278 Theorems and Problems 
For each of the bodies P, Q, R, moves in 
absolute space, namely, in the direction Gg, 
with the velocity y, common to them all; 
consequently the sum of their vires vive in 
this direction is as (a+b+d) y*, by cor. 1. 
theor. 4. But the vires vive of P, Q, R, in 
the relative space PQR are respectively as 
au*, bV* and dv’ by theor.3 ; hence the sum 
total of these forces is as (a4)4d). y*+au? + 
bV*+d v’, cor. 1. theor. 4. Q.E.D. 
Cor. 1. If P, Q, and R be at rest in the 
relative space PQR, they move only in abso- 
lute space with the velocity y; that is the vis 
viva of the system is equal to that of its centre 
of gravity; because u=V=v=o; and the 
figure of the system undergoes no change; 
because P, Q, and R preserve their relative 
positions unaltered. | 
Cor. 2. But if u, V, and v be real quanti- 
ties, the vis viva of the system exceeds that of 
its centre of gravity, by the theorem. For 
the same reason, the bodies P, Q, and R are 
not at rest in the relative space QRP; that is 
the figure of the system is undergoing a 
change ; consequently if P, Q, and R react 
upon each other from any cause whatever, the 
foregoing excess of vis viva will be exerted to 
overcome this reaction; which will continue 
