1916-17.] Adelphic Integral of Differential Equations. 
97 
Now consider the connection between the oo 1 members of the family 
of geodesics which belong to the same line of curvature. It is known * 
that if 
</> (q v q 2 , Vv> Pi) = Constant 
is an integral of a dynamical system, then the infinitesimal contact-tra#f ^^ 3 1 
formation which is defined by the equations 
1 1 JUN Z5 
d(f> 
= k- 2 = 
dpi dp ^ 
&lh = - e jT~ , &P-2 = 
d(f> 
dc h 
'dcp 
4k -A 
(where e is a small constant) transforms any trajectory into an adjacent 
curve which is also a trajectory. If we apply this theorem to the motion on 
the ellipsoid, we find without much difficulty f that the infinitesimal trans- 
formation which corresponds to the integral 
pd = Constant 
transforms any geodesic into another geodesic which belongs to the same 
line of curvature. 
Summing up, we see that the oo 2 geodesics on an ellipsoid may be classi- 
fied into co 1 families, each family consisting of oc 1 geodesics : the members of 
any one family are either all closed or all unclosed : and a certain continu- 
ous group of transformations , which is closely associated with the integral 
pd = Constant, transforms any geodesic into all the geodesics which belong 
to the same family. 
Besides these geodesics which can be arranged in families, there are 
on the ellipsoid three other closed geodesics, namely, the three principal 
sections of the ellipsoid. These have quite a different character: they 
are solitary, instead of belonging to families : and the infinitesimal 
transformation which has just been mentioned tranforms them not 
into other geodesics but into themselves — that is, they are invariant 
under the transformation. This last property suggests a resemblance 
with the theory of “ singular solutions ” of ordinary differential 
equations of the first order: for if a differential equation of the first 
order admits a particular infinitesimal transformation, then this 
infinitesimal transformation changes the ordinary integral-curves into 
each other, but it leaves invariant the singular integral-curve. On 
account of this resemblance I propose to call a periodic solution 
(of a dynamical system with two degrees of freedom) ordinary if 
* Of., e.g., my Analytical Dynamics, § 144. 
t As this problem of motion on an ellipsoid is only a special case of the general theory 
which is given later, I do not give the analysis relating to it in detail. 
VOL. XXXVII. 
7 
