SURFACE ON WHICH A SHIP MAY BE IMAGINED TO ROLL. 
621 
and (Acos^;?+Bsin^;?) sin^^={A + (B— A) 
(A cos^;?+B sin^jj) sin^^=A sin^^+(B— A) sin% ; 
by equation 13, 
U(^, Q=W(HiipH 2 ) vers ^+^iM/{A sin^^+(B — A) sin%}, .... (15.) 
by which formula the dynamical stability of the ship is represented, both as it regards 
a pitching and a rolling motion. 
TT 
If in equation 13 the line in which the plane PQ (parallel to the deck of the 
ship) intersects its plane of flotation is at right angles to the length of the ship, and 
we have, since in this case (see equation 14.), 
U(g = W(H.+HOversJ+|f<,BsW(. ( 16 .) 
which expression represents the dynamical stability, in regard to a pitching motion 
alone, as the equation 
*U(^)=W(HiI^H 2) vers^+^iM/Asin^^ (17-) 
represents it in regard to a rolling motion alone. 
16. If a given quantity of work represented by U(^) be supposed to be done upon 
the vessel, the angle 6 through which it is thus made to roll may be determined by 
solving equation 17 with respect to sin We thus obtain 
.;^2i_ W(W + H,)+f^A- ^{W(Hi + H,)+i^^AP -2/.A.U($) 
2 2 j(aA ^ 
17 . If PR and QS be conceived to be straight lines, so that POR and QOS are tri- 
angles, then w.z, taken in respect to an element included between the section CAD, 
and another parallel to it and distant by the small space dx, is represented by 
1 
4 sia &dx(mg-\-nh ) ; 
or, since 
mg-\-nh=:^-^y^ sin d, 
by Y 2 ^ s\ifey\yfx ; 
wz=^i/j%w?djy\yflx, 
and, equation 1 1 
U(^, Q=W(H.+H2) vers^-f^^sin"^f?/2C?a;, (19.) 
which formula maybe considered an approximate measure of the stability of the 
vessel under all circumstances. 
* This formula may be verified experimentally by a method similar to that applied to equation 6. See 
Art. 10. 
