246 KANSAS UNIVERSITY SCIENCE BULLETIN. 



These ten equations are not independent of one an- 

 other, for, expanding the vanishing fourth-order deter- 

 minants of the matrix 



Xi X-i Xi X4 X5 



2/1 2/2 2/3 2/4 2/5 

 Xi X2 X3 Xi Xa 

 2/1 2/j 2/3 2/4 2/s 



we find that the p's satisfy the following five conditions: 



(1) Pi- p,n +Pn r'42 +Pu Pl3 = 



(2) P12 P35 + Pl3 P52 + Pl5 P23 = O 



(3) P12 P45 +Pl4 P52 +Pl5 P24 = O 



(4) Pl3 P45 +Pl4 P53 +Pl5 P34 = 

 (B) P:3 P45 +P24 Pol + P25 P34 = . 



Now any pair of these five equations is equivalent to 

 any other pair. Multiply (l)hyps,, (2) by p^^, and 

 subtract. This gives 



(1) Pi:) P42 P,i5 + Pl4 P2:i P.ir. + fij P25 P34 + Pi!, Pil P34 = O . 



Then if we multiply (4) by p^^, (5) hy Psi, and sub- 

 tract, we get identically (1). 



Hence the p's are subject to three independent con- 

 ditions. 



§ 6. In the same manner it can be shown that the 

 g's undergo a dependent linear transformation, whose 

 coefficients are the third-order determinants of (a) and 

 whose determinant = A" . The ten equations of this 

 transformation are not independent, but are tied up to 

 the same extent as the equations on the p's. 



Similarly we have a transformation on the s's of 

 which the coefficients are the fourth-order determinants 

 of (a) and whose determinant is = A*. 



Part III. 



Collineaiions in Normal Form. 

 § 7. In this section we shall develop through geo- 

 metrical considerations our collineation in what Pro- 

 fessor Newson has called the normal form. In this 

 form the parameters of the collineation are explicitly 



