4.58 



HYDRODYNAMICS. 



E<iuilibri. arising from multiplying the distance by the solid it- 



umand AB 3 x*rf~z 



.!i.iluy of se ]f w iu be ^ t an j t h e sum of the products 



corresponding to the whole line z will be fluent of 



this quadratic equation, n- n 



a~s 



and 



Floating 

 Uodies. 



y a t 

 4- -^ . 

 ' Oc* 



KquUtbii* 

 urn and 



Stability of 

 Floating 



since A represents the volume of the Cor. I. From this proposition, we may infer, that 



part immersed by inclination, and also of the part ele- 

 vated by inclination, the distance of the centre of gra- 

 vity, of the immersed part or c X, also of the part ele- 

 vated, or b X, will be fluent of aTTT * and the dis- 

 tance between the two centres of gravity in the line 



A B** x s d z 



t> c will be twice this quantity, or fluent of - . 



1 A. 



Substituting this value for b in the general equation, 

 we have GZ == fluent of ,-^rv h s , which is 



figure. 



the axis of motion and the part immersed by inclination, 

 and the solid will float permanently ; and, 2. That if 

 the first member is less than the second, the line of sup- 

 port q z will be on the contrary side of the axis, and the 

 body will overset. Hence it is obvious, that be- 

 tween these limits, we must have the equilibrium of in- 

 difference which takes place when fluent of - - 



t=JU 



If the solid has an uniform figure and dimensions, then 

 putting D for the area of any of the sections immer- 

 sed under the fluid, the solid contents of the volume 

 immersed will be Dz, hence V=Dz; and since AB is 



now a constant quantity, ' a * - c 



we have fluent of = 



4* 



8D 



PROP. IV. PHOB. 



12Dz 

 GZ = 



12D 



To determine the limits of stability and instability 

 in a parallelepiped depending on the dimensions and 

 specific gravity of the solid. 



Case of a 

 floating pa- . 



with one of surface EF upwards, and IK the surface of the fluid, 

 iu Hat sur- Through its centre of gravity G draw SGL parallel to 



In applying the preceding expressions to a parallele- 

 piped, let EFDC be its vertical section, with its flat 



faces up. 



PLATE 



rccxv. 



Fig. 2. 



CE, and let us take 



c = CE 



c = CD 



n= specific gravity of the solid, or n : 1 = SN : SL. 

 Then if O be the centre of gravity of the part immer- 

 sed, and since n : 1 = SN : SL, or CL, we have SN = 



nc; GO = , and ABCD = acn. Substituting 



' 2 



these values in the general expression already found, 



a 3 * *XC nc 



we nave GZ = ; and since the e- 



\2acn 2 



quilibrium is one of indifference, when the first mem- 

 ber of the expression is equal to the second, or when 



a 3 .? s x c n c 

 fa = > we " ave > ty the resolution of 



whenever -,-7-j 

 o c 



is less than i, or when the heiht c of 



the general expression required. 



Application Now it follows, 1. That if the first member of this e- 

 ot the for- A B 3 X sdz 

 nmla to the quation, viz. fluent of - -^ , is greater than the 



case of uni- 

 formity of second h s, the line of support QZ will be between 



the solid has a greater proportion to the base than that 

 of i/ 2 to K/ 3, two values may be assigned to the 

 specific gravity of the solid, which Will cause it to float 

 in the equilibrium of indifference. If, for example, c=0, 

 we have n =: \ :: */ J _ JL, which gives 



= i + 0.28868 = 0.788G8 

 w = | 0.28868 = 0.21132. 



Cor. 2. If the specific gravity of the solid is very 

 small compared with that of the fluid, the term ---- 



must be greater than 



2 



, and the solid will float 



permanently with the line EF parallel to the horizon. 



Cor. 3. If the specific gravity of the solid is increa- 

 sed beyond .21132, then, since this is the limit at which 

 it ceases to float with stability, if it is placed with its 

 flat surface upward, its equilibrium will be instable, 

 and it will therefore assume a position of permanent 

 equilibrium. By increasing the specific gravity from 

 .21132 to .78868, the instability increases at first, and 



reaches its maximum when =: -r ; it then diminish- 

 es and vanishes at the second limit when n = .78868. 

 When n is between .78868 and 1. the body will float 

 permanently with its flat surface EF horizontal. The 

 maximum of instability is found by putting the least 



a } s s x c en 



increment of the quantity =: 0, 



Via en 2 



considering n as variable, and making a=c. 



Cor. 4. If the height SL of the parallelepiped is in a 

 less proportion to its base CD than that ot ^/2 to ^/3, 

 there is no value of n at which the stability will vanish ; 



for in this case the quantity \/ V become im- 



6c2 



possible. The solid will therefore always float perma- 

 nently with its surface EF horizontal. 



PROP. V. 



To determine the limits of stability and instability of 

 a square parallelepiped when one of its diagonals is in 

 a vertical position. 



Let EDCF be a vertical section of the parallelopi- p LATE 

 ped floating on the surface AB of the fluid, and let G, CCCXV. 

 as formerly, be the centre of gravity of the solid O, Fig. 3. 

 that of the part immersed, and n the specific gravity of 



the solid. Then if DC=rt, we shall have GC = -=. 



But since HB=HC, we have ABC=HB 2 ; and since 

 ABC : DEFC = : 1, we have ABC = a'li, HB = 



OC= 



and GO= ^r 



a X a < 



