492 
ME. A. COHEN ON THE DIFEEEENTIAL COEEEICIENTS 
Chaptee IV. 
47. As soon as we pass from the statics or dynamics of a particle to the statics or 
dynamics of a system of particles or of a rigid body, we find that two forces which are 
equal and parallel to one another are not equivalent to one another, and that we have 
to take into account the position as well as the magnitude and direction of a force. 
Notwithstanding this, we are enabled by means of an elementary principle of statics to 
confine our operation and our notation to lines passing through one and the same point. 
For suppose a force P to act at a point m of a rigid body, and apply at the origin 0 two 
equal and opposite forces P and — P, then P at m is equivalent to P at O and the couple 
whose forces are P at m and — P at O. Let the axis of this couple be denoted by G; 
the couple, being completely determined by G, may be called the couple G. It is 
extremely convenient to have a name for the line G, indicating briefly its connexion with 
the force P at m, and I shall adopt that given to it by French writers*, and shall call G 
the moment-axis about O of the force P at m. 
It has been proved in section 25 of Chapter II., that the line G, being the axis of the 
couple whose forces are — P at O and P at w, is equal to det( — P, P), where K is the 
radius vector of the particle. Hence we have 
G=det (-P, K)=:det (R, P). 
We thus see that the force P at m is completely represented and determined by the 
two lines P and G drawn from the origin, G being the moment-axis with respect to the 
origin of P at m, and being equal to det (R, P). 
48. Suppose now that we have a system of forces Pj, Pj, &c. acting respectively at 
points mj, m^, &c. of a rigid body. Then it is clear from statics that the given system 
of forces is equivalent to a force P at the origin O and a couple whose axis is G, where 
P is the complete sum of the forces Pj, Pj, &c. supposed to be collected at the origin, 
and G is the complete sum of the moment-axes (G,, Gj, &c.) (about the origin) of the 
forces Pi at m,, P 2 at Wa, &c. 
49. We will now apply the above considerations to dynamics. Since the acceleration 
is the complete difierential coefficient of the velocity, it is evident that the line which 
represents the moving force of the particle is the complete differential coefficient of the 
line which represents the particle’s momentum ; or, more briefly, the moving force is the 
complete differential coefficient of the momentum. 
Let now P represent the moving force, and U the momentum of a particle m, P and 
U denoting straight lines ; then, if we treat the moving force and momentum as if they 
were statical forces, it is clear that P at m is equivalent to P at the origin O and a 
couple G, where G is the moment-axis about O of P at to ; and similarly, the momentum 
U at TO is equivalent to U at O and a couple of momenta whose axis is H, where H is 
the moment-axis about O of U at to. We have just seen that P is the complete differ- 
ential coefficient of U, and we will now prove that in like manner G is the complete 
* See Delauxat’s ‘ Meclianics,’ page 254. 
