AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



2 3 4 5| 

 13 4 5 1 



225 



"12 3 4 5 1 



'1 1 



3 4 



5 1 





1 1 

 1 J 





1 

 2 







2 



1 











where, as evidence of the appropriateness of the notation, it is most important to 

 observe that if any element of a square array be taken it will be found to be such that 

 its row-numbers taken along with the numbers of the bordering element in the same 

 row with it form the full set 12 3 4 5, and its column-numbers taken along with the 

 numbers of the bordering element in the same column with it form the set 12 3 4 5 



123 



also. Thus the element 



] 24 



element in the same row with it, and 



2 

 column with it ; and 



'12345 1. 



in the square array of D has l| 4 5 | for the bordering 



'| 3 5 | for the bordering element in the same 



in E in like manner has the bordering elements l| 1 3 4 5 | and 



(11) In the verification of the foregoing developments it is very interesting to 

 observe the necessity which arises for using a linear relation like Kronecker's between 

 co-ordinate minors of an axisymmetric determinant. In the case of the 4th order the 

 term B, and in the case of the 5 th order the term D are dependent upon it for certain 

 of their details. 



Let us consider the mode in which the details of the term D are ascertained, these 

 being the Italic-letter coefficients of 6 2 , 6(f> , . . . in the development of the determinant 



a h+v g— 



h- v b f-\ 



g + p. f-X t 



r—T q + a p- 



n + \}r 7ft — x M 



The case of such a product as 6<p presents no difficulty. We have only got to delete 

 the 4th and 5th rows and the 3rd and 5th columns, and from the resulting minor 



a h + v /■ + t 



h — v b q — a 



g + ix f-X p + p 



strike out all the Greek letters. But when we come to the case of 6X , or 0/x , or dv the 

 process is not so simple. Fixing the attention on the in the place (4 , 5) we see that 

 it can go along with not only the X in the place (2,3) but also with the X in the place 

 (3,2). In the former case its Italic-letter cofactor is 



a 



h+v 



g-v- 



t+t 



n — \fs 



h — v 



b 



f+x 



q-a 



m+ x 



ff+H 



f-x 



c 



P + P 



l-<j> 



r — T 



q+a- 



p-p 



cl 



k+d 



7l + \fr 



m- x 



/+</> 



k-6 



e 



a h r 







9 f P 



n m k 



or 



13 5 

 12 4 



VOL. XXXIX. PART I. (NO. 7). 



2 L 



