( 724 ) 
GS, =m, G@ =m cr, © =mvD 
for the three components. 
§ 18. What precedes has prepared us for the consideration of 
the fundamental principles by which the motion of the system under 
the action of given furces is to be determined. We may in the first 
place start from the following assumptions: 
a. The system will have the acceleration f, if the force is preci- 
sely © = mf. 
b. Two forces %, and § may have the same influence on the 
motion. For this it is necessary and sufficient that the force 
8a — Be 
should be perpendicular to all pessible displacements. 
Let the system be subject to the force § with the components 
Bor Bi, Se, and let the acceleration be f. Then, by the first principle, 
& has the same influence as © = mf, and by the second principle 
S—© must be perpendicular to all possible displacements. This 
amounts to the same thing as 3; = ©), 5. = @, or 
5, = mvc, Fo=moD. .. . . . « (24) 
It will be immediately seen that the above assumptions are equi- 
valent to D’ ALEMBERT’s principle. We might also have replaced 
them by the following rule: 
Decompose the acceleration into two components f, and f', the 
first perpendicular to all possible displacements, and the second in 
the direction of such a displacement. Decompose the force § in the 
same way into the components 8) and 6’. Then the equation of 
motion will be 
k= mf’. 
This leads directly to the equations (24), by which it is clearly 
seen that the change in direction of the path is determined by the 
component 8), and the change in the value of the velocity by the 
component 8. It is to be kept in mind that the first of the for- 
mulae (24) is a vector-equation. In general the free curvature, as 
well as the force 5, may have different directions, in some cases a 
great many of them. The equation does not only show us to what 
amount the path deviates from one of least curvature, but also to 
which side the deviation takes place. 
If §=0, we have cr-=0 and v=0; we are then led back to 
the fundamental law of Hertz. 
§ 19. Let us now return to the equation (21), taking for the 
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