DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
305 
functions of fli, &c., and again, through them, of the variables. We have then 
/ d,Ci , dci ,, 
and if in this equation the values of a', &c. be introduced from the formula (E.) of 
the last article, the following expression results : 
Ci={Z, cJ—P, cj) 
(in which the symbol {/>, q\ is used to denote 
y \ dp dq _ dp 1 
’'\dbk da^ dau db^} 
SO that by (59.) (Theorem V.) we have {p, g} = — [p, g] ; but in [p, q] p and q are 
considered as functions of a^, &c., b^. See., whilst in \_p, q~\ they are considered as 
functions of x,, &c., y-^. See.). Now, considering j», q as functions of Ci, &c., and through 
these, of «i, &c., we have (by reasoning exactly similar to that employed in deducing 
equation (24.), art. 9.) 
(the summation referring to all binary combinations of the indices a, jS). 
have, putting q—Ci, 
{p, = 
Hence we 
. (64.) 
and consequently the above expression for c- becomes 
e;={Z,c,} + 2.2,({c„c,}(Q,g-P,|)), (F.) 
an equation which is easily seen to become identical with (E.), art. 52, when C 1 ...C 2 ,. 
represent 
54. The simplest case is that in which the system of equations (63.), whose inte- 
grals are sought, are of the canonical form ; that is, where 
p _^W ^ _ dW 
dyt ’ dxi ’ 
W being a given function of the variables (with or without t). 
formula (E.) becomes 
,_dZ_d^ 
‘ dbi dbi 
> 
dz.(m 
dui dtti - 
In this case the 
(65.) 
whilst (F.) is easily found to be reducible, by the help of (64.), to either of the fol- 
lowing forms : 
c-={Z, c,} — {W, cj 
( 66 .) 
