276 Theorems and Problems 
PS or F is in constant proportion to au, 
Mechanics, Prop. 6; hence. as F: f:: au? ; 
bv*. Q.E.D. 
Cor. 1. As PS: ps::au?:by?. 
Cor. 2. Let gravity be the motive force 
acting on a, then P=a 3 also put c=16,3, feet, 
and we have w=32+ feet=2c, and it will be, 
by cor. f, as ac *ps::4ac*: bv*; hence 
Acps=bv’. 
Cor. 3. If P=p, we have F's f:: S:s::u:v. 
For F:f:: PSips::8:5:: au*>:bv*; but 
au=bv, Mechanics, prop. 4th; therefore as 
Fifi::Sis::usv:: 6:a@; and this corollary 
is true when the force P or Pp is variable; for 
in this case it will be as Fi f >: Siaiss eats 
bv», 
Cor. 4. If m be the momentum of b, f will 
. . 2 
be in constant proportion to ” or to mv; 
b 
hence if b be one of a system of bodies in 
motion, its vis viva will be affirmative in all 
cases ; because m? is affirmative, and the signs 
of m and v are always alike. 
Tuerorem IV. If a, b, d, &ec. be the 
masses of any number of bodies moving with 
the velocities u, V, v, &c. which they have 
acquired by the uniform action of the motive 
forces P, p, gq, &c. in passing through the 
spaces S, s, t, &e,; we have 4 ex (PSt+pst+ 
 @t Ke. )=au? +bV2+d v2, &e, . 
