109 
1916-17.] Adelphic Integral of Differential Equations. 
and now p Y and p 2 do not occur except in the arguments of trigonometric 
functions and in the terms (cq Pi + a 2 p 2 ). 
Now the equations 
_0W 0W R _0W a _dW 
qi ~Wi’ 
define a contact-transformation from the variables (q 1} q 2 , p x , p 2 ) to the 
variables (cq, a 2 , /3 t , /3 2 ) : so in terms of these new variables the differential 
equations take the form 
d aj 0H da. t oil d/3 1 0H d/3 2 0H nil 
dt d/3 } ’ dt d/3., ’ dt 0a j ’ c It 0a 
But we know that cq = Constant and « 2 = Constant are two of the integrals 
of the system, since l l and l 2 are constant : and therefore 
®= 0 , 
¥ ’ 
0H 
¥■2 
= o, 
so when H is expressed in terms of (cq, a 2 , /3 1 , /3 2 )> if w iH found to involve 
aj and a 2 only : and then the second pair of equations (11) give 
/3,= - + tl 
0a 1 
j 3 2 = - nH(a i’ + f2 
0a 2 
where e 1 and e 2 are arbitrary constants. 
Thus we have the complete solution of the dynamical system expressed 
by the equations 
0W 
0a j 
0W 
dpi 
0H(a 1 , a 2 ) 
0a l 
9ti 
t + 
: i » 
0W _ 
d P2 
0W 
0a o 
Q 
2 3 
dH(ai, a 2 ) 
0a o 
t + fo J 
where W is given by equation (10), and the four arbitrary constants of 
integration are (cq, a 2 , e x , e 2 ). On referring to the form of W, we see that 
these equations enable us to express and q 2 as purely trigonometric 
series, the arguments of the trigonometric functions being of the form 
m(3 l + wd 2 , 
where m and n are integers (positive, negative, or zero) and where /3 1 and 
/3 2 are linear functions of the time, given by equations (12). We have 
thus obtained expressions for the co-ordinates in terms of the time, by 
means of series in which the time occurs only in the arguments of 
trigonometric functions. 
It is moreover evident that a change in e v in which the other constants 
of integration (e 2 , cq, a 2 ) are left unaltered, does not affect either of the 
