606 
ME. A. COHEN ON THE DHFEEEENTIAL COEEEICIENTS 
fact is the reason why the equations of vis viva can be obtained by multiplying Eulee’s 
three equations byts-^, respectively, and by adding together the products so 
obtained ; for in performing those operations we are in reality finding the sum of the 
moments of the forces about the instantaneous axis whose direction-cosines are 
CT ’ CT ’ ■ST ’ 
70. If Gi be the sum of the moments of the impressed forces about the instantaneous 
axis, then by D’Alembeet’s principle Gj equals the sum of the moments of the moving 
forces about the instantaneous axis. Hence it follows from the preceding section that 
This equation is often useful. 
For instance, since 2(m'y^)=2(m'55-V), where r is the distance of a particle from the 
instantaneous axis, it follows that if I denote the body’s moment of inertia 
about the instantaneous axis. Therefore 
Now, if the instantaneous axis were fixed in space, we should evidently have 
Therefore the only cases in which we can take moments about the instantaneous axis 
as if it were fixed in space are when ^ ^=0, or when the moment of inertia about the 
instantaneous axis is constant. This proposition is useful in solving problems concerning 
rolling cones, and is usually deduced by analysis from Eulee’s equations. 
71. I will give two more examples of the advantage of the method I have employed 
in these pages. 
Let be the angular velocities of rotation of a body about three rectangular 
axes which are fixed in and move with the body, and let «, 5, c be the direction-cosmes 
with respect to those axes of a line which is jixed^ in sjpace. Take on the latter line a 
point P at a unit of distance from the origin. The velocity of the fixed point P is zero. 
Now, as its components along the moving axes are a, b, c respectively, it follows from one 
of our elementary propositions that 
da 
bvs^ 
equals the component of P’s velocity along Ox^ and therefore equals zero. 
