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KANSAS UNIVERSITY SCIENCE BULLETIN. 



To show that (5) and (7) are identical collineations, 

 make (5) also homogeneous and take for the invariant 

 pentahedroid the frame of reference v^hose vertices are 



a — o o o o 1 



b — 1 o o 



c — o 1 o o o 



d — o o 1 o o 



e 



o 



Then both (5) and (7) reduce to the same canonic 

 form : 



(8) 



i> a-, 



J' Xi 

 i' Xi' 



= k Xi 



= k' X. 

 = k"' Xi 



and the two collineations must be identical. 



§8. The determinant of the form (7) consists of 

 twenty-five terms, each term being a determinant of 

 the fifth order, and factors easily into 



D = kk'k"k"' 



where Aij is the cof actor of a,-, in the first determinant 

 of this product. 



We can write this product in the following way with- 

 out altering its value : 



1 o o o o 

 k o o o 

 o o k' u o 

 o k" o 

 0000 k'" 



X 



The matrix of A is equal to the product of these 

 three matrices, and we have therefore factored our 

 transformation, which we may call T,, into the product 

 of three transformations with matrices (A), (k) and 

 (a). Now (A) is the inverse transformatian of (a). 

 Hence we have T, = Tr' K T, . 



