DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
77 
and 
dX dQ, - 
^+^■=0, as before*; 
but the equations corresponding- to (10.) will not subsist unless X be homogeneous 
with respect to the r variables x lf x 2 , ... x r . 
5. Let us now suppose that the function X contains, explicitly, besides the n vari- 
ables jt 13 <z* 2 , ... another variable t, and also w constants a ls a 2 , ... a n ; and that these 
last are contained in such a way that the n equations 
dX fiX , dX_ 
da x 13 da 2 25 da n n 
( 11 .) 
would be algebraically sufficient to determine a„ a 2 , ... in terms of b x , b . 2 , &c., &c. 
Then taking X 4 = — (X) + (a 1 )^ 1 -b(a 2 )& 2 -|--- + («»)&„ 
(the brackets indicating that a,, a 2 , &c. are to be expressed as above supposed), we 
shall have, by the theorems of arts. 2 and 3, 
dX ± _ dX>_ dX,_ 
db 1 — a ' 3 db z ~ a ” "" db n ~ • V 12 -) 
and also, for all values of i, 
dX b dX , x 
o hi dxi ( 13 -) 
to which we may add 
dX,,_ dX_dY 
dt dt dt ( • ) 
Now assuming the 2w equations (5.) and (11.), namely (for all values of i), 
dX_ dX 
dxi y*’ dcii *’ 
we may suppose each of the 2 n variables x x , x 2 , ... y x , y 2 , .... to be expressed by means 
of them as a function of the 2 n constants a x , &c., b 1} & c., and t ; or, conversely, each 
of the 2rc constants to be expressed as a function of the variables x 13 &c.,y 15 &c., and t. 
On the former hypothesis each of the variables x 13 ...y 13 .... is given as an explicit, 
and on the latter as an implicit function of the single variable t, which we will con- 
sider as independent ; and total differentiation with respect to t will throughout this 
paper be denoted by accents, which will be used for no other purpose. Thus, p being 
any function of all the variables, we shall have 
^ dp 
P dt'dx x 
For the rest, we may, when necessary, distinguish the meanings of the various partial 
* Although these theorems, as stated in the text, are more general in form than those of the preceding 
article, they may, under another point of view, be considered as particular cases of them, and may in this way 
be best established. 
