PONCELET'S SECOND THEOREM.

To approximate to the value of Va2-b2, let aa 36 be the formula of approximation, then will the relative error be represented by

Now, let it be observed that a2 being essentially greater than b', >1;

therefore, be represented by cosec. 4, then will the relative error be

value in the preceding formula, and observing that -a secẞ tan. ==

Assuming, and to represent the values of 4, correspording to the

a

greatest and least values of and observing that in this case, as in the

preceding, the values of a and 3, which satisfy the conditions of the question, are those which render the values of the error corresponding to these limits equal, when taken with contrary signs, to the maximum error, we have

-1+a sec. 4ẞ tan. 4, = 1 — Vaˆ—3* (14).

α

1- —a sec. 4+ẞ tan. 4—1—a sec. 42+ẞ tau 42. . . . (15). The latter equation gives, by reduction,

And a sec. 4-ẞ tan. 4, ẞ cot. (41 + 41⁄2) . . . . (17).

....

Substituting these values in equation (14), and solving in respect to

cos. § (41+42) + √ cos. 4, cos. 41⁄2

The maximum error is represented by the formula

(18).

..(19).

These formula will be adapted to logarithmic calculation, if we assume

§ (41 +42)='F1, and

cos. (4-4)

=cosec. ; we shall thus obtain from

sin. (+42)

equations (16) and (17) a = 3 cosec 2, Va-323 cot. Y, and a sec. v. -ẞ tan. 3 cot. ; therefore, by equation (14),

The form under which this theorem has been given by M. Poncelet is different from the above. Assuming, as in the previous case, the limiting

a

values of to be represented by cot. 4, and cot. 42, and proceeding by a b

geometrical method of investigation, he has shown that if we assume tan. cos. 1, tan. 42 cos. w2, w1+w2 = 271, wi -w= 28, and cos. 1⁄2=

If the least possible value of a be 16, and its greatest possible value be infinite as compared with b, M. Poncelet has shown the formula of approximation to become

Na2b2 = 1·1319a-0·726366..... (23),

with a possible error of 0.1319 or

nearly.

If the least possible value of a be 2b, and its greatest possible value infinite compared with b; then

Vab2 = 1.018623a-0-2729448 . . . . (24),

....

with a possible error of 0186 or d nearly.

NOTE C.

ON THE ROLLING OF SHIPS.

(First published by the Author in the Transactions of the Royal Society for 1850, Part II.)

Let a body be conceived to float, acted upon by no other forces than its weight W, and the upward pressure of the water (equal to its weight); which forces may be conceived to be applied respectively to the centre of gravity of the body and to the centre of gravity of the displaced fluid; and let it be supposed to be subjected to the action of a third force whose direction is parallel to the surface of the fluid. Let AII, represent the vertical displacement of the centre of gravity of the body thereby produced*, and AHI, that of the centre of gravity of its immersed part. Let moreover the volume of the immersed part be conceived to remain unaltered ✦ whilst the body is in the act of displacement. If each centre of gravity be assumed to ascend, the work of the weight of the body will be represented by W.AH,, and that of the upward pressure of the fluid by + W.AH,, the negative sign being taken in the former case because the force acts in a direction opposite to that in which the point of application is moved, and the positive sign in the latter, because it acts in the same direction, so that the aggregate work Eu, (-ee equation 1, p. 122.) of the forces which constituted the equilibrium of the body in the state from which it has been disturbed is represented by

Moreover, the system put in motion includes, with the floating body, the particles of the fluid displaced by it as it changes its position, so that if the weight of any element of the floating body be represented by w1, and of the fluid by w, and if their velocities be v, and v2, the whole vis viva is represented by

*When a floating body is so made to incline from any one position into any other as that the volume of fluid displaced by it may in the one position be equal to that in the other, its centre of gravity is also vertically displaced; for if this be not the case, the perpendicular distance of the centre of gravity of the body from its plane of flotation must remain unchanged, and the form of that portion of its surface, which is subject to immersion, must be determined geometrically by this condition; but by the supposition the form of the body is undetermined It is remarkable what currency has been given to the error, that whilst a vessel is rolling or pitching, its centre of gravity remains at rest I should not otherwise have thought this note necessary.

†This supposition is only approximately true.

If the centre of gravity of the body or of the displaced fluid descends ( property which will be found to characterise a large class of vessels), AH, in the one case, and ▲H, in the other, will of course take the negative sign.

In the extreme position into which the body is made to roll and c which

wo=0,

or if the inertia of the displaced fluid be neglected,

U(0)=W.(AH,—▲H,) . . . . . (27).

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 immersed part; so that other things being the same, that ship is the most stable the product of whose weight by this difference 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.

This conclusion is nevertheless in error in the following respects:

1st. It supposes that throughout the motion the weight of the displaced fluid remains 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 ris vira may cease at once when the principal oscillation terminates. The frame of a ship and its masts are, however, elastic, and by reason of this elasticity there cannot

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 destroying the equality of the weight of the displaced fluid to that of the body. From this Inequality there results a vertical motion of the centre of gravity, and another axis of rotation.

but result oscillations, which are independent of, and may not synchro nise with, the principal oscillation of the ship as she rolls, so that the vis vica of every part cannot be assumed to cease and determine at one and the same instant, as it has been supposed to do.

3rdly. No account has been taken of the work expended in communicating motion to the displaced fluid, measured by half its vis vira and

1 2g

represented by the term Ewo in equation 26.

From a careful consideration of these causes of error, the author was led to conclude that they would not affect that practical application of the formula which he had principally in view in investigating it, especially as in certain respects they tended to neutralise one another. The question appeared, however, of sufficient importance to be subjected to the test of experiment, and on his application, the Lords Commissioners 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 Master Shipwright of that dockyard, and Mr. Rawson, were kind enough to undertake them.

These experiments extended beyond the object originally contemplated by him; and they claim to rank as authentic and important contributions to the science of naval construction, whether regard be had to the practical importance of the question under discussion, the care and labor bestowed upon them, or the many expedients by which these gentlemen succeeded in giving to them an accuracy hitherto unknown in experiments of this kind.

That it might be determined experimentally whether the work which must be done upon a floating body to incline it through a given angle be that represented by equation 27, it was necessary to do upon such a body an amount of work which could be measured; and it was further necessary to ascertain what were the elevations of the centres of gravity of the body and of its immersed part thus produced, and then to see whether the amount of work done upon the body equalled the difference of these elevations multiplied by its weight.

To effect this, the author proposed that a vessel should be constructed of a simple geometrical form, such that the place of the centre of gravity of its immersed part might readily be determined in every position into which it might be inclined, that of its plane of flotation being supposed to be known; and that a mast should be fixed to it, and a long yard to this mast, and that when the body floated in a vertical position a weight suspended from one extremity of the yard should suddenly be allowed to act upon it causing it to roll over; that the position into which it thus rolled should be ascertained, together with the corresponding elevations of its centre of gravity and the centre of gravity of its immersed part, and the vertical descent of the weight suspended from the extremity of its arm. The product of this vertical descent by the weight suspended