6 
notes on lagrange’s theorem. 
[101 
of variables x, y, z... are connected with as many other variables v, v, w ... by 
the same number of equations (so that the variables of each set may be considered 
as functions of those of the other set) the quotient of the expressions dxdy ... and 
dudv ... is equal to the quotient of two determinants formed with the functions which 
equated to zero express the relations between the two sets of variables ; the former 
with the differential coefficients of these functions with respect to u, v..., the latter 
with the differential coefficients with respect to x, y .... Consequently the notation 
^d d ma y be considered as representing the quotient of these determinants. This 
being premised, if we write 
x — u — hd (x, y ...) — 0, 
y — v — k(f> (x, y ...) = 0, 
then the formula in question is 
F(x, y ...) 
dxdy ... 
dudv ... 
8 hS,, 8 
U U U V 
ghÛT-kÿ... ^ 
if for shortness the letters 6, F denote what the corresponding functions become 
when u, v, ... are substituted for x, y, .... Let denote the value which , 
considered as a function of x, y..., assumes when these variables are changed into 
u, v, ..., we have 
V = 
1 — h8 u 0, — h8 v 0 ... 
— fc8 u (f>, 1 — Jc8 0 <f)... 
By changing the function F, we obtain 
F(x, y ...) = 8 u hS/ ‘ 8 v kdk ... e h6+k *- F V ; 
where, however, it must be remembered that the h, k,..., in so far as they enter into 
the function V, are not affected by the symbols h8i ly k8 k , ... In order that we may 
consider them to be so affected, it is necessary in the function V to replace h, k, &c. 
h k 
by r- , ^ , &c. Also, after this is done, observing that the symbols h8 u 0. h8„6 ... affect 
o u o v 
a function d ld+k4>+ - F, the symbols h8 u 6, h8 v 6,... may be replaced by 8 U 9 , 8 v e ,..., where 
the 6 is not an index, but an affix denoting that the differentiation is only to be 
performed with respect to u, v ... so far as these variables respectively enter into 
the function 0. Transforming the other lines of the determinant in the same manner, 
and taking out from 8 u hS '‘ S/ 5 * ... the factor 8 U 8 V ... in order to multiply this last 
factor into the determinant, we obtain 
F (x, y ...) = 8 u hs >~ 1 8 “‘- 1 ... e h6 + k *- F □ ; 
where 
□ 
8 —8 6 —8 * 
U U U U 9 L U y • • • 
— $ * s - a <t> 
v v , V v u v ,