304 
PROFESSOR DONKIN ON THE 
{Hyp. II.), c becomes also a function of the same ; and we have 
y ! dc d7l 
dtti dbi 
dc dTi 
dbi dtti. 
( 61 .) 
(see the last article). But, by Theorem V., this becomes 
§=[^.Z] («2-) 
It is worth observing that both this equation and (60.) might have been obtained 
• • dc 
indirectly as follows. Since c is constant, we have c'=0 ; that is, ^^+[Z, c]=0 (see 
(32.), art. 22.); this gives (62.), since [Z, c] = — [f, Z], and again, by Theorem V., is 
changed into (61.); and if, in the latter, we put successively c=a,, c=Z»,-, we obtain 
the system (60.). 
Section V. — On the Variation of Elements. 
52. The following general problem includes, I believe, all the cases which occur in 
practice. Let P,, ... P„, Qi, .... Q„ be any functions whatever of the 2w variables 
r,, ... i/i, ...pn and t. It is required to express the 2w integrals of the system of 
2n simultaneous differential equations of the first order 
y'i=Qi (63.) 
in the same form as the integrals (supposed given) of the canonical system 
(I-) 
by substituting functions of t for the constant elements of the latter system. 
Suppose a normal solution (see end of art. 50.) of the system (I.) to be employed. 
The elements hi represent the same functions of &c., y-^, &c. and t as before, but 
are now variable ; consequently we have 
! ddi 
[dtti , dtti da, 
dt 
+2, 
[ 1 dxj ^^dyjy 
with a similar expression for &■. But, by equations (57.) and (60.), these are imme- 
diately transformed into the following: 
dZ 
dxs 
iy, 
dbi 
dbi 
^_p 
dui ^ dai\ 
(E.) 
where Z, Q^, P^, y^ in the second members are supposed to be expressed {Hyp. I.) 
in terms of the elements and t. Thus the system (63.) is transformed into a system 
involving the new variables hi, instead of the original variables a?,, 
53. If, instead of employing a set of normal integrals of the pattern system (I.), we 
take any complete set of integrals c,, Cg, ... Cg,,, then c,, &c. may be considered as 
