On the Vis viva. 275 
and this is true whether P is a motive or re- 
tarding force, by def. 6. 
Cor. 3. If gravity be the motive force, and 
a, b, the masses acted on, it will be money: 
aS: bs; for itis as P: p::a:b in this case. 
Cor. 4. If the vitoestis of the bodies be 
equal, we shall have by theor. 1.as F: f::a:6; 
hence by cor. 2. asa: b:: PS: ps, and p-equal 
bPS~as; therefore if gravity be the motive 
force acting on the body whose mass =a, we 
have P=a; and p—bS~s, or the weight of a 
body which is equal to the resistance opposed 
to the given body whose mass =b, by the 
medium, which it penetrates ; this follows front 
the 3d axiom. 
Tuerorem III, Put u« ui v for the velo- 
cities of the bodies a@ and 5b, and we have as 
F :f :: au’: bv* universally. 
Case 1. Let P and p be constant forces, 
and it will be by cor. 2. theor. 2. as F: f:: 
PS: ps; but as PS: ps:: au*:bv*, Emer- 
sons’s Mechanics, prop. 6; hence as F': f:; 
au”: bv’. 
Case 2. Let one of the forces P and p be 
variable, namely, p; then by cor. 2. theor. 2. 
as F: f::PS:ps; but ps is in constant 
proportion to bux, by Emerson’s Fluxions, 
(sect. 3, prob. 2, Ist edit. ); thereforethe fluent, 
or f,is in constant proportion to 6 v* ; moreover 
Mm2 
