( 723 ) 
forces, in the usual sense of the word, and let X, be the rectan- 
gular components of these. We shall take together all these forces, 
so that we may speak of them all as of one thing, but in doing 
so we shall slightly depart from the way in which we have defined 
the velocity and the acceleration. We begin by multiplying each 
. . . m ' Ld . . . 
individual force by —, m' being the mass of the point on which it 
m S 
acts, and m the mass of the whole system, and we understand by 
the force & acting on the system the complex of these new vectors. 
The elements of & are therefore zin DGN 
m, 
Assigning to $ a definite direction and a definite value will of 
course imply that all the forces acting on the material points of 
the system are given in direction and magnitude. 
The definition of the force $ has been so chosen that the work 
of the forces in an infinitely small displacement, i. e. the expression 
3n 
) v Xyday, 
1 
becomes equal to the scalar product (3. d 8). 
§ 16. Every force 8 may be decomposed into one component 
So, perpendicular to all possible displacements, a second component 
$1, having the direction of a possible displacement and perpendicular 
to the path, and a third component 9, in the direction of the path. 
One can conduct this operation in two steps. Replace first (§ 7) 
& by 8, perpendicular to all possible displacements, and 6’, in the 
direction of such a displacement. This being done, we have to 
decompose (§ 5) %' into a force 33, along the path, and a force $j, per- 
pendicular to it. The latter component will have the direction of a 
possible displacement, because §' and 82 have such directions. 
For a given force the three components are wholly determinate. 
§ 17. We may imagine each material point to be acted on by a 
force in the direction of the acceleration of the point and equal to 
the product of the acceleration and the mass. We shall denote by 
& the force acting on the system in this special case, by So, ©), 6, 
its components in the above mentioned directions. 
Now we have in the supposition just made Xy = my ay, from which 
we find me, for the elements of ©, © =mf for the force itself, 
and, by (23), | 
