'miF) = = 1 T(ll'm 2 m' 2 F) 

 3 T(ll'mF) + 3 Ti(ll'imiF) = = 1 T(ll' a m 8 m'aF) 

 3 T(ll'mF) + n T(l /i F) ==»Ti(ll / imF) 

 3 T(ll'mF) -f u T(1im F) == 3 Ti(l 2 r 2 mF) 

 3 T(ll'mF) ■+ n T(lmAF) == 3 Ti(ll'mF) 

 3 T(ll'mF) + u T(m'AP) = 1 T(ll'm 2 m' 2 F) 

 3 T(ll'mF) + 10 T(ZZ>F) = 3 Ti(ll'inF) 

 3 T ( 11'mF ) + 10 T ( U\ tx. F ) == 3 Ti ( ll' 2 raF ) 

 3 T(ll'mF) + 10 T(Wi/*F) == 3 T 1 (l 2 l' 2 mF) 

 3 T(ll'mF) + 10 T(«im'XP)= = 1 T(ll / mm , »P) 

 3 T(ll'mF) + 10 T(mim'\F) == 1 T(ll'm,m',.F) 

 3 T(ll'mF) -f 9 T(lmpj5F) = 3 Ti(U' 2 mF) 

 3 T(ll'mF) + 9 T(lmp^F) == 3 Ti(ll' 2 m 2 F) 

 3 T(ll'mF) + y T(limpi6F) = = 1 T(l»l'smm' 8 F). 



§(5. Type I. 



27. The resultant of a transformation of type I with any other 

 transformation is, in general, of type I. Moreover, a transformation 

 of type I may be resolved into components of any one of the five 

 types discussed in this paper. 



In all the work of the preceding articles, it must be borne in mind 



