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KANSAS UNIVERSITY SCIENCE BULLETIN. 



p"X"x = 



which solved for x"i gives 



Xl X-i Xi Xi Xh 



All All A\" A\k A\h a'ii 



A21 A22 A-i-.'. A'i\ Aih a'i2 



Ai\ Ail Ass A34 A-^i a'\i 



All Aw A43 A44 A« a'ii 



Asi A.',2 Ar,3 Am Ass a'js 



This is evidently T^ , the resultant of T and Tj . 



§11. Cross-ratio of the Resultant. The form of the 

 cross-ratio of the resultant of two collineations can 

 easily be shown by considering the canonic forms of 

 the collineations. 

 Suppose T and Tj in canonic form are 

 T : pXi' = k^'^ Xi, 

 T, : p a;/' = A;*** x/ . 

 By eliminating x', we have 



that is to say : the cross-ratio of the resultant of two 

 collineations along a line invariant under both collinea- 

 tions is equal to the product of the corresponding cross- 

 ratios of the two collineations. 



§ 12. Roots of the Characteristic Equation. It was 

 shown in § 5 that there are five values of p for which a 

 point is left invariant under T. Our normal form of 

 T gives an easy solution for these five roots of the char- 

 acteristic equation, showing their relation to the cross- 

 ratios and the invariant points. The five roots must 

 satisfy identities of the following form, where p„ is the 

 value of p for which a point a is left invariant. 



Pa^i 



ai a-2 as a* as 

 ai a-2 as at as ai 

 61 bi hi hi hb k hi 

 c\ C2 ci C4 Co k' Ci 

 di di di di db k "di 

 ei 62 es ei es k'" e\ 



