128 KANSAS UNIVERSITY SCIENCE BULLETIN. 



changed in equation (29). the value of A2 is not changed; but if A 

 and Ai are interchanged in (29), the value of A2 is changed, thus 

 showing that T and Ti are not commutative in G2(A). 



When T and Ti have both invariant points in common and ki = -jp 

 their resultant is the identical transformation ; but when T and Ti 

 have only one invariant point in common and ki = -£-> the resultant is 

 of type II. For, putting ki=-j- in (29), we find A2 = o; thus the 

 two invariant points of T2 coincide, and it is of type II. The value of 

 the constant t of T2 is found as follows : 



k,-l __!• kk t A-kA + kA.-A^ 



t== lim ^pL = li m 



j£j — 1 A 2 A A x 



(ir-xX 1 -*)- ( 31 ) 



Theorem 13. The group G2(A) contains oo 1 subgroups Gi(AA') 

 and one subgroup Gi(A). The transformations in G2(A) are not 

 commutative. The resultant of two transformations of type I in 



G2(A), for which ki = -^-and A'i not equal to A' is of type II. 



The three-parameter group G%. — It was shown (Theorem 1) that 

 the resultant of T and Ti, any two projective transformations of the 

 points on a line, is again a projective transformation. From this we 

 infer that all projective transformations of the points on a line form a 

 group. This is called the general projective group G3. It is a group 

 of three parameters ; for the equation of T contains three independent 

 parameters, viz., a : b : c : d. If these coefficients, a, b, c, d, be made to 

 vary continuously, all the resulting transformations belong to the 

 group G3 ; and conversely all transformations belonging to the above 

 group are obtained by continuously varying the coefficients in T. 

 Such a group is evidently continuous. If the equation of T be put 

 into the normal form, 



x, — A' _ 1 x — A' 



-> 



x, — A x — A 



the three natural parameters, A, A', k, may be made to vary con- 

 tinuously, thus generating the group G3. The group G3 contains oo 1 

 two-parameter groups G2(A), one for each point on the line. It con- 

 tains, as we have already shown, oo 2 groups Gi(AA') and oo 1 groups 

 Gi(A). 



§ 5. Transformations op Pencils of Lines and Planes. 



The theory sketched in the foregoing pages applies equally well to 

 the one-dimensional transformations of the lines of a flat pencil or the 

 planes of an axial pencil. There are two varieties of such transforma- 

 tions, viz., those with two invariant elements and those with only one 

 invariant element. 



In the first case let O be the vertex of a flat pencil, A and A' the two 



