AUTOMOKPHIC LINEAR TRANSFORMATION OF A QTJADRIC. 



221 



Bearing this in mind it is not difficult to arrive at the final development in any- 

 given case. Thus, for the determinant of the 3rd order, we look first for terms of the 

 degree 3 + , and then for terms of the degree 1 + 2 . The aggregate of the former is 

 readily seen to be 



h < 



f 



9 



and, each of the latter being composed of an Italic letter with a Greek cofactor, the 

 aggregate is almost as easily seen to be 



X 



M 



V 



a 



h 



9 



h 



b 



f 



9 



f 



c 



We have thus 



a h + v g — fi 

 h- v b f+\ 

 g + /u f-\ c 



= 



a h g 





h b f 





9 f c 



+ 



\ 



M 



V 



a 



h 



9 



h 



b 



f 



9 



f 



c 



X' 



an identity which may also be viewed as giving an expression for the sum of a ternary 

 quadric and its discriminant. 



Next taking the determinant of the 4th order we first look for terms of the degree 

 4 + , then for terms of the degree 2 + 2 , and lastly for terms of the degree + 4 . 

 The aggregate of the first kind lies to hand as readily as before : those of the second 



kind are most easily obtained by seeking for the coefficients of p 2 , pa- , a- 2 } , 



which are seen to be 



a 



h 





a 



9 





a 



9 



h 



b 



i 



h 



f 



t 



9 



c 



and the aggregate of those of the last kind by dropping out all the Italic letters from 

 the determinant which thus becomes zero-axial skew with the Pfaffian equivalent 



V —fA T 



X — <r 



or 



| X ft v 



P <* 



T 



When we come to determinants of the 5 th order it is seen that there are still only 

 three kinds of terms, the degrees of which are 5 + 0, 3 + 2, 1+4: and in obtaining the 

 aggregates no new consideration is necessary. 



