356 
PROFESSOR DONKIN ON THE 
The definitions of the new coordinates |i, &c. will furnish m equations (which may 
explicitly contain t) by means of which |i, ... may be expressed as functions of 
.Cl, ... (with or wdthont t ) ; and conversely, by means of these m equations, together 
with the r equations of condition Lj=0, &c., the n variables X 2 , ... x„ may be 
expressed as functions of li, ... with or without t. When x^, ... x„ are so expressed, 
let them be represented by [x^), ... (x^). We shall have then 
x.= 
d{xi) d{Xi) , 
dt 
+ 
SO that x\, &c. are expressible (and in only one way) as functions of &c., 1), &c. 
If then the formula (D.) be transformed by expressing j?,, &c., x[, &c. in this manner, 
it becomes, as is well known, 
where (W) represents the result of transforming W as above ; and since &c. are 
now independent, this formula breaks up into the m separate equations 
/d{W)\' d(W) 
\ )~ d^i 
d{xi) , 
{x'.) 
(F-) 
Moreover, if we now define by the equation 
</(W) 
and put 
— (Vi ) + (Si)j?i+ ... + (|,„)^m, 
where (|i), &c. are expressed in terms of >?i, &c., we know already (art. 18.) that the 
system (F.) becomes 
/7\!/ //W 
(G.) 
’-dnr 
, d^ 
„ and of the n old variables 
Now let P be a function of the m new variables li, . 
yi, ... (with or without t), defined by the equation 
P= ('^i):^! + + . • . + Myn- 
d(W 
Since , and since (W) contains I-, &c., only through x'i, &c., we have, ob- 
, dW 
serving that -^=3/;, 
but from the equation {x'.) we have 
consequently 
d{x^) , d[x^ d{xn) 
’>‘=y^-ds;+y^s+-+y--^’ 
d^ 
an expression evidently equivalent to Thus ni may be d^ned by the equation 
