THE DIFFERENTIATION OF A CONTINUANT. 



211 



the differential-quotient with respect to an}^ element is the same as the cofactor of that 



element. Hence instead of 



SK 3 2 K 



da,' da h da„ +1 ' 

 we may write 



cof a ln cof a h a, l+l . 



(6) When the two elements used as independent variables are consecutive in the 

 main diagonal, the result is the same as is got by one differentiation with respect to an 

 element of the minor diagonal. For, taking dK/da h as above, we have 



S 2 K 



da h da 



»™/!+l 



\«! a 2 . . . a,,_! 



,fflft+2 



a„_, a, 



dK 



db h 



Corroboration of this is found m the fact that one of the two-line minors of K is 



'' " or a h a, l+1 + b h , 



~ l a h+\ I 



thus ensuring that the cofactor of a h a h+1 is the same as the cofactor of b h . 



(7) When the continuant concerned has unit elements in the minor diagonal the 

 rule for mechanically obtaining the differential-quotient is still simpler, the only element 

 to be deleted being that which is taken as the independent variable. Thus if 



K = (a v a 2 , a s , a 4 , a 5 , a 6 , a.) 



we have 

 and 



3K / W N 



— = (a v a 2 , a s , a 4 ) (a , a 7 ) , 



9 2 K 



3a ft 3a 3 



= (Oi, « 2 ) • K) • («6. %) • 



(8) The foregoing considerations have been suggested by a curious theorem which 

 has turned up recently in the course of an investigation connected with Hessians. 



The Hessian, it will be remembered, is always axisymmetric : and, if the originating 

 function be linear in each of the variables, its second differential-quotient with respect 

 to any variable will be 0, — that is to say, the Hessian will be zero-axial as well as axi- 

 symmetric. In this case, consequently, there arises a semi-quadrate array which seems 

 worthy of study. 



(9) Taking the case where the originating function is K(a,b,c,d), i.e. abed + ad + cd + 

 ab+l we have 



I 32R 3 2 K 8 2 K 



da.db da.de da.dd 

 8 2 K 3 2 K 

 db.de db.dd 



a 2 K 



dc.dd 



cd + 1 bd be+1 

 ad ae 

 ab + 1 



