HYDRODYNAMICS. 



457 



<,..,. 



l-r:. 

 ifcilicy of 



loat-r..; 



.-, when one bo>ly of a system is removed from 

 its place, the corresponding motion of the common 

 centre of gravity, estimated in any given direction, is 

 to the motion of the body moved and estimated in the 

 same direction as the weight of that body is to the 

 weight of the whole system. Hence, considering I KM N* 

 as a system of bodies whose common centre of gravi- 

 ty is E, and that the body IWX, who*e centre ofgra- 

 \ ity is a, has moved, or been transferred to N X 1', whose 

 centre of gravity is /) we shall have volume WRMP or 

 AIM1B: rotamlWI or \\\l ) = bc: ET, the motion 

 e body, the motion of the centre of gravity of the 

 system. Calling, therefore, 



V = volume of the part of the floating body im- 



mersed, 

 A = the volume NXP, or the part immersed in con- 



sequence of the inclination, 



ft GO the distance between the centre of gravity 

 of the whole solid and that of the part im- 

 mersed, 



t sine of the angle of inclination KGS, 

 b =. be the space through which A has been trans- 



ferred. 

 Then, by the proposition b : ET = V : A, and ET = 



external force to revolve round in axis of motion, and Equilibri- 

 pass through different positions of equilibrium, the po- Su *?V, , 

 sitions of stability and instability must alternate, and ' 

 no position of either species can follow a position of the 

 same species. 



In determining, therefore, the position which a solid 

 will assume after it has been overset from any situa- 

 tion of instable equilibrium, we must ascertain the 

 angle of inclination through which the solid must re- 

 volve, so that the distance GZ may become evanes- 

 cent, and we must also determine whether any position 

 of equilibrium originally given is stable or instable. 

 This may easily be done from the value of GZ already 

 given ; for if we take any point / in the line Ell, and 

 through / draw qt i parallel to GO, then it is obvious, 



1 . That while ~y- = ET is greater than h t = 



ER, the part Z and the line of support OZ, will be 

 between the axis and the parts of the solid immersed 

 in consequence of the inclination which gives liable 

 equilibrium. 



2. That while 4- = ET is leu thsn h t = ER, 



ButF.R:EGorGO = 



Pior. III. 



To find an expression for the stability or instability of 

 floating, when the floating solid is not of an uniform fi- 

 gure and dimensions with respect to its axis of motion. 



Let us suppose in Fig. 1. that another section of the fi AI i 

 solid is drawn parallel to ADHD, and very near CCCXV. 

 it A small portion of the solid will be comprehend- fig- ' 

 ed between these planes; now, since the sine t of 

 the angle KGS is evanescent, and since \VXl>=IX W. 

 and the angle NXP= the angle IXW, the point X will 

 bisect the line A 13, and the points I', B, N will be co- 



incident. Hence, the evanescent area NXI'= ' - 



the part a and the line of support q will be on the op- 

 we have ER = posite side of the axis, and will give an instable e<]ili- 

 . brivm. Hence we can always determine, from t! 



tti consequently, RT=ET ER, or GZ= i la* of GZ .what, is the particular kind of equilibrium 



V with which the body will float when the angle of in- 



clination, and consequently its -ine, are assumed to be 

 If the floating solid should be of an irregular form, 

 the same demonstration will hold good; but we must, 

 in this case, consider that the volume or space immersed 

 by the inclination will no longer be WXP. but a space 

 which most be obtained by r*lmlst*on front the shape 

 and ilimiaiiniis of that volume. The centre* of gra- 

 vity of the volumes I'\ V I \ \V, will in that case be no 

 longer a and /'the centres of gravity of the areas, but 

 must be found by the usual rules. This proposition is 

 applicable either to homogeneous or heterogeneous bo- 

 dies, and enables us to determine the stability of ves- 

 sels or other bodies, at any angle of inclination from 

 given position of equilibrium ; for the stability is mea- 

 ured by a force equal to the upward pressure of the 

 fluid, or the weight of the loaded vessel applied per- 

 pendicularly at the end of the lever GZ moving round 



Uit a\. uf motion. 



POP. II. 



To ascertain the different positions of equilibrium in 

 which a body will float permanently on the surface of a 

 fluid, and to discover in which of these position* the 

 equilibrium is permanent or stable, and in which of 

 them it is momentary or instablc. 



order to do this, we must attend to the species of 

 equilibrium in winch tile *olid is placed previous to its 

 it. Atiuming that the body is in a 

 state of stable equilibrium, let it be inclined through an 

 angle till it is again placed in a position of equilibrium. 

 Then since during this inclination the upward pressure 

 of the fluid acts with a force proportional to GZ to di- 

 minish the angular distance from the primitive position 

 of equilibrium, it follow*, that the same force must act 

 on the solid, so as to increase the inclination or angular 

 distance from the second potitionof equilibrium to which 

 the body arrive* after revolving through the angle A 

 or any part of e it is obvious that the second 



= -- , and if we represent by 



O 



a line drawn 



through the middle of the solid, on a level with the 

 surface of the fluid, and parallel u> the larger axis, the 

 traneacent portion of the solid comprehended between 

 the adjacent vertical planes ADI IK, and the one tup- 



posed extremely near it, will be - -' x d:; the per- 



pendicular distance of the centre of gravity of this 



AR 

 evanescent solid from the point X is . In order to 



find the distance from X of the centre of gravity of 

 the whole volume immersed by the inclination, or tin- 

 common centre of gravity of the elementary solids 



- corresponding to the length :, we must 

 8 



multiply each elementary solid by the distance of 

 its centre of gravity from the horizontal line passing 

 through X, and divide the sum of these products by 

 the sum of the elementary solids. Hence, in the pre- 

 sent case, since the distance from X of the centre of 



^ of equilibrium mus instability and, j rf ^ eleinellUr y wlid U ** u, e product 



m general, that when a floating body is caused by an ' 



3 M 



VOL. XI. TAUT II. 



