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Proceedings of the Royal Society of Edinburgh. [Sess. 
To the adelphic transformation there corresponds an integral of the 
dynamical system : this integral we shall call the adelphic integral of the 
system. 
As there is only one really distinct adelphic transformation of a 
given dynamical system with two degrees of freedom, so there is only 
one really distinct adelphic integral : all other adelphic integrals may 
be obtained from this by combining it in various ways with the integral 
of energy.* 
In practically all the known soluble problems of dynamics with two 
degrees of freedom, the integral which enables us to effect the solution is 
an adelphic integral. Thus, when the trajectories are the geodesics on an 
ellipsoid, the adelphic integral is the equation pd = Constant. When the 
problem is that of two centres of gravitation, the adelphic integral is Euler’s 
integral of that problem. When the solubility of the problem is due to the 
presence of an ignorable co-ordinate, say q 2 , the corresponding integral 
(namely p 2 = Constant) is adelphic. 
In the present paper we shall find the adelphic integral for the general 
dynamical system with two degrees of freedom, and make this the basis 
from which to complete the integration of the system. It will appear that 
by this procedure we are enabled to overcome the difficulty formulated in 
Poincare’s celebrated theorem, that “ the series of Celestial Mechanics, if 
they converge at all, cannot converge uniformly for all values of the time 
on the one hand, and on the other hand for all values of the constants 
■comprised between certain limits.” This unsatisfactory feature of the 
usual series springs from peculiarities which are deep-seated in the 
nature of the problem, and which are difficult to discern by the 
methods of solution employed in Celestial Mechanics. By fixing our 
attention in the first place on a single integral of the dynamical 
system, rather than attempting at once a complete solution, we shall 
find what these peculiarities are ; for they manifest themselves very 
clearly in connection with the adelphic integral, and (as we shall see) 
may be so taken account of in its determination, that they no longer 
remain to trouble us in the final stages of the complete integration 
of the dynamical system. 
§ 3. Tlte form of the Hamiltonian function . — We now proceed to 
inquire how the adelphic integral of a dynamical system with two degrees 
of freedom may be determined. 
* The integral of energy corresponds to that infinitesimal transformation which changes 
every orbit into itself, each point of an orbit being displaced in the direction of the tangent 
to the orbit. 
