622 
AXIS ABOUT WHICH A SHIP ROLLS. 
If, as in the case of the experiments of Messrs. Fincham and Rawson, the vessel be 
prismatic and the direction of the disturbance perpendicular to its axis, 
^=constant = a, and sin d; 
.'. sin and 
U(^)=W(Hi+H 2) vers aw sin d. 
Mr. Rawson has obligingly undertaken the verification of this formula by com- 
paring it with his experiments on the cylindrical model. The following is the 
result ; — 
No. of 
experi- 
ment. 
W. 
W. 
Hi- 
e. 
U(^) by 
formula. 
U(^) by 
experi- 
ment. 
3 
lbs. 
253*43 
lbs. 
17-294 
3*903 
4*800 
o 
9 
1*760 
1*766 
4 
255*43 
46*84 
4*02 
4*82 
26 
13*478 
13*5015 
5 
197-18 
37*98 
0*80 
3*80 
26 
6*807 
7*3761 
18 . ^ rigid surface on which the vessel may be supposed to rest whilst in the act of 
pitching and rolling. 
If we imagine the position of the centre of gravity of a vessel afloat to be continu- 
ally changed by altering the positions of some of its contained weights without alter- 
ing the weight of the whole, so as to cause the vessel to incline into an infinite num- 
ber of different positions displacing, in each, the same volume of water, then will the 
different planes of flotation, corresponding to these different positions, envelope a 
curved surface, called the surface of the planes of flotation des Jlotaisons),'w\\o&e 
properties have been discussed at length by M. Dupin in his excellent memoir, Sur 
la Stabilite des Corps Flottants, which forms part of his Applications de Geometric*. 
So far as the properties of this surface concern the conditions of the vessel’s equili- 
brium, they have been exhausted in that memoir, but the following property, which 
has reference rather to the conditions of its dynamical stability than its equilibrium, 
is not stated by M. Dupin : — 
If we conceive the surface of the planes of flotation to become a rigid surface, and 
also the surface of the fluid to become a rigid plane ivithout friction, so that the former 
surface may rest upon the latter and roll and slide upon it, the other parts of the vessel 
being imagined to be so far immaterial as not to interfere with this motion, but not so as 
to take away their weight or to interfere with the application of the upward pressure of 
the fluid to them, then will the motion of the vessel, when resting by this curved surface 
upon this rigid but perfectly smooth hori%ontal plane, he the same as it was when, acted 
upon by the same forces, it rolled and pitched in the fluid. 
In this general case of the motion of a body resting by a curved surface upon a 
horizontal plane, that motion may be, and generally will be, of a complicated cha- 
* Bachelier, Paris, 1822 . 
