AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC 



227 



(12) The procedure followed in § 9 is equally effective in finding the like develop- 

 ment of the primary minors of the determinant. For example, let us consider in the 

 determinant of the 4th order the cofactor of the element (13), viz. : — 



h — v b q — <r 

 y + M f-X p + p 

 r—T q + a d 



Here the aggregate of terms of the degree 3 + is 



h 



9 



r 



The aggregate of terms of the degree 2 + 1 



h i 



1 



9 / 



P 



r q 



d 





k 



I -a 



1 



h . 



V i 



— vbq 















9 f p 



I q . 



+ 



g -X p \ + 

 r a- d 



M- f P 

 —rq d 



J 





= - 



- p 



h r 



- a 



h r 



— X h r | — t 



b q 



~ M 



b q 



— V 



f P 1 







b q 





f P 



9. 



d \ 



f P 





q d 





q d J 



and as the coefficients of the Greek letters here are determinants of the 2nd order 

 formable from the 2nd and 4th rows of D, this may be written 



h b f q 



r q p d 



The aggregate of terms of the degree 1 + 2 



-V b 



(p , <r , X, t , m , v). 



— <7 

 P 



+ 



-r q 



a 





— v 



• 1 



P 



+ 



P- 



— X p 



, 





— T 



a d 



— a 



— T 



V 



h 



b 



9. 



9 



f 



P 



r 



2 



d 



where the borders p , <r , X and — o- , 



v are got from the Pfafhan 



v — fX 



X 



by deleting the 1st and 3rd frame-lines respectively and changing one sign. 

 Lastly, the aggregate of terms of the degree + 3 



-v 

 M 



-T 



■A 



— <r 

 P 



