DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
301 
Theorem II. — Suppose the function X to contain explicitly, besides the n variables 
X, ...<r„, another variable t, and also n constants a,, aj, ...a„; and in addition to the 
equations (50.), let the following be assumed : 
dx_ 
da^ '^’’""dan 
(54.) 
where 6 i, ... ^„are n other constants ; so that, by virtue of the 2 w equations (50.), ( 54 .), 
the 2 w variables Xj ... x„, 3/1 may be considered as functions of the 2 w constants 
^ 1 , and t. Then if from the equations (50.), (54.), and their total differ- 
ential coefficients with respect to the 2 w constants be eliminated, there will result 
the following 2 w simultaneous differential equations of the first order ; viz. — 
, , dTi 
^*~dyi^y^~~dxi'> 
where Z is a function of (which will in general also contain t expli- 
citly), and is given by the equation 
dX 
In this equation represents the partial differential coefficient of X taken with 
respect to t so far as t appears explicitly in the original expression for X in terms of 
x,...x„, a,...a„ and and the brackets indicate that «i, ... a„ are afterwards to be 
expressed in terms of the variables by means of the equations (50.), (arts. 5 , 6 .) 
Theorem III. — Let the supposition that the 2 w variables Xi...x„, y^...y^dxe, expressed 
in terms of the 2 w constants and t, be called Hypothesis I. ; and the converse suppo- 
sition that a,...a„, are expressed in terms of the 2n variables and t. Hypothe- 
sis II.; then will the following relations subsist : 
dxi 
r 
duj 
1 
#1 
ph— 
dbj dyi dttj 
dttj 
dxi dbj dXi 
( 57 .) 
(In each of these equations the first member refers to Hyp. I., and the second to 
Hyp. II. ; and since there is no connexion between the indices of the variables and 
those of the constants, the case of i=j has no peculiarity.) 
Theorem IV. — Let the symbol [jp, q~\ be an abbreviation for the expression 
t dp dq dp dq\ 
^\dyi dxi dxi dyij 
(where p, q are any functions of the 2n variables, which may also contain any other 
quantities explicitly ; and the differentiations are performed only so far as Xi,&c., 3 /„&c. 
appear explicitly in p, q)\ then if a^,... a„, b^,... be expressed {Hyp. II.) in terms of 
the 2n variables and t, the following equations subsist identically : 
[tti, i,] = — [bi, aj = 1 , [cii, y = = [pi, =0 . . . (58.) 
2 s 2 
