AJfD DETERMINANTS OE LINES. 
493 
differential coefficient of H. If, namely, R denotes the radius vector of the particle, 
G=det (R, P), and similarly H=det (R, U). Now, if we differentiate the last equation, 
H=det (R, U), we obtain, according to section 28, 
D,(H)=det (R, D,(U))+det (A(R), U) (1.) 
But Di(R} is identical with the particle’s velocity, and is therefore in the direction of 
the momentum U. Whence it follows, from the very definition of a determinant, that 
det (D,(R), U)=0. 
Therefore the above equation (I.) becomes, since D^(U)=P, 
D,(H)=det (R, D,(U))=det (R, P)==G. 
This is an important result. It shows that the moment-axis about any point of the 
moving force of a particle is the complete differential coefficient of the momentum, and 
that therefore the moment of the moving force about any line is the differential coefficient 
of the moment of the momentum. 
The above result may also be easily deduced from the identical equation 
d^y d'^x d ( dy dx\ 
^ W~y M~y It)' 
50. The proposition which we have just proved may be easily extended to a system 
of moving forces and momenta of the particles of a rigid body. For, according to section 
43, the system of moving forces is reducible to a moving force at the origin O, and a 
couple G. And the system of momenta may be similarly reduced to a momentum U, 
and a momentum-couple whose axis is H. Now we have seen that P is the complete 
sum of the moving forces, and that each moving force is the complete differential coeffi- 
cient of the corresponding momentum. It therefore evidently follows that P is the 
comjDlete differential coefficient of the complete sum of the momenta, or of U. Hence 
P=I)<(U). Moreover we have seen that G is the complete sum of the moment- 
axes about 0 of the moving forces, and that each of these moment-axes is the complete 
differential coefficient of the moment-axis of the corresponding momentum. Plence it 
follows that G is the complete differential coefficient of the complete sum of the moment- 
axes of the momenta. Hence G=I)^(H). 
This result may be also easily proved by means of the identical equations 
m 
CL^X 
df- 
m 
d^y dt^x\ d ^ 
^~d?'~y 
dy dx 
^Tt-y-dt 
51. The science of the dynamics of a rigid body is founded upon D’Alembeet’s prin- 
ciple, which asserts that the moving forces of a body’s particles are together statically 
equivalent to the impressed forces acting on the body. If therefore these external forces 
be reduced to a force P at the origin O and a couple G, then P and G are equal 
MDCCCLXII. 3 x 
