604 
ME. A. COHEN ON THE DIFEEEENTIAL COEFFICIENTS 
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n 
moving about a fixed point without assuming the principle of virtual velocities, and is in 
fact a very simple deduction from D’Alembeet’s principle. ^ ^ 
Let O ^ be the instantaneous axis, about which the body is rotating 
at time ?{■ with an angular velocity is. Letm be a particle of the body, 
let its mass be its distance from O e, r, and its velocity v in the 
direction m n. 
Suppose now a force P to act at m. Then, since m w is perpendicular on the plane 
i O m, it is clear from statics that the moment of P about O ^ is the moment of the 
resolved part of P along m n. It is therefore, if 9 be the angle which the direction of P 
makes with m equal to rP cos or - P cos (p. 
Suppose then P to be the moving force of the particle m. Then its resolved part 
along the velocity mw is of course so that P cos and therefore the moment 
V dv 
of the moving force about O ^ equals Consequently the sum of the moments of 
the moving forces about O i equals "Z (^nv . But this sum, by D’Alembeet’s prin- 
ciple, equals the sum of the moments of the external forces about O i. Now we have 
already seen that the moment of any force P about O z is - P cos p, so that, if P represent 
an external force acting on the body, the sum of the moments about O ^ of the forces 
acting on the body equals ^ 2 (P^; cos <p). Hence we have 
i2(P®cosf)=i2(m®§)- 
Therefore 
Z{mv^) = 2 j(Z^S(P'y cos (p). 
This equation embodies the principle of vis viva ; for it is evident that, if the compo- 
nents of P be X, Y, Z, then 
dx , dy , dz 
Pv cos <p—X. ^ + Y Z 
68 . The same result may also be obtained by analysis; and it may be worth while to 
notice that each step in the analytical proof is exactly equivalent to the corresponding 
step in the above geometrical proof. This correspondence between the steps in analytical 
and geometrical demonstrations is one of the most striking features of modern analytical 
geometry, and would, as we have already attempted to show, present itself generally in 
analytical mechanics, if more attention were paid to the interpretation of the equations 
and formulae which are employed. 
The analytical proof is as follows : — 
Let X, Y, Z be the components of any one of the forces acting on the body, and sup- 
pose that force to act at a point {x, y, z) of mass m. Let be the angular velocities 
