508 Dr. T. Muir on the Jacobian of the 



being in fact the quotient of the former by the latter, and 

 then of course evaluating these and performing the division. 

 The expressions which he arrived at for his subsidiary 

 Jacobians are not very pleasing in form, viz. : 



1 I i=l 



(-i)*- ,1= n(i-»i). \r,. I -("-''RfclffH,, 1 . 



but the quotient to which they lead is less forbidding, viz. : 



(-i)w-« | n.|"-VnM . 



A glance at this suffices to suggest that a loss of simplicity 

 may have occurred through specialization ; and a little 

 examination of the form of the elements of the derived 

 determinant | p\ n | makes clear that the more appropriate 

 and more promising object of investigation is that which is 

 indicated in the title of the present paper. 



(2) In a general axisymmetric determinant of the nth 

 order there are %n(n + l) different elements, and the same 

 number of different primary minors ; consequently the 

 Jacobian of the latter with respect to the former must be a 

 determinant of the order %n(n + 1). Further, as each primary 

 minor is of the (n — l)th degree in the elements involved in 

 it, its differential-quotient with respect to any one of the 

 latter will be of the (n — 2)th degree, and therefore the 

 degree of the Jacobian in question will be not higher than 

 ^n(n+l)(n — 2). It will be seen presently that this degree 

 is attained by the Jacobian containing as a factor the 

 jt(n + J)(?i — 2)th power of the original determinant. 



(3) For the purposes of proof it is necessary to draw 

 attention to two results regarding determinants of special 

 form. The one, given by Ferrers* about the year 1857, 

 is almost self-evident, viz. : 



l + a 2 1 ... 1 

 1 l + oa ... 1 



/. 1.1. 1 \ 



= a,a 9 . . . a,i 1+ + +...+ — ) 



I + Cln 



The other is not so easily stated. It concerns a determinant 

 of the order -^(w-fl) whose elements are derivable in a 

 peculiar way from those of a determinant of the ?ith order, 



* Ferrers, N. M., " Two Elementary Theorems in Determinants/' 

 Quarterly Journ. of Math. i. p. 364. 



