CHAIN ON A FIXED PLANE CURVE. 1074 
Differentiating with respect to ¢, of whichs, s,, and y are functions, 
dv ds ds 
m lv Pomel a fees 
d. - 
but — v throughout the chain, and therefore 
mlv a 
U ha NJ (Y2—t) Vv 
or 
dv q Pepe 
Te (Y2 — Y), as before. 
Again, 
Emlv = myfyds = mqly + C, 
where y is the ordinate of the center of gravity of the chain, and therefore 
v= 297 + 0; 
hence if the chain descend from rest, its velocity in any position is that which would be 
acquired by a particle falling freely through the depth descended by the centre of gravity. 
Differentiating the last equation with regard to ¢, we obtain 
Cp dy 
dt 7 at 
and therefore 
dy aK g 
Ds v Te (Y2— In) 
or 
D  @& 
Weare (Y2— Yr). 
This result is however applicable to the general case, as appears in the next article. 
3. The chain being uniform, we have geometrically 
ly = je yds. 
oh 
: 2 FD : z ; ds 
Differentiating this with regard to ¢, and remembering that the value of a when s = s, or 
8, is still v, 
Cp 
a, — Gi— LE 
So also, 
dx À 
l a = (Ly — di) Vv. 
Therefore, squaring and adding, if + be the velocity of the centre of gravity, and c the 
length of the chord of the chain, 
; pa Le a SVM PTE 
or : 
DOUÉ 
Sec. III, 1882. 3 
