HYDRODYNAMICS. 



159 



If we now apply the general expression GZ = By inserting these values in tlie general formula Kquilibri. 



vad AB^X* AB J * , uui and 



- ki to the present case, we shall obtain GZ = j ^ ft t, we have AB 3 =Sa 3 x 1 n 1 ; D = Stability of 



nX3 v's*" SnSvi i* Brdics ' 



andconse. a'n, and since GO=rf we obUin GZ= - X I~ * ^~~' 



|a**t 



. a X 33* 



In order, 



x 1 



12o*n 



. Thi* value of 



vTx 3 



sJEZL 

 vTx s 



therefore, to obtain the limit between the stability and GZ being put =0 to obtain the limit, and the whole 

 instability of floating, we must make - = hs, as being multiplied by _. .__? we shall hare 



,. 8 axS S8n 



m Prop. I\ . or making = , 



' 120*11 v'Tx 3 



shall have | vT = ~ ~ 



. 



pen t 



nece&tvily I. 



-3. If * : 1 the 



equilibrium . : 



ratio than tr...- .he * 



if* exceed* '. 

 nentK , with tlie .. 



Paor VI. 



1 = 2 av/l , and GH = a x ^ . 



Now if h I* the centre of gravity of the area A KB 



len from the property of the centre of gravity we have 



X area EDCF = area ABDCFA x OH area 



AEB x HP, ora X VT v'l = a x OH 

 'Tir, and by reduction 



gJ<A ^'8 X S^ + 



T 10 X H 



tracting from this expression the value of IIG = 

 a X v' x _ v^HT" = 3 



we shall have the line 

 _a X 3-3 



X M 



we 



I =T- 7-^-, and I n= : 



or =,, = . 28 123, the 



j X \ r i _ 

 ri, the limi 



specific gr ! rium of indifference 



l>eg'i -pecific gravities at 



whic?. the solid w.. 1oat wit!: .--nd instability, 



-rom the value of GZ given above, 

 .'. or extremely small, the solid 

 ' 



with the angle up- 

 v;due of GZ must 



will float with the 



elbreif n: 1 inale** 



I will overset ; but 



I float perma- 



To determine the limit of tal ili'._> nd inttability in a 

 square parallelepiped with one of it* angle* upward*, 

 when its tpecific gravity is greater than one half of the 

 ipecific gravity of the fluid. 



Let EDCF, Plate CCC \ \ I . ;. be the square pa. 

 XV. rallelopiped, which having a greater specific eravitv 

 than 1. that of the fluid will >ink so that the duuronal 

 FI) i* below the surface of the water IK. {': 

 same symbol* as in the last proposition, we have the 

 ansa ABDCFA = a , and the area EAB = EH' = 

 a' a* , and consequently EH =aSlm, nod AH = 



required. 



Cor. Hence from this, and the prrcediiKj pro] 

 tion*. we have the four^ limiting values of the specific 



gravities, via. 4 </^ j; ^ ; | and .J + v/^I', nr 



.211, .291, .718, and .789 ; that is, if the specific*^ 

 i* less than 21 1, the parallelepiped, with its flat surfair 

 upward and horizontal, will float permanently in that 

 position, but will overset if the specific gravity is great- 

 er than .21 1 and less than .789- If the parJU-lopipeil 

 ha* one of its angles upward, when the specific gravity 

 i* ks* than 'Ml,- it will overset ; if greater than .281, 

 and leu than .7 1 8, it will float permanently with an angle 

 upward; but if the opecific gravity exceeds .718, it will 

 ovenet when placed in the fluid with an ang.le upward. 



P.OP.VH. 



If the parallelepiped ia placet! in the fluid in a posi- 

 tion of instable equilibrium, so as to o\ er.-t t or change 

 its position, it is required to ascertain the position whirli 

 it will assume when it continues to float permanently ; 

 or to ascertain through what angle the solid will re- 

 volve till it* centre of gravity and tlie centre of gravity 

 of the part immersed are again in the same vertical 

 line. 



Let F.FDC, Fig. 5. be the vertical section of the pa- PIATB 

 rallelopiped when in it* petition of instable cquilibn- 

 um ; IK the surface of the water, G iu n-ntre of gra- K * i 

 vity ; <) the centre of gravity of the part immersed. 

 Let the *oh<i, after ovenettina:, have re\o!\c<l through an 

 angle L'GS into the position YWVH, the part immersed 

 will j,ow be ZHVR, and QXR will be the part now 



immersed 



PZ and , , , 



X a=> X m, and X/= J X n, a and/ will be the cen 



in consequeuce of the inclination. Bisect 

 in m and n, join m\, n X, and taking 



e 



tresofgra 

 fall the pe 



applying the general equation 



, 



triangle* PXZ, and yXK. I , 

 rs a b, / c upon the line AB. Jn 



=^ ft to tin- 



\ 



pTMent cw, wehaveQXR=A ; ZHVRor ACDB=V 

 bc=b ; OG=/ ; sine of U(iO =1 . Make / equal tht- 

 tangent of the angle I 'GO. Then from the similari- 

 ty of triangle. QX= M' /I'^QK; XX^XIt. Mak- 

 ing SL=r, and VU or XQ=a : thenQR =a / ; and 



Now, since 



X 



* 4 



f r = rin. H XR : sin. XR n, consequently 



Riior Q : Xasrsin. XR; XRn, wehave-r-l 



