99 
1916-17.] Adelphic Integral of Differential Equations. 
where and /ul 2 are functions of cq and a 2 only, and and e 2 are the two 
remaining constants of integration. 
Periodic solutions evidently arise when the constants cq and a 2 are 
such that fi x is commensurable with /x 2 : the period of the solution is 
then 27r/r, where v is the largest quantity of which ^ and jul 2 are integer 
multiples. 
Suppose then that cq and a 2 have such values. Then if the constant e x be 
varied continuously, we obtain a family of periodic solutions, each having 
the same period (since this does not depend on e j). The constant of energy 
depends only on cq and a 2 , and is therefore the same for each of these 
periodic solutions. The family is therefore a family of “ ordinary ” periodic 
solutions. 
It might hastily be supposed that by varying e 2 as well as we should 
get a family of oo 2 periodic solutions. But it is easily seen that the trans- 
formation which is obtained by varying e 2 may be obtained by combining 
the transformation which consists in varying e ± with that which consists 
in adding a small constant to t. Now this latter transformation merely 
transforms every orbit into itself (each point being displaced in the direction 
of the tangent to the orbit), and so may be disregarded. The e 1 and e 0 
transformations are therefore to be regarded as not distinct from each 
other.* 
Singular periodic solutions are found chiefly in domains where the 
solution by purely trigonometric series is not possible. 
§ 2. Definition of the adelphic integral . — Having now distinguished the 
“ ordinary ” and “ singular 11 periodic solutions of a dynamical system, we 
shall consider those infinitesimal transformations which change each 
trajectory of the system into an adjacent trajectory, in such a way that 
every ordinary periodic solution is changed into an adjacent periodic 
solution of the same family , i.e. having the same period and the same 
constant of energy. In the notation we have just been using, this trans- 
formation corresponds to a small change in e r This transformation will 
be called the adelphic transformation.]- The adelphic transformation 
changes any solution of the dynamical system, whether periodic or 
not, into one of oo 1 other solutions which stand in a particularly close 
relation to it, being in fact derived from it by a change of the constant 
e 1 only. 
* The only case of exception is when all the orbits of the system are periodic, 
t From &5eA .<Pik6s, brotherly , because these orbits stand in very close relation to each 
other, and also because the integral corresponding to the transformation stands in a much 
closer relation to the integral of energy than do the other integrals of the system. 
