( 714 ) 
§ 1. We shall consider a system, consisting of x material points 
and we shall determine its position by the rectangular coordinates 
of all these points. The coordinates of the first point will be re- 
presented by zj, #2, #3, those of the second point by «4, #5, zo, etc., 
and any one of the coordinates by «,, the index v varying from 
1 to 3n. We shall write m for the mass of the system, and m, for 
that of the point to which the index v belongs. This implies that 
any one of these quantities m, has the same meaning as two other ones. 
§ 2. We shall determine an infinitely small displacement of the 
system by the increments dr, (or, as we shall write in some cases, 
Ow,) of the several rectangular coordinates. We shall ascribe to such 
a displacement a definite length, to be denoted by ds, and defined 
as the positive value that satisfies the equation 
3n 
mds. >, My tr. ee a eee 
PS ) 
The displacement of the system may be considered as the complex 
of the displacements of the individual points, and the rectangular 
components of these last displacements, i.e. the differentials de,, may 
be called the elements of the displacement of the system. We shall 
also call ds the distance between the positions of the system before 
and after the infinitely small displacement. 
§ 3. Let P, P, P" be three positions, infinitely near each 
other, ds, ds’, ds" the lengths of the displacements PP, PP", 
P'>P", It may be shown by (1) that any of these lengths can 
never be greater than the sum of the other two, so that we may construct 
a triangle, having ds, ds’, ds” for its sides. By the angle between the 
displacements PP and PP" we shall understand the angle 
between the sides ds and ds’ of this triangle. If we denote it by 
(s, s'), the elements of the first displacement by da, and those of 
the second b dza',, we shall have 
3n 
m ds ds' cos (8, 8'). == Ys my da, day. re = = Ry 
Ee 
In special cases the angles of the triangle may be on a straight 
line, so that (s, s') = 0 or 180°. 
The above may be extended to two displacements, having the 
elements dz, and da',, the lengths ds and ds’, whose initial positions 
do not coincide. In this case, just like in the former one, we calcu- 
late the angle between the displacements by the formula (2). 
