1415 
3. Motion relativated without the aid of the hypothesis of Förrr.. 
First we will give a solution of the second problem of $ 1, 
without applying the hypothesis of Förer. 
We assume the same as in § 2, so we start from the differential 
equation (2), for which hold the relations (4a) and (46), we assume 
moreover that the forces, of which X, and Y, are the components 
only depend on the relative position of the points (e.g. Newton force). 
Take now a new system of axes XY with its origin permanently 
in an arbitrary material. point, which we will call point 1, and 
with its axis of X permanently through a second arbitrary-point 2 
of the system of points. The new co-ordinates w and y are now 
relative co-ordinates as meant in § 1. After transformation of the 
former co-ordinates to the new system of axes’), the equations of 
motion (2) pass, considering (4a) and (45), into: 
oe 8 NE jn 
BW WE — 2wy == 
my m, 
(8a) 
oe ¢ C Ys, V2 
nw wy, | 2we, = — 
My m 
1 
in which w is given by: 
au Abn os Os Nine fap NSD) 
a, 6 and c are functions of the a, y, x, y and a, y of the n points 
a= Emmet + y?) —(S me)? — (Z my). 
b= EmEm(ee + yy) — Sma Eme — Emy = my. (8c) 
e= Sm > m (ey — yx) — (E me E my — TS my Sma). 
The sign = includes all points. 
There are 2n—3 co-ordinates 2,,y, and according to this 2n—3 
equations (8a) of the second order, the auxiliary quantity w occurs 
in one equation (85) of the first order in w. So (8a) and (86) form 
together a system of order 2 (2n—3) + 1 = 4n—5. After elimination 
of w there remain 2n—38 co-ordinates, so that e.g. the system can 
be reduced to 2n—4 equations of the second and one of the third 
order: In these equations only the relative co-ordinates, their deri- 
vatives and the quantities X, Y, occur. As we supposed the com- 
ponents X,Y, dependent on the relative position of the material 
points only, the problem is solved. 
) See § 5.6. 
