POND: COLLINEATIONS IN FOUR DIMENSIONS. 243 



All points of the space V ^ go into the points of 

 the space at infinity. 



Since T~' transforms points of the space V'=0 to 

 infinity, T must transform the points of the space at in- 

 finity into the points of V' = 0. T and T~' may be 

 written in homogeneous form as follows: 



(1) T: p x'i = a,; .T^ + tt;, X, + a,-.* X; -\- a^ x^ + a,-., x^ , 



\Z) 1 . pXi ^= A liX i-\- AgiX ,-\- A^iX g-r A.^iX ^-\- 

 A,iX, {1 = 1,2,3,^,5), 



where Aij has the same meaning as before. 



§ 2. T transforms lines into lines ; for if x, y, and 

 z, are three collinear points, a relation exists : 



Zi = c,Xi + c,yi {i = l,2,3,U,5) . 



Transform the three points by T into x', y', and z'. 



Then it appears that the same linear relation exists 

 between x/, y/, and z/, namely : 



Z/ = c,Xi' -{- c,y/ , 

 which means that x', y', and z' are collinear points also, 

 and therefore lines go over into lines under T, and 

 hence T is a collineation. A similar argument shows 

 T sends planes into planes and spaces into spaces. 



§ 3. By means of T we can transform any six points 

 of the R^ into any other six points ; for, if we lay down 

 the conditions that a set of six given points be trans- 

 formed into any other six points by T we obtain a set 

 of thirty equations in thirty-one unknowns : the twenty- 

 five parameters of ( a ) and six p's. 



There is a solution of these 31 equations for which no p 

 is and all other solutions are proportional to this one. 

 Hence six points can be transformed into any other six 

 points in one and only one way, and in doing so we de- 

 termine the collineation uniquely. 



2— Univ. Sci. Bull., Vol. VII. No. 14. 



