318 
PROFESSOR DONKIN ON THE 
variables defined by the equations 
dW 
[It is to be observed that this last transformation is not, in general, equivalent 
to expressing the original Z in terms of the new variables ; for the original Z 
is — W+2(j?i3/i) (art. 66.), and the analogous expression derived from (81.) is 
— W+2(||;?i), which is not, in general, equivalent to the former. It will be seen 
presently that the two expressions are equivalent when the equations connecting 
Xi, ...x„ with li, ...L do not involve t explicitly.] 
68. Following the.analogy of the dynamical equations, I shall adopt the following 
as the 
Definition of a transformation of coordinates. 
The original equations (I.), art. 64, having been changed into the form (80.), art. 66, 
let ll, ^25 be n new variables connected with the n variables x,, Xj, ... x„ by n equa- 
tions, which may also involve t explicitly ; and let be w other new variables 
dW 
defined by the equations (where W has been expressed in terms of |i, &c., &c.). 
By means of the 2n assumed relations, the 2n original variables x,, ...x„, 3/1, ...?/„ can 
be expressed as functions of the 2w new variables ?7 i, Let this substi- 
tution be called a ^^transformation of coordinates." 
It has been seen in the preceding article that the original equations (I.) are changed 
by a transformation of coordinates into a system of the same form, which however 
cannot in general be obtained by merely expressing Z in terms of the new variables. 
But we are not at liberty to assume (and it is not generally true) that a change of 
the system (I.) into another of the same form is a normal transformation (art. 62.). It 
has already been stated, however'(end of art. 65.), that this is true in the present case ; 
a proposition which I proceed to establish. 
69. Every transformation of coordinates is a normal transformation. 
To prove this theorem, we have to show that every transformation of the kind 
described in the last article is also of the kind defined in art. 62 ; in other words, 
that it is possible to assign a function of li, ... |„, 3/1, (with or without t), such 
that the given relations between Xj, &c., &c., which define the transformation of 
coordinates, shall be equivalent to the system of equations 
dV dV 
Take P =(^1)3/1+ (•^2)3/2 +..- + (a?«)3/« (82.) 
(the brackets indicating that x„ ... x„ are to be expressed in terms of ... |„ ; so that 
P is a function of ...|„, 3/,, ...3/„, with or without t according as the equations con- 
necting x„ ...x„ with ...|„ do or do not contain t explicitly). Then P is the func- 
