310 
MR. R, S. HEATH ON THE DYNAMICS OF 
The determinant equated to zero gives the relation between the coordinates of the 
screws, and these final equations determine the ratios of the twists on them in order 
that they may produce rest. 
In the case of simple rotations or translations the coordinates become the coordinates 
of lines, and therefore, 
( 0 , 0 ) = 0 ( 1 , 1 ) = 0 ... 
(0,1) will be the moment of the lines of action of the translations or rotations, and 
the conditions that the system may produce rest are of the same form as before. 
Kinetics. 
40. The definitions of acceleration, momentum, and kinetic energy of a particle are 
taken to be exactly the same as in ordinary space, and do not need further comment. 
We shall next consider the measure of force. To determine a force completely we 
require a line of action and a measure of its magnitude. We assume that a force may 
be applied at any point of its line of action. As in ordinary space w’e shall take 
Newton’s second law of motion as the basis of the measurement of forces. This law 
states that “change of motion is proportional to the impressed force, and takes place 
in the direction in which the force acts.” Hence, if a force act on a particle of known 
mass it will cause a change of momentum in the direction of its action, and the 
measure of the force will be proportional to the change of momentum per unit of time. 
By the proper choice of the unit of force we may deduce the equation 
P =mf 
which gives the dynamical measure of any force. 
Since a linear acceleration f along any line is exactly the same thing as an angular 
acceleration f about the polar line, it follows that a force P along any line is the same 
thing as a couple P about the polar line. 
41. Newton’s law further implies the physical independence of forces; i.e., if a 
number of forces act on a body each produces the same effect as if it alone acted on 
the body. Hence if a number of forces act on any number of particles each force 
produces its own effect, and to calculate the resultant of the forces we must calculate 
the resultant change of momentum per unit time; in other words, the laws by which 
forces are combined are the same as those for velocities and accelerations. If, there¬ 
fore, a force P act along a given line whose coordinates are a, b, c, f g, h its effect is 
the same as if forces Pa, P b, . . . acted along the edges of the tetrahedron of reference; 
and the component force along any other line a', b', c,f', g , h' is 
¥{aa + bb’-\-cc +ff +gg’+hh') 
