152 ON MR. EWART’S PAPER ON THE 
quantities of elementary work, which name M. 
Poisson adopts. The sums =Pdp and 2Qdq 
will be the quantities of elementary work per- 
formed during the same instant by all the moving 
forces and all the resisting forces; and their in- 
tegrals /2Pdp and />Qdg¢ will express the entire 
motive work and the entire resisting work, from 
the commencement of the motion to the time in 
question. 
The equation shows that, in any machine in 
motion, the increment, during any time whatever, 
of the half sum of the living forces of all its parts, 
is always equal to the excess of the motive work 
over the resisting work during the same time. 
If there were no original velocity in the ma- 
chine, and the object was to consider the effect of 
friction as a separate retarding force, the following 
equation is given by M. Poisson for the purpose, 
$=mv’=/ = Pdp—/=Qdq—=fNas, 
where f is the co-efficient of friction, NV the 
mutual pressure of the parts which rub against 
each other, ds the element of the curve described 
by their point of contact, and /=Qdgq the useful 
