494 
ME. A. COHEN ON THE DIEFEBENTIAL COEFFICIENTS 
respectively to what was denoted in the preceding section by P and G ; and we have 
therefore 
P=D,(U), G=D,(H). 
In other words, if we treat the momenta as statical forces, and reduce the system of 
momenta of a body’s particles to a momentum U at O and a momentum-couple whose 
axis is G, then the external forces acting on the body are equivalent to the force I)i(U) 
at O and the couple of forces whose axis is D^(H). 
Since G=D^(H), the resolved part of G along any fixed line will be the difierential 
coefficient of the resolved part of H along that line ; or, in other words, the sum of the 
moments of the external forces about any line equals the difierential coefficient of the 
sum of the moments of the momenta of the body’s particles. 
The above results may be easily deduced from the ordinary equations 
2(X) = S (: 
/ d'^x\ 
( dx\ 
y^^dt) 
&c., 
But it will be generally found far better not to use those six equations at all, and 
simply to bear in mind the fact which they alone express, namely, that P= 0^(11), 
G=D,(H). 
52. It will be convenient to recapitulate once for all the notation and phraseology I 
shall constantly use in the sequel. The system of momenta of a body’s particles, or 
what may be called the body's moment a-system^ is reducible, if we treat the momenta as 
forces, to a momentum at a point O, and a couple of momenta. The former I call the 
body's momentum^ and denote it by U ; the latter I call the body's mo^nentum-couple about 
O, and denote its axis by H. U and H may both be represented by straight lines 
through the origin O. It is to be observed that U remains the same wherever O be 
taken, but that H changes with the position of O. 
The components of U and H along the axes of coordinates will be denoted by 
U^, Uj,, U^, H^, H^, respectively, the magnitudes of all these quantities being repre- 
sented by the corresponding small letters. Thus h^ will be equal to the resolved part of 
H along the axis O^, and will therefore equal the sum of the moments of the momenta 
about Ox. 
The system of moving forces is reducible to a moving force Bi(U) at O, and a couple 
whose axis is D^(II) ; and it follows from D’Alembert’s principle that, if the external 
forces be reduced to a force P at O and a couple G, then 
P:z.D,(U), G=D,(H). 
63. The next step will be to investigate the expressions for U and H. 
In the first place, U can be easily found. For, let B, denote the radius vector of a par- 
ticle of mass m; then, if 2 denote the operation of taking the comjy lete sum of lines, we have 
U=2mD,(Il) 
=D^2;(mK). 
