314 
PIIOFESSOR DONKIN ON THE 
[It follows from Theorem I. art. 49, that the system (73.) is equivalent to each of 
the following- : 
dQ dO. 
dxi d^i 
in which Q= — and is expressed in terms of x^, ... |i, ... or 
in which R= — and is expressed in terms of 3 / 1 , rt^, ... rin\ or lastly, 
dS dS 
dr}i dXi 
in which S= — P+SiCj^i^i+g,;?,), and is expressed in terms of .Tj, ... ... v 
Any one of these forms might be used ; but I shall employ the form (73.) for 
reasons of convenience.] 
63. Inasmuch as the equations (73.) of the last article are of the same general 
form as the system (54.), art. 49, all the conclusions deduced from that form will 
subsist, mutatis mutandis ; so that Ave may apply the Theorems (II.), (HI.)? (IV.), (V.), 
art. 49, by merely changing X into P, and 
... 3 / 1 ’ ••• ... K, respectively into 
il? ... ‘Snj ^15 ... y ... 3^n5 .^15 ... 
observing that instead of x\, y\ we must now write We thus obtain the fol- 
lowing relations : 
^ d^^ ^ d'^ . 
d.t dfl: d.t. dP; v/ •> 
dt drti' dt d^i 
where T is a function of li, &c., ni, &c. and t, defined by the equation 
=-( 
dtj’ 
(75.) 
the brackets indicating that the expressions for y^, ... y^ in terms of the new variables 
&c., rii, &c., are to be substituted in after the differentiation with respect to t ; 
which is performed so far as t appears explicitly in the original expression for P as a 
function of ... L, 3/1 ... 3 /„ and t. (See Theorem II.) 
We have also the system 
d^i dxj d^i dyj ' 
dj/j dr]i dxj drii 
> 
drii dxj dtii dyj 
dyj d^i dxj d^i ■ 
(76.) 
* For in the original theorems is the same thing as the dilFerential coefficient of Xi taken with respect to t, 
as t appears explicitly in the expression for Xi in terms of «p &c., b-y, &c. and t ; the analogous quantity in the 
present case is therefore the differential coefficient of taken with respect to t, as t appears explicitly in the 
expression for in terms of Xy, &c., yy, &c. and t. But this must not now be denoted by inasmuch as 
x^, &.C., yy, &c. are themselves afterwards to be considered as functions of t. 
