626 
TIME OF THE ROLLING OF A SHIP. 
by equation 20, 
1 "w r 1 
W;(H,+H 2 )(cos cos ^0+ 2 I^A(cos" 0- cos^ ^0 sin^ 
F+(Hi + ?)^sin®9 
^ + Ha) (cos 6 - cos ^i) + ^ ^ (cos^ A - cos^ Sj] 
d& 
F + (Hi + §)2sm2 0 
1 + H 2 + ^^(cos^ + cos5i) j.jcos6- cos 6 ^ 
d&. 
Assuming 6 and to be so small that cos cos ^1 — 2, and observing that 
cos 6 — cos ^ 1 = vers 6 ^ — vers 
m= 
A^+(Hi + g)^sin^9 
vers vers 5 
. d&. 
Supposing, moreover, to remain constant between the limits 6^ and and inte- 
grating as in Art. 20, equation 21, 
^(^0 = 
(23.) 
where ^ is to be taken with the sign according as the sutface of the planes of flota- 
tion is above or below the load-water line, and H^, according as the centre of gravity 
of the displaced fluid ascends or descends. 
Since the value of sin" \ 6, is exceedingly small, the oscillations are nearly tauto- 
chronous, and the period of each is nearly represented by the formula 
t(6,)= , (24.) 
The following method is given by M. Dupin for determining the value off*: — 
“ If the periphery of the plane of flotation be imagined to be loaded at every point 
with a weight represented by the tangent of the inclination of the sides of the vessel 
at that point to the vertical, then will the moments of inertia of that curve, so 
loaded, about its two principal axes, when divided by the area of the plane of flota- 
tion, represent the radii of greatest and least curvature of the envelope of the planes 
of flotation.” 
Iff be taken to represent the radius of greatest curvature, the formula 25 will 
represent the time of the vessel’s rolling; if the radius of least curvature (B being 
also substituted for A), it will represent the time of pitching. 
* Applications de Geomdtrie, p. 47. 
