EXPERIMENTS ON THE STABILITY OF FLOATING BODIES. 
613 
or if the inertia of the displaced fluid be neglected, 
u(^)=w.(ah:-ah,) (6.) 
Whence it follows that the work necessary to incline a- floating body through any 
given angle is equal to that necessary to raise it bodily through a height equal to the 
difference of the vertical displacements of its centre of gravity and of that of its im- 
mersed part, so that other things being the same, that ship is the most stable the product 
of whose weight by this d fference is the greatest. 
In the case in which the centre of gravity of the displaced fluid descends, the sum 
of the displacements is to be taken instead of the difference. 
8. This conclusion is nevertheless in error in the following respects ; — 
1st. It supposes that throughout the motion the weight of the displaced fluid re- 
mains equal to that of the floating body, which equality cannot accurately have been 
preserved by reason of the inertia of the body and of the displaced fluid*. 
From this cause there cannot but result small vertical oscillations of the body about 
those positions which, whilst it is in the act of inclining, correspond to this equality, 
which oscillations are independent of its principal oscillation. 
2ndly. It involves the hypothesis of absolute rigidity in the floating body, so that 
the motion of every part and its vis viva may cease at once when the principal oscil- 
lation terminates. The frame of a ship and its masts are however elastic, and by 
reason of this elasticity there cannot but result oscillations, which are independent of, 
and may not synchronize with, the principal oscillation of the ship as she rolls, so 
that the vis viva of every part cannot be assumed to cease and determine at one and 
the same instant, as it has been supposed to do. 
Srdly. No account has been taken of the work expended in communicating motion 
to the displaced fluid, measured by half its vis viva and represented by the term 
■^^w^v\ in equation 5. 
9. From a careful consideration of these causes of error, I was led to conclude 
that they would not affect that practical application of the formula which I had 
principally in view in investigating it, especially as in certain respects they tended to 
neutralize one another. The question appeared however of sufficient importance to 
be subjected to the test of experiment, and on my application, the Lords Com- 
missioners of the Admiralty were pleased to direct that such experiments should be 
made in Her Majesty’s Dockyard at Portsmouth, and Mr. Fincham, the eminent 
* The motion of the centre of gravity of the body being the same as though all the disturbing forces were 
applied directly to it, it follows, that no elevation of this point is caused in the beginning of the motion, by 
the application of a horizontal disturbing force, or by a horizontal displacement of the weight of the body, 
which, if it be a ship, may be effected by moving its ballast. The motion of rotation thereby produced takes 
place therefore, in the first instance, about the centre of gravity, but it cannot so take place without de- 
stroying the equality of the weight of the displaced fiuid to that of the body. From this inequality there 
results a vertical motion of the centre of gravity, and another axis of rotation. 
