On the Vis viva. 289 
b.bo? :: F.AO:OB?. (b+k+1), ibid; 
hence O B* . (b+k+/)=b.bo0° th. ho*tl, 
1o?, and BO = | (Libor tt- bolt fo") Q.ELL. 
al bk-I 
Cor. 1. The centre of gyration of a system 
b, k, I, is also the centre of its vis viva ; 
that is, if a material point, B, whose mass 
=b+k+l, &c. revolve round the centre O at 
the distance O B with the angular velocity of 
the system 5}, k, l, the vis viva of B is equal 
to the vires vive of the particles, 6, k, 1, &c. 
i by cor. 4, prob. 3, or theor. 4. 
Cor. 2. Ifo, the centre of rotation, coin- 
cide with the centre of gravity of 6, k, J, the 
system has no momentum, (mechanics, 
prop. 50); but it has a quantity of vis viva 
equal to that of B, by the last corollary ; 
hence if the parts of a system move amongst 
themselves, it has a quantity of vis viva by 
this cor. and theor. 4, whatever may be the 
state of the centre of gravity. 
Cor. 3. Let G be the centre of gravity of 
the system b, k, 1; jom oG, in which pro- 
duced, take o R=O B, the radius of gyration 
to the point o; also make G r= the radius of 
gyration to the point G; puto R=R, 0 G=g, 
G r=r, then g?+7r°=R’, by mechanics ; but the 
system revolves with equal angular velocities 
about the points o and G; therefore the abso- 
lute velocity of R may be resolved into the 
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