Ill 



Miir. 



Mill'. 



612 



UbUity ; but the triangle OKS i rimilar to A'MD, all the side* of one 

 being jKTjwudiciiUr tn those of the other : therefore 



A'M : OH : : OK : os. 



Again. A'M varies with DM ; therefore, OK vane* with a 8 and v.og 

 or I.I, 1 : that U V.OB in in mi i alio the momentum of the ship's stabi- 

 lit;. The point 8 U called the meta-centre, and it indicates the mot 

 elevated pc*iti<.n which the centre of gravity o can have consistently 

 with the stability ; for when u coincide* with 8, it is evident that the 

 above expression vanishes. 



Since the depth of the ship doe* not enter into the expression l.l> 3 , 

 it may be inferred that when ships have equal lengths and breadths, 

 they will have equal stabilities, whatever be their depths. Again, when 

 tin V have equal breadths, their stabilities will vary with their lengths ; 

 aud when they have equal lengths, the stabilities vary with the cubes 

 .] breadths. It follows also that, in general, the stability of a 

 ship is directly proportional to its length and to the cube of its breadth, 

 whatever be its depth. 



In the next place, in order to find npproximatively the relations be- 

 tween the velocity of a ship and its dimensions, let the ship ) 

 represented by an isosceles triangular prism having the greatest rect- 

 angular surface uppermost and parallel to the surface of the water, and 

 the triangular ends perpendicular to that surface, so that the fore and 

 after parts are rectangular planes inclined to the same surface ; then 

 for the present purpose we may assume that the resistance experienced 

 in moving through the water is expressed by r. 3 A sin* I. [HYDRODY- 

 NAMICS!, where r is the velocity of the vessel, A the superficies of the 

 inclined front, aud I its inclination to the surface of the water. But 

 the resistance of the water is directly proportional to the moving 

 power, that is, to the pressure of the wind on the sails, and this last 

 varies with the area of the sails, which may be represented by a : 



therefore, r 1 ex 



Again, the momentum of the ship's sta- 



A. sin.* I 



I'ility is directly proportional to the momentum of the wind on the 

 sails, and the latter is expressed by the product of the area of the sails 

 into the height of the centre of pressure (centre of gravity of the sails) 

 above the axis of rotation : now this height varies with the height of 

 the sails ; that is, with the square root of their area, the sails in differ- 

 ent ships being supposed to be similar plane figures. Therefore, the 



momentum of stability varies with a*, or o oc (moment, of stability) 3 . 

 Let / be the half length of the vessel, ft its breadth, and d its depth ; 

 tin n the momentum of stability as above may be represented by /.ft 3 , 



A will be equal to 6(1* + d*)* and sin* I to Hence, 5 oc 



_ '; and this being simplified, neglecting I*, which may be 



considered as small when compared with P, we have r x 



d* 



This expression indicatea that the velocity will be increased by 

 diminishing </, which may represent the ship's draught, or the depth 

 to which she is immersed. It is evident also that the like effect will 

 take place if the length or the breadth, or both, are increased (the 

 breadth of the sail being supposed to increase with that of the ship) ; 

 and since, in this case, the stability will at the same time be increased, 

 this will permit the ship to carry a greater quantity of sail ; but the 

 number of the crew being supposed to vary with the area of the sail, 

 an increase of the latter is in general attended with a corresponding 

 increase of expense. The factors in the value of v being different 

 powers of A, /, and <l, it thence follows that ships having the same pro- 

 portions possess unequal sailing properties; it may be perceived 

 indeed that a small ship built according to the proportions of a large 

 one which is known to sail well will not possess the like good quality. 

 On the other hand it may be inferred that a vessel having the same 

 proportions as a good one of smaller dimensions will be superior to the 

 fatter. If two ships carry sails proportional to their stabilities, and if 

 the height of the lower tier of guns above water be the same in both 

 when the ships float upright ; then, the inclinations of the planes of 

 their masts being also supposed to be equal, the lower guns of the 

 mailer ship will be farther from the water than those of the larger 

 ship, and, in action, the latter might be in danger when the other 

 would be safe. Consequently, if the greatest possible quantity of sail 

 be given to the smaller ship, a smaller quantity relatively to its 

 stability ought to be given to the larger. 



The discovery of the elements on which depend the stability and 

 the sailing properties of ships will probably be made rather by study- 

 ing the proportions of such as from experience have been found to 

 possess the desired qualities, than by purely scientific researches ; and 

 as a step preparatory to this study it will be necessary to be acquainted 

 with the methods by which, in a body like a ship, which cannot be 

 considered as corresponding to any geometrical solid, the areas of 

 sections, the volume of the whole or of the part immersed in the 

 water, the position of the centre of gravity of the hull or the sails, &c., 

 are found. 



The method of equidistant ordinatea is generally used for these 

 purposes; and in finding the area of a section, some line in it being 

 taken as an axis, an uneven number of lines as ordinates are drawn or 



supposed to be drawn at equal distances from one another perpen- 

 dicularly to that line till they meet the surface of. the ship. The 

 lngth. o f these ordinates being known by actual admeasurement, or 

 by the scale of the drawing, and also the distance between them ; and 

 the curve line in which the plane of the section meeU the surface of 

 the ship being considered, between every three consecutive ordinates, 

 as an arc of the common parabola ; by Sterling's rules, the area will be 

 found a follows: let the figure represent the half plan, divide it into 

 any number of equal parts, say eight. 



,,.,. 



Then area = !!(A+ 4p-r2o.) = (- + Zr + g) in which 

 3 3 \2 / 



A = sum of first and last ordinatea, 



p = sum of even ordinates, 



Q = sum of remaining ordinates, 



r = common distance between the ordinates ; 



or, according to his second rule, the ordinates may be drawn so that 

 the equal intervals are made some multiple of the number 3, in which 

 case the 



Area = -T(A + 2p + 3(j) = -I(- + r + IJq ) where 

 8 4 \2 I 



A - - sum of first and last ordinates, 

 I- = sum of 4th, 7th, 10th, 4c., ordinates, 

 <J = sum of remaining ordinates, 

 r = common distance. 



Either rule may be used, for whether the curve be considered as a 

 common parabola, or as a cubic parabola, will scarcely affect the 

 practical result. 



By either of these rules the area of a vertical section through the 

 middle of the ship's breadth in a longitudinal direction, or through 

 any part of the ship's length in a transverse direction, and also the area 

 of a horizontal section coincident with or parallel to the surface of the 

 water, may be found. The same formula may be applied to determine 

 the volume of the whole hull, or of the displacement ; for this purpose 

 vertical sections may be supposed to pass through the equidistant 

 ordinates drawn perpendicularly to the longitudinal axis of the ship, 

 and their areas to be computed as above. The formula: may also be 

 employed to determine the momentum of any part of the ship with 

 respect to some line aboyt which it may be made to turn, or with 

 respect to a plane passing through such line. 



In order to find the centre of gravity of the displacement, that is, of 

 the immersed part of the ship (considered as a homogeneous solid) 

 when the ship floats upright ; imagine that immersed part to be divided 

 by equidistant vertical planes perpendicular to the ship's length, and 

 also by equidistant horizontal planes ; then the area of each horizontal 

 section between every two vertical planes being multiplied by the 

 vertical distance of that horizontal section below the surface of the 

 water, and all the products being added together by either of the above 

 formulae, the sum will be the momentum of the immersed part with 

 respect to the horizontal plane at the surface of the water. This 

 momentum being divided by the volume of the immersed part com- 

 puted as above, gives by mechanics the distance of the centre of gravity 

 below the surface of the water. . In a similar manner may be found 

 the position of the centre of gravity with respect to a vertical plane 

 passing perpendicularly to the length of the ship, suppose at one of its 

 extremities; and thus its position may be completely determined. 

 From the symmetry of the ship's figure on each side of a plane passing 

 through its masts and keel, the centre of gravity of the whole ship will 

 always be in that plane ; and that of the immersed part will be in the 

 same plane when the ship floats upright. 



The position of a vertical line passing through the centre of gravity 

 of the whole ship may be found by determining that of a vertical line 

 passing through the centre of gravity of the displacement ; for, by 

 hydrostatics, when the ship floats with its masts upright, these lines 

 are coincident. But the determination of the place of the centre of 

 gravity in this line is a problem of considerable difficulty on account of 

 the complexity of the subject, arising from the form of the hull, the 

 positions of the masts, rigging, guns, lading, &c. : and it can only be 

 found mathematically by ascertaining the place of the centre of gravity 

 and the weight of every separate object constituting the mass of the 

 ship and its lading. The momenta of all these objects being computed 

 with respect to any one plane, as that of floatation (the horizontal 

 plane at the level of the water), the difference between the sums of the 

 momenta of the objects above and below that plane is, by the nature 

 of the centre of gravity, equal to the product of the weight of the 

 whole ship into the distance of the required centre of gravity from the 

 same plane. Hence the situation of this point might be found. 



The most simple mechanical method of finding the centre of gravity 

 of a ship is probably that which was proposed by Mr. Major in the 

 ' Annals of Philosophy,' June, 1826. It consists in making the ship 



