The object of Mr. Atwood in the investigation, of which the results are recorded in the preceding ta ble, was to estimate the effects produced on different bodies, by assuming different forms for their sides; and for this purpose he preserved all their other ele ments constant. Thus, as the table illustrates, the breadth of the water section was in all cases denoted by 100; the distance of the centre of gravity of' the en tire body, and of the volumes displaced by 13; the area of the section of the volume displaced by 3600, and the angle of inclination of the constant magni tude of 15'. In the two last columns will be found the measure or stability for the different forms.
The table furnishes several remarkable conclusions. For example, by comparing the results of No. 6. with No. 3. the singular fact is disclosed, that if two isosceles wedges having their surfaces inclined at the same angle, have also the same breadth at the water's surface, and the distances between their centres of' gravity, and of the volumes displaced equal, as also the weights of the bodies themselves, then will the stabilities of the two bodies, when inclined to the same angle from the upright, be always the same. The same principle may also be remarked by com paring the form No. 2. with No. 9. and likewise No. 3. with No. 10.
But the circumstance here adverted to possesses a much more general character, it being equally true, whatever be the nature of the figure assumed for the sides, provided the surfaces below the water line in one vessel, are similar equal, and similarly disposed with respect to the water section, to the sides of another vessel above the same section. This remark able property may be demonstrated as follows: Let QC110, Fig. 15. represent a vessel, the sides of which above the plane of the water section pro ject outwards. and the sides below tile same plane in ward•. the vessel in this position heing`in a state of permanent equilibrium. Suppose the vessel to he deranged from that position by the action of any force, and let CH be the position of the water's sur face, in consequence of the inclination. Let also .AS11 and SBC be the equal areas produced, the former being immersed in the fluid, and the latter elevated above it; and let M and I be their respective centres of gravity.
Suppose now the entire body to revolve round the line All as an axis, and to perform half a revolution or 180'; then will the positions of its sides be entire ly reversed; those parts of them which in the original position of the body projected outwards, above the water's surface, in the new position, inclining in wards, below the same surface; and the other parts of them which in the position of permanent equilibrium inclined inwards being now found inclining outwards.
Let Fig. 16. denote this new condition of the body, and ch the position of the fluid surface, when the solid is inclined to the same angle as denoted by Fig. 15. Now since b, it follows that ABCO be ing applied to a b c o, so that the point A may coin cide with a, AB with a b, and consequently the point B with the point b, the two sections will be identical and equal in all respects. Also since the lines CH, c b, are equally inclined to the lines AB, a b, and cut off the areas ASH, ash, respectively equal to the areas BSC, b s c, it follows when the line AB coincides with a b, the points S and s must coincide also, and likewise the areas just mentioned. As a necessary consequence, the centres of gravity NI and I will co incide with the corresponding points m and i; the line 1\IL with nil, IK with ik, and consequently KT, with k 1. And since the area ASH is equal to the area BSC, and a s h to b s c, it follows that the four areas are equal to each other. Hence since the volume im mersed, is by the supposition in each case the same, , A v 7it follows that El. — and e t A v and there V fore ET = e t.
Now this equality between the lines ET and c t be ing independent of the positions• of the centres of gravity of the entire bodies, and also of the positions of the centres of gravity of the immersed volumes, it follows that if the distances of those centres be the same, that ER will be equal to c r, because by the hypothesis, the angles at which the bodies are inclined are the same. if therefore from El' we subtract ER, and from c t take c r, there will remain RT or GZ equal to r t or g 7.; and from which we infer that the sta bilities of the two bodies are the same.
Another property demonstrated by Mr. Atwood is, that when the vertical sections of one vessel arc ter minated by the arcs of a conic parabola, and the sides of another vessel are parallel to the plane of the masts above and below the plane of the water section, the stabilities of the vessels will be equal at all equal in clinations from the upright, when the breadths of the water sections, and all the other conditions are the same in both cases—a coincidence which could scarce ly be supposed to exist in bodies so dissimilar in form.