280 Theorems hua Problems 
system ; put wand v forthe absolute velocities of 
APE and B@E respectively; then the vis viva 
of the system is as au*+bv* by theor. 4; but 
the velocity of et by dynamics, where 
the sign, connecting the terms of the nume- 
rator, is affirmative when the bodies APE, 
B@E move in the same direction, and nega- 
tive when they move in contrary directions; 
a* u* + 2aubo +570”, 
a+b 
by theor. 3; but the excess of au*+bv* com- 
pared with this expression is as the vis viva 
which acts on the figure of the system by cor. 
2. theor. 5; which excess =a b* (“+ 242 +07) 
a+b ; 
now the vis viva of G is as 
now when the bodies move in the same direc- 
tion u—v=k and when they move in contrary 
directions wtv=k; therefore the excess in 
question —“2", Q. E.D. 
a+b 
Cor. 1. The momenta of the bodies APE, 
BQE are equal, in the relative space APQB ; 
because they add nothing to the momentum of 
the centre of gravity G; therefore the vis viva 
of APE before concussion is to the vis viva of 
BQE, as the velocity of the former, to that of 
the latter, or inversely as their masses, by 
cor. 4. theor. 3: moreover the velocities of 
APE, B@E are respectively equal to 
ok asd 
a+b a-+-b° 
