1931 - 2 .] Dr Muir on the Theory of Jacobians. 
183 
where i, k may each have the values 0, 1, 2, . . . , ra, we see 
that from a previous result we have 
B* 
Afn- 
if fn+i within the brackets involves /, f \ , ... , f n _ x in place of 
x , x 1} ... , x n _ v From this it immediately follows that 
j J ±bb[ .... C = 
<fn m bfn+l 
dx n dx n+1 
bf n+m \ 
dx n+m ) ' 
But the theorem already obtained regarding the factorisation of 
any functional determinant gives 
Consequently, by substitution we have finally 
y ±bb[ ... b (m) = B m . V + d J- tfn+m 
" m ^ bx dx 1 dx n+m 
a result which Jacobi says is of frequent use in dealing with 
questions regarding determinants.* 
* A curious interest attaches to this result. On the right-hand side are 
two determinants whose elements are differential-quotients ; but the first, B, 
being a minor of the second, the total number of different elements is simply 
the number in the second determinant, vis., ( n + m + l ) 2 . On the left-hand 
side is a compound determinant of the (m + l) th order, each of whose elements 
is a determinant of the (?i + l) t]1 order ; nevertheless the number of different 
elements is again {n + m - f l) 2 and not (m + 1 2 (w + 1) 2 , because all the (m+1) 2 
elements of the compound determinant have the n 2 elements of B in common, 
and of the 2n + 1 which border these n 2 elements, only one, viz., the cofactor 
of B, is different throughout, each of the 2 n others being repeated m + 1 times, 
so that the total number of different elements is n 2 + {m + Yf + < 2n{m + l 2 ) 
(m + 1). Further, on both sides the degree in these {n + m + l) 2 differential- 
quotients is the same, being clearly (w + 1) (m + 1) on the left, and mn + 
{n + m + 1) on the right. It is thus at once suggested to us that the identity 
is not necessarily an identity connecting differential-quotients only, but is 
true of any {n + m + l) 2 elements whatever; and the suggestion is readily 
verified when the ubiquitous presence of B as a coaxial minor raises the 
suspicion that the identity must be an £ extensional.’ The case where n = 2 
and m = 3 is given on p. 215 of my text-book {Treatise on the Theory of 
Determinants ) in the form — 
