ABC + EFG - GDE - HR r =0, and which condition furnishes for the value of the area of the triangle sought, 'IR r --ABC + EFG -CDE = 72 + 118 - 108 = 82.
Hence we shall have by proportion, as 82 X or numerically 119 : 82 : : : - 1 ; 119 whence H = 13.4 11.21, and consequently A r = AN + II r = 125.6 + 11.21 = 136.81.
Taking now the moments of the triangles ABC, CDE,'EFG, and Il r R in relation to the line R r, we shall, in the first place, have for the positions of the centres of gravity, rb -A b = 136.81-16.3=1:20.51 A d=136.81 - 59 = 77.81 rf=rA -Af= 136.81- 94 = 42.81 11,21 4 7* H = = 3.74 • and, secondly, for the moments required 120.51 X + 72 = + 8676.72 77.81 x - 108 = -8403.48 42.81 x + 118 = 5051.53 3.74 x - 82 = -306.63 and since the values of the positive moments amount to 13728.30, and of the negative moments 8710.16, it follows that the definitive moment will be the positive quantity 5018.14.
Here, therefore, as in the vertical plane passing through Pp, the sections which arc infinitely near to R r, will have their weights less than the resis tance of the water they displace; and the moments of the same sections act in a contrary direction to that of the total moment. Hence, by the second theorem, the positive moment 5018.14, above ded need, is a max M um.
At the extremities A and 0 of the vessel, the sum of the moments being zero, must furnish likewise two 2/1//a/Mthl values; and if we therefore collect together the series of maximum and minimum values of the moments having a tendency to arch the vessel, they may be represented as in the following table: According to Dr. Young, (see the preceding note,) the moments estimated at every 22 feet from the stern to the stem, may be represented as in the following table: II we now refer to the maximum and minimum sec tions which pass through A p and A q, in the inves tigation of Dupin, we shall perceive, that from the former to the latter, there must be a continual de clension in the value of each moment; and that con sequently, at the distance of 86 feet front the origin at A, the magnitude of the moment must be greater, than at the distance of 88.53 feet from the same point xvhere the maximum section ohteins; a conclusion which arcs with the thtory of Dr. Young. 111.: 111:11111CV, by ref,•rring to the maximum sections deduc ed by Dupin, we shall find that they are greater than the sections nearest to theta in the investigation or Dr. Young.
But the same col responienee in the results does not take place, if we compare the position of the maximum section of Dr. Young with the deductions of Dupin; since the former makes that section to exist at the distance of I feet from th:: after part of the wa ter line, and producing a strain equivalent to 5261 tons, acting at the distance of one foot; whereas the last mentioned philosopher estimates the strain at a similar point to he 4920.3 tons. This subject there
fore, like most others connected with naval architec ture, requires a more rigorous and extended investi gation; and we regret that our limits prevent its from entering farther into so interesting au inquiry at the present time.
The preceding investigation has been conducted on the supposition, that the causes of arching are due "entirely to the unequal distribution of the weight and pressure; but there is, in fact another cause for this important derangement of a ship's figure, arising from the longitudnal mid horizontal pressure of the water. According to Dr. Young, the partial pressure of the water in a longitudnal direction, affects the lower part of the ship only. compressing and shortening the keel, while it has no immediate action on the upper decks.
The pressure, thus applied, must obviously occasion a curvature, if the angles mad • With the decks by the timbers, arc supposed to remain unaltered, while the keel is shortened, in the same manner as any soft and thick substance, pressed at one edge between the fin gers, will become concave at the part compressed; and this strain, upon the most probable supposition resit( cling the co.nparative strength of the upper and lower parts of the ship, must amount, Dr. Young thinks. to more than one-third as much as the Mean value of tle- former. being equivalent to the effect of a NS eight of about tolls, acting on a lever of one foot in length, while the strain, arising front the un equal distribution of the weight and the displacement, amounts, where it is greatest, that is, about 37 feet from the head, to 526). in a seventy-four gun ship of the usual dimensions: and although the strain is con siderably less than this exactly in the middle, and throughout the aftermost hall' of the length, it is no w here converted into a tendency to "sag," or to be come concave. it must, however, be remembered, says Dr. Young, that w hen arching actually takes place from the operation of these forces, it depends upon the comparative strength of the different parts or the ship and their fastenings. whether the curvature shall vary more or less from the form, which results from the supposition of a uniform resistance through out the length. An apparent deviation may also arise from the unequal distribution of the weight through the ship: thus the keel may actually sag. under the step of the mainmast, even when the strain, as here calculated, indicates a contrary tendency with respect the curvature of the whole ship.