244 KANSAS UNIVERSITY SCIENCE BULLETIN. 



§ 4. What points, if any, are left invariant under Tl 

 Any such points must satisfy the conditions 



pXi = anXi + ai-^x,. + ai X3 + aitX, + a,5Xs (i = l, 2,3, J,, 5) , 



or, 



(3) (aii = /J) xi+ai2X2-\-aijX3 + aMXi + aiiX5 = o 



a^iXi + (ai2-l>)x2 + a-nX3 + a-^Xi + 035X5 = o 

 031 a;i 4- a32a;2 + (033 — l')Xi + a-nX, + 0352:5 = 

 O41 Xi + 042X2 + 0)3*3 + (044 — /')a;4 + aj5X5=o 



O51 Xi + 052X2 + 053X3 + 05, X4+ (O55 — /')X5=0 . 



The necessary and sufficient condition that this sys- 

 tem of equations has other solutions than x^ — x^^ 

 Xs = x^=Xs = o is that their eliminant = o . 



This is a quintic in p and in general has five distinct 

 roots, for each one of which we get a separate solu- 

 tion of (3) . 



Hence, in general, T leaves invariant a figure con- 

 sisting of five distinct points and their joins, which are 

 ten lines, ten planes, and five spaces. 



The equation in p is called the characteristic equa- 

 tion of the collineation, and in a later section we shall 

 investigate it more closely to discover what variations 

 may occur in this invariant configuration. 



Part II. 



Dependent Transformation^. 



§ 5. By the same argument that is used for space of 

 lower order we can show that the ten determinants of 

 the second order of the matrix 



Xi Xa X3 X, X5 



2/1 3/2 2/3 ?/4 2/6 



determine uniquely the line through the points x and 

 y and can be used as the homogeneous coordinates of 

 the line. 



