( 623 ) 
the method of Hamilton-Jacobi in order that the constants we obtain 
may form a canonical system. 
If «!, « a , ft and ft are the canonical constants then by substitution 
of the expressions found for x and y in R this R will become a 
function of a lt « 3 , ft, ft and t. The variability of the a’s and fts 
with the time is then given by: 
detj_ dR da 2 _ dR dft _ dR dp 2 _ dR 
dt ~~ dft ’ dt dft ’ dt ~ da 1 ’ iU ~ da, ‘ ’ ^ 
In case 7? is a function of the a’s and the fts alone, and conse¬ 
quently does not contain t explicitly, the system has as an integral: 
R = constant .(7) 
§ 5. If now we solve the equations 
y+4n,- J( = oj 
dR . dR 
arising from (5) by omission of the terms — and —, 
the method Hamilton-Jacobi we may arrive at: 
according to 
y = - 
^nft-^ft); 
where a 1} a,, ft, ft form a canonical system of constants. We must 
suppose a x and a 2 to be of order h .* as the amplitudes of the a;-and 
y-vibrations must be of order h. 
Substitution of (8) in R= — dj?y furnishes 3 terms: 
aVa. 
S (2n 1 « + 4« 1 ft), 
8» t * 
► {4n x t + 4» t (ft -f ft)} and 
s 4«! (ft ft)* 
each term multiplied by — d 2 ■ 
The first two terms contain t explicitly; setting aside the variability 
of the a’s and fts we can say that those terms are periodical, whilst 
the period is comparable to that of the principal vibrations. The 
last term, however, does not contain t explicitly. Only this last term 
is of importance for the first approximation; the two others we omit 
(we shall revert to this in $ 6). 
We therefore take: 
