1916-17.] Adelphic Integral of Differential Equations. 
95 
VII. — On the Adelphic Integral of the Differential Equations 
of Dynamics. By Professor E. T. Whittaker, F.R.S. 
(MS. received September 21, 1916. Read November 20, 1916.) 
§ 1. Ordinary and singular 'periodic solutions of a dynamical system . — 
The present paper is concerned with the motion of dynamical systems 
which possess an integral of energy. To fix ideas, we shall suppose that 
the system has two degrees of freedom, so that the equations of motion 
in generalised co-ordinates may be written in Hamilton’s form 
dq x 0H dq 9 0H dp v 0H dp 9 0H . 
dt dp l ’ dt dp 2 ’ dt dq l ’ dt dq 2 
where (q v q 2 ) are the generalised co-ordinates, (p v p 2 ) are the generalised 
momenta, and where FI is a function of (q v q 2 , p v p 2 ) which represents the 
sum of the kinetic and potential energies. 
The successive states of the system may be illustrated by the motion 
of a point whose co-ordinates referred to the axes are (q v q 2 ) : the curve 
described by such a point is called a trajectory. Particular interest 
attaches to those trajectories which are closed curves : these are known as 
periodic solutions. 
I wish to draw attention in the first place to a distinction which should 
be made in regard to these periodic solutions ; the matter may perhaps be 
elucidated most readily by considering a particular problem, namely, that of 
the motion of a particle on the surface of an ellipsoid under no external 
forces. The particle describes a geodesic on the surface, so the periodic 
solutions are simply those geodesics which are closed curves. Now for a 
geodesic on. an ellipsoid we have Joachimstal’s equation 
pd — constant, 
where p denotes the perpendicular from the centre of the ellipsoid on the 
tangent-plane at the point, and d is the diameter parallel to the tangent 
to the geodesic at the point. The same equation holds for the lines of 
curvature on the ellipsoid ; so that every geodesic may be associated with 
a line of curvature, namely, that line of curvature for which pd has the 
same value as it has for the geodesic. We shall speak of the geodesic as 
“ belonging to ” the line of curvature. There is only one line of curvature 
having a prescribed value for pd, but there is an infinite number of geodesics 
having this value for p)d, so that an infinite number of geodesics “ belong 
to” each line of curvature. Now the line of curvature consists of two. 
