242 



KANSAS UNIVERSITY SCIENCE BULLETIN. 



Part I. 



General analytic foryn of collineation in the i?^, 

 § 1. Consider the substitution T: 



X': 



Y' 

 Z' 



U' 



Oil X 4- Or: y + QiH Z + Un U + a 



asi X + o.-,2 y + a.,3 z + a^^u + Oi 

 a2i x + ai-,y + a^-.t z + Oj, u + a- 



Osi X + as-, y + a-,j z -{- a^^u + a-. 



a-ii X + 0.32 y + a:,A z + a,, v. + Oji 



Ooi a; + 052 y + a.5:i z + a^iU + Os, 



O41 X + a42 ?/ + 043 « + a44 u + a. 



asi X + a:,-2 y + asj z + as, u + 055 



X 

 F 

 Y 

 V 



V 



This substitution changes the set of numbers x, y, z, 

 u, into the set x' , //', z' , u' . If we take x, y, z, u, to 

 be the Cartesian coordinates of a point in the i?^ re- 

 ferred to four mutually perpendicular spaces as spaces 

 of reference, then the substitution transforms the point 

 whose coordinates are x, y, z, u, into the point x', y', 

 z' , u' . If we solve the four equations of T for x, y, 

 z, iL, we get a transformation of the same form as T, 

 viz., T-' : 



' x' + X21 y' + A:„ z> + ^41 ui + A.,1 __ X^ 

 Ai5 X' + A-,-, y' + A,r, z> + A45 W + Ai.-, ~ V' 



x = 



with three similar expressions for y, z, and u, where 

 A ;j is the co-factor of a^j in the matrix 

 (a) = 



Evidently 7" and T~' set up a one-to-one correspond- 

 ence between the points of the 11^. T' always exists 

 if A 9^ 0, A being the determinant of the matrix (a). 

 By T the point X=Y^Z=U=0 is transformed into 

 the origin x' = y' = z' — u' = o. The origin goes over 

 into the point 



/ "is/ 



/ass' 



(125/ 



/ass ' 



(135/ 



045/ 



:j- 



