Motion of a System of Bodies. 53 
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vee 3 
By (w),(since a? Se are each=0,) if we suppose a’ =0, y’=0, 
J/ dx? +d 2 qe 
we shall have 2! o/ p” += a hid =the velocity of a 
\ 
point ae is on the axis of 2’, at the distance 2/ from the origin, also by 
(47), 5 = ca ae 1 —c,2 =the sine of the angle made by the axis 
2 -b eee Tr saad aay bi 
of 2’ with the momentary Sa =the perpendicular 
IP iateeed a cht. ! 
from the extremity of z’ to the instantaneous axis; put w=the angu- 
Jar velocity around the instantaneous axis, and we have 2'V p? + q? 
el/ p+ 
Ta p? eq? ae paEEnS Oh Reina peta apnea hae). 
By (47) and (48), p=a,w, q=b,w, r=c,w, (49), where p, q, 7 
are evidently the momentary rotations around the axes of wv’, y’, 2’ 
respectively ; hence it is evident that rotary velocities are compound- 
ed and resolved by the same rules as rectilineal velocities. 
Remarks.—It is evident that if the origin of the coOrdinates is at 
the centre of gravity of the system, all the formule which we have 
found will apply, whether the centre is at rest or in motion; for (19) 
which have the same forms as (18), are applicable whether the cen- 
tre is at rest or in motion; hence by proceeding with (19), as we 
have done with (18), we shall obtain the same results as before ; .*.by 
placing the origin at the center of gravity, all the above formule will 
apply when the system is free; and the motion of the centre will be 
found by (4). 
Again, it is supposed in (4’) that the bodies are so small, that 2’, 
y’, 2’ may be considered as having the same values for all the points 
of each, but should not this be the case, we must change m into dm, 
then find the value of A for each body, by taking the integral relative 
to its mass; then the sum of all the values thus found, will be the 
complete value of A; and in the same way we must find the com- 
plete values of B, &c.; but should the system be a continuous sol- 
id, we must find the values of A, B, &c. by integrating relative to 
its mass. 
