AUTOMORPHIC LINEAR TRANSFORMATION OF A QUADRIC. 



223 



1 123 



Loo being used to denote the minor obtained by deleting the 4th and 5th rows and 



4th and 5th columns of the determinant preceding D , and where 



+ ... 



E = a| X <r x 



2 + 2h\ X a X 



•1 P t y\r 



P <t> 

 6 



P <t> 



e 



P 

 6 



(10) The law of formation of the bipartites B and D, when properly looked at, is 

 very simple. The square array in each case consists of the secondary minors of the 

 axisymmetric determinant which immediately precedes it, — in other words, the 

 elements of the square array of B are exactly the elements of the second compound of 

 the determinant which precedes it, and the corresponding elements of D are the 

 elements of the third compound. The bordering elements, of D say, are the elements 

 of the quasi-Pfaffian 



\ V fl t y[? 

 X a- x 



P <t> 

 & 



taken in a backward order, viz., in the order ; <£ , p ; \, <r , \ ; \Ja,t,/k,i>. 



The law of formation of E is equally simple. Each term is the product of three 

 factors, the first being an element of the preceding axisymmetric determinant, and the 

 two others being primary minors of the quasi-Pfaffian just referred to. The Pfaffian 

 factors which are to accompany any element are determined by the position of that 

 element in the determinant-array, the law being that when the element belongs to the 

 place (r,s) we delete the r tb frame-line of the quasi-Pfaffian to obtain the one factor and 

 the s th frame-line to obtain the other. Consequently, if we denote the quasi-Pfaffian by 

 ^fand the minor of it obtained by deleting the r th frame-line byff {r , we may write each 

 term of E in the form 



We are thus further led to see that E also is a bipartite having the beautifully simple 

 form 



ff*ff9ff»ff*ff<* 



ff {5 > 



and it is suggested to inquire whether there be not a mode of writing D and E, and 

 perhaps even the term preceding D , which will put in evidence the fact that they are 

 members of a series advancing according to a definite law. 



a 



h 



9 



r 



n 



h 



b 



f 



1 



m 



9 



f 



c 



P 



I 



r 



1 



P 



d 



k 



n 



m 



I 



k 



e 



