618 
FORMULA REPRESENTING THE DYNAMICAL STABILITY 
Therefore the work done upon the vessel by the disturbing weight in the act of tliis 
inclination is represented by 
w{h vers ^+<^sin 6), 
which expression ought, therefore, neglecting the inertia of the fluid, to equal that 
(equation 7-) which represents U(/^), or 
w{h vers sin vers 6, 
whence we obtain 
tani^=w;^^^w;V^ 
In the vessel experimented upon, W 2 = 0 , 
\ . wc . , 
/. tan2^— (9.) 
which is the formula used in calculating the eighth column of the Table, p. 615. 
15. The dynamical stahility of a vessel of any given form subjected to a rolling or 
pitching motion. 
Conceive the vessel, after having completed an oscillation in any given direction, 
— being then about to return towards its vertical position — to be for an instant at 
rest, and let RS (fig. 5) represent the intersection of its plane of flotation then, and 
PQ of its flotation when in its vertical position, with a section CAD of the vessel per- 
pendicular to the mutual intersection O of these planes. The section CAD will then 
be a vertical section of the vessel. 
Let G be the projection upon it of the vessel’s centre of gravity when in its vertical 
position. 
H, that of the centre of gravity of the fluid displaced by the vessel in the vertical 
position. 
g, that of the fluid displaced by the portion of the vessel of which QOS is a sec- 
tion. 
h, that of the fluid which would be displaced by the portion, of which POR is a 
section, if it were immersed. 
GM, HN, gm, hn, KL perpendiculars upon the plane RS. 
W= weight of vessel or of displaced fluid. 
weight of water displaced by either of the equal portions of the vessel of 
which POR and QOS are sections. 
H,= depth of centre of gravity of vessel in vertical position. 
depth of centre of gravity of displaced water in vertical position. 
AHi= elevation of centre of gravity of vessel. 
AH 2 = elevation of centre of gravity of displaced water. 
6 =inclination of planes PQ and RS. 
rj = inclination of line O in which planes PQ and RS intersect, to that line 
about which the plane PQ is symmetrical. 
