On the Vis viva: 29% 
for, the velocity of the point R or the centre 
of gyration, is as R; therefore the vis viva of 
t 
7 P} 2 : yi : 
the same centre is as le by theor. 3; 
’ & zt 
m R?, 
therefore the vis vita of the system is as ——- 
t 
cor. 1, prob. 4, or as ae by the last 
corollary. . ! 
Proriem VI. Let there be two cylinders 
A and B of the same ductile matter whose 
diameters are a and 0, and heights ¢ and d, 
respectively ; aud let these cylinders be drawn 
out ia length until. their diameters become 
and 4. what is the ratio of the forces 
ae ? 
m nN 
F and f, required to produce these changes ? 
-Soxurron. When the cylinders have 
been drawn as directed in the problem, the 
length of A= m*c; length of B= n* d; and 
the heights of their centres of gravity above 
the plane, on which they stand, are as their 
lengths, or as m2 eto n?d; but the. heights 
of the centres of gravity of A and B above the 
same plane were as ¢ to din their first. shape; 
therefore the spaces through which their 
centres of gravity move, while their figures are 
changing, are as (m*—1) .¢ to (n?—1) a3 
consequently as Fi f+: (m*—1) . ¢: 
(n?—1). d, by cor. 3; theor. 7; where the 
diameters .a, and b, are not found in the 
proportion, Q.E.L. 
