( 545 ) 
Let us represent the solution by: 
% =, \t) and = ay = Wo (t) 
then ps and yw, are the solution of the problem between the moments 
To 
ti a. V and ti a. =< 
S 6. From a mathematical point of view perhaps the function- 
solutions only may be considered as genuine solutions of the problem. 
We seek namely a quantity y as function of 2, which is in a cer- 
tain relation to a quantity y, which represents the same function 
of x, but for a different value of «. If in the general solution y is 
equated to wi (we), y' is represented by another function of «, namely 
w(x). It is not essential that we take y, only between the limits 
em and z, wo between the limits zg and z3: we might also have 
taken as a solution : 
y= Yi (2) + wa (2) + Ws (a) + ete. 
We must take care, however, that if for a given value of z we 
take y on a branch belonging to the curve y = Wn (2), we take y' on 
a branch of the curve y= Wn +1 (2). 
From a physical point of view, however, the general solution 
with FourIER's integrals is in fact the genuine solution. For let 
us imagine that we have only one system with n degrees of freedom 
which is made to move after a certain method, and which after 
the moment ¢ is suffered to move freely. The motion after the 
moment ¢ will then be determined only by the x generalized coor- 
dinates and their » fluctions. On the other hand, if we have two 
systems which act on each other with forces which propagate with 
finite velocity through a medium, and if we suffer these after the 
moment ¢ to move freely, the motion after the moment ¢ will depend 
not only on the zn coordinates and their fluctions, but also on the con- 
dition of the medium. The condition of the medium at the moment 
t gives an accurate image of what has happened with the systems 
during some time preceding the moment ¢, that is to say it gives 
an accurate idea of the way in which the systems are made to move. 
Let us for instance take the problem of GALITZIN, and let us 
imagine that the molecules are set vibrating by the collisions, and 
a 
that a collision lasts a very short time 4, so that 0< 5: Let 
