1901-2.] Dr Muir on the Theory of Jacolians. 
The immediate result, on dividing both sides by R m , is 
173 
y + 0 Z df i . . . . y + dx . d fi ^ = y + ¥m+i m% ¥n m 
" — dx'dxj dx n " — 3/ 3/-L 3/ m " ~ dx m+1 dx n 
The theorem 
p _ V . *fl *fn 
tx ’ d f~Zj-dx 1 dx n ’ 
already obtained, is the special case of this where m = 0. Another 
special case is at the other extreme, viz., where m = w, when we 
have 
dx dx-j 
dx n 
W, 
dxdx. 
9 fn 
dx„ 
the generalisation — or the analogue, as Jacobi would seem to prefer 
to view it — of the theorem 
dy 1 
dx dx 
dy 
The next part (§ 10) of the subject relates to the case where 
the functions, whose functional determinant is sought, are not 
given explicitly in terms of the variables — where, in fact, we have 
n + 1 functions of x , x 1 , . . . , x n , /, f Y , . . . , f n , viz., 
F = 0, Fj = 0 , .. ., F n = 0, 
and where we are asked to find 
y + 'jt §/» 
~dx dx l dx n 
Differentiating any one of the F’s with respect to any one of the 
xta we have 
0F, 0F 3/ ; . . > dFjdf, 
dx k df'dx k dffdxje + df n 'dx k 5 
which may be viewed as giving an expression for - — • Using 
dx k 
this expression (n + l) 2 times we obtain for 
^ 
dx dx^ dx n 
an equivalent determinant each of whose elements is the sum of 
n + 1 products, and which from the multiplication-theorem we 
know to be equal to 
