358 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



LDecembbb, 



frravity, as comparoil with that of water is, as 11 to 20 ; and let 

 tlie side^i of tlie transverse trianiriilar section be respectively as 

 follows : — viz., A B ^ 36-5 inches ; A V ^ ■H-2 inches ; and 

 B\' = 53-1 inches. 



Then, since the prism is uniform throughout the whole length, 

 the weiglits and solidities of tlie floating body and the immer- 

 sed part of it, will be truly represented by the areas of the 

 triangles ABV and CDV. Now, by the riiles of mensuration 

 and by logarithms, the area of the triangle A B V is I'ouud as 

 follows. 



Side B V = 531 Inches; 

 Side AV - 44-2 inches , 

 Side AB = 3li5 inches. 



,BV + AV+AB = l;'.as. sum of the sides of the triangular sect'nn of the prism ; 



J(BV+AV+AB)= 61 9. half sum of t e sides .. h.g. 1 ,s354.!>l 



la-8. hjlf sum minus H V .. log. l-l;i!'H;nl 



;;27, half sum minus A V .. log. l-3;')602"«9 



iiU'4, half sum minus A B ,, log l'4Sl'873(j 



log. 6-8(l42047, Sum ; 

 Area of the whole triangle ABV — 7a,S 1S3 square inches. ,J log. 2-9(121023, J Sum ; 



and, oonseipiently, the area of the immersed triangle CDV, being 

 to the whole area as 11 to 20, is 



20 : 11 :: rOSlSS : 439-00065 square inches. 



Bisect the sides of the triangles A B and C 1), in the points at P 

 and P ; draw V F and V P, and from the vertex V, set off V G 

 and V tj respectively equal to two-thirds of V F and V P ; then, by 

 mechanics, G is the position of tlie centre of gravity of the 

 triangle ABV, and g, tliat of the triangle CDV; join the centres 

 G and (/ by the straight line Gg; then, according to the second 

 condition of equilibrium, Gg is a vertical line. 



Since the area of the triangle C D V is known, the horizontal line 

 C D touches a given hyperbola described with the asymptoles A V 

 and B V; and C D is bisected by that curve in P, the point of con- 

 tact. Join P F, then P F is parallel to G g, and because G^ is 

 vertical, P F is also vertical, and consequently perpendicular to 

 C D, which is horizontal ; it is likewise perpendicular to the hyper- 

 bola Q P R which C D touches in P. Therefore, since the position 

 of the point F is known, the position of the straight line P F can 

 he found ; and for each perpendicular thiit can be drawn to the 

 curve of the hyperbola from the point F", there will be a position 

 in which the prism, whose transverse section is the triangle A B V, 

 can float in equilil)rio with the verte.x downwards; .and the differ- 

 ent positions of P F wliich satisfy the conditions of equilibrium, 

 may be determined, either by the solution of an algebraic equation 

 of the fourth degree; or geometrically, by the intersection of two 

 , hyperbolas, of which the elements of construction are known. 



When a body, floating permanently on the surface of a fluid 

 specifically heavier than itself, has its equilibrium of flotation dis- 

 turbed by the action of some extraneous force — that is, when the 

 centres of gravity of the whole floating mass, and of the immersed 

 part, are not in the same vertical line ; if a vertical plane be made 

 to piiss through those centres, the body will revolve upon an axis 

 perpendicular to that plane, and pa.ssing through its centre of 

 gravity; for when the impulse communicated to a body is in a 

 line passing through its centre of gravity, all the parts of the 



body move forward with the same velocity, and in lines parallel 

 to the direction of the impulse com-iunicated. But when the di- 

 rection of the impulse does not pass through the centre of gravity, 

 as is the case in the present instance, the body acipiires a rotation 

 on an axis, and also a ])rogressive motion, by which the centre of 

 gravity is carried forward in the same straight line, and with the 

 same velocity, as if the direction of the impulse communicated 

 had actually passed through the centre of gravity ; and it is a 

 curious mechanical fact, that the rotatory anil ]irogressive motions 

 thus cominunicateil, are wholly indejiendent of one another, each 

 being the same in itself as if the other did not take place. 



This follows from the general mechanical principle or law, th<it 

 the quantity of motion in bodies estimated in a given direction, is 

 not afl^ected or changed l)y the action of the bodies on one another. 

 The revolution of a body on its axis is produced by an action of 

 this kind, and therefore it can neither increase nor diminish the 

 progressive motion of the whole m.ass moved. ^Vhen a single 

 impulse only is communicated to the body, the axis on which it 

 begins to revolve is a line drawn through its centre of gravity, 

 and perpendicular to the plane whicli passes through that centre 

 and the direction in which the impulse is communicated. 



It is the nature of some floating bodies, when their eqiiilibrinm 

 of flotation has been disturbed, to return to their original ])osition, 

 after making a few oscillations backwards and forwards, upon an 

 axis similar to that above alluded to. But others, again, when 

 thqjr equilibrium of flotation is ever so little disturbed, do not 

 resume their original position, hut continue to revolve on an axis 

 passing through their centres of gravity, until they attain another 

 position, when they are again in equilihrio. In the former case, 

 the equilibrium is s said to be stable, and in the latter it is unstable, 

 and the body oversets. 



When the floating body is made to revolve from the position of 

 equilibrium, by the action of some external force ; if the line of 

 support''' move, so as to be on the same side of the line of pressure,f 

 as tliat part of the body, which becomes depressed below the surface 

 of the fluid in consequence of the inclination from the state of equi- 

 librium ; then, the ei[uilibrium is stable, and the body will restore 

 itself; that is, it will resume the position which it occupied before 

 it was submitted to tlie action of the deflecting force. But if the 

 line of buoyancy, or the line of support, be on the same side of 

 the line of pressure, as the emersed or elevated part of the 

 floating body, then the equilibrium is unstable, and the body will 

 recede farther and farther from its original position, until it finally 

 oversets. 



When a body floats upon the surface of a fluid specifically 

 heavier than itself, the force which tends to make the body revolve 

 about its centre of gravity, is equal to the weight of the body, 

 acting on a lever, the length of wliich is equal to the horizontal 

 distance between the line of pressure and the line of buoyancy ; 

 and when this distance vanishes, that is, when the centres of 

 gravity of the whole body and the immersed part of it are in the 

 same vertical line, the force tending to cause the body to revolve 

 is equal to nothing. 



When the floating body is any how inclined or deflected from 

 the position of equilibrium, and when the line of buoyancy falls 

 on tlie same side of the centre of gravity of the whole flo.iting 

 mass, as that part of the body which becomes depressed below the 

 surface of the fluid in consequence of the deflection, the lever by 

 which the force acts is said to he affirmative, and the force tends 

 to establish the equilibrium, or to restore the body to its original 

 position. But on the other hand, when the line of buoyancy is on 

 the same side of the centre of gravity of the whole body, as that 

 part of it which becomes elevated aliove the surface of the fluid in 

 consequence of the deflection, the lever by which the force acts, is 

 said to be negative, and the force tends to overset the body. 



These are the chief principles necessary to be known in taking 

 a cursory view of the subject ; and we shall now proceed to show 

 in what manner the momentum of stiibility is to be calculated. 



Let the vertical transverse section of the floating body be uni- 

 form, or the same from end to end ; then put — 



« =: area of the transverse section of the immersed part of the body ; 

 d = distance between centre of gravity of tlie wln)le and immersed part ; 

 / =: iensth of the water-line, or the base of tbe immersed section ; 

 (j> z= small angle of incliuation or deliectjon ; 

 w = whole weight of the Soating mass ; 

 m — momenlum of stability. 



* The vertical line which passes through the centre of gravity of the immersed part of 

 the floating body, is called " the line of buoyancy," or '* the line of support." 



t The vertical Hue which passes through the centre of gravity of the whole floating 

 mass, is called " the line of pressure." 



