AND DETEEMINANTS OE LINES. 
505 
of rotation about the axes of coordinates, which are here supposed to be fixed in space. 
Then it is clear that 
doc dtf dz 
Therefore 
s(x|+y|+z§) 
may be put into the form 
2(X^- Zip)^,+S(Ya:-XyK+X(Zy - (I-) 
But, by D’Alembert’s principle, 
2(X5-Za;)=Sm 
2(Zy-Yj)=2m ^). 
Substituting these expressions in (I.), we obtain 
(d'^x d'^s\ _ /d'^y dx'^\ ^ (d'^s d'^y \ 
XVT, 
which again can be put into the form 
(d^x, . d^y. . d^z, 
^'^[dF 
/d^xdx dl^ydy d^zdz\ 
—^'^\dFdt'^dFdi’^~diJt)' 
Hence it follows that 
/ ^ d^y^ d^zdz\ 
dt^^ dt] ~^^\df dt '^df dt '^df dt ) 
and therefore 
2™((: 
dt 
dt 
=2^dx{yidx-\-Ydy-\-7.dzl 
which is the equation of ms viva. 
It may be observed that the same proof may be quite easily extended to a body moving 
freely, by decomposing the original motion into a motion of translation with the velocity 
of the centre of gravity and a rotation about the centre of gravity. 
69. The two proofs which have just been given of the principle of vis viva are both 
founded on the fact that the sum of the moments of the moving forces of the body’s 
particles about the instantaneous axis, is equal to ^ 2 ^ , or to ^ ^ 2 (wu^). This 
