( 718 ) 
is an equal number of linear equations, i of them expressing (§ 6) 
that the first component has the direction of a possible displacement, 
and the remaining 3n equations, that the elements of the given vector 
are the sums of the corresponding elements of the two components. 
§ 8. The path of a moving system is determined by the positions 
it occupies one after the other. It may be considered as a succession 
of infinitely small displacements, which we shall call the elements 
of the path. The length of any part of the path is defined as the sum 
7 ds 
of the lengths ds of the elements of which it consists. 
The direction of a path in one of its positions is given by the 
direction of an clement. 
We shall always think of the system as moving along a path in 
a definite direction. Then the coordinates z,, and all other quantities 
that have determinate values for every position in the path, may be 
regarded as functions of the length s of the path, reckoned from 
some fixed position. Accents will serve to indicate differentiation of 
such quantities with respect to s. 
From what has been said in § 4 it follows that the quantities «', are 
the direction-constants of the path; they will always satisfy the 
relation 
3n 
Nima =m, EMP NT er (ll) 
1 
as appears from (6). Using (3), we see that a vector whose elements 
are «', has the value 1. This vector of value 1, in the direction 
of the path, may be called the direction-vector. We shall represent 
it by ®. 
§ 9. We define the curvature of a path as the vector c, given by 
dd 
= 
the numcrator being the difference between the vectors D at the 
beginning and at the end of an element of the path of length ds. 
The elements of D being «',, we see at once that those of ¢ are «",; 
accordingly, in virtue of (8), the value ¢ of the curvature is given by 
3n 
Mm SR a eye ah inl le, MRE Us Vee AD 
l 
By differentiating (11) one finds 
¢ en 
kkn 
a” — 
