HYDRODYNAMICS. 



461 



i- thehorizontallineDE,thenmakinffPQ=A,sin.BXE=i; 

 d tang. BEX = /, and EC = a, we have CU = < a ; CB= 



ccexv. 



I /'"* . rr PH 

 V -g~; *-<->_ T -n_ 



ta' 



art a 



ABC = CH' = -=-. Putting area BXE = w, we have 



la* 

 are* CDE or ABC : area BXE = 1'Q : OT or : = 



6:OT = ^, andOG = ^. AddingCOtoCG, 



we obtain CG =: j~f g m 



we have CV = GC x ^~2 and CV = 



_ 

 ^ 9 "~ 



3ta*t 



But since 1 : n = area CAHB: IFCV, or as CH' : C V-, 



wehavevT= TV and /^t ' = 



- 



If we awume the angle OC.T or BX and 



take (I. = 1. we .hall fin<l BXE = = .039393. and 

 I'V = 6=0.750*9, which being Mibrtituted in the 



above value rf ., give. ^Ts 



Fig, It, 



0.51094 and n = 0.361, the specific gravity with which 

 the solid will float in stable equilibrium, when it* di.i- 

 gonal 'n inclined 1 .'> to a vertical line. 



The preceding solution if applicable to all case* in 

 which n i* between ,', and _ T , ; and by a simil. , 

 ces* we may obtain an equation for the ca*e when tli 

 specific gravity is between \\ and : J. In tin's case, 

 the solid will floct permanently with the line 1C up- 

 ward , but inclined to the vertical at some angle be- 

 tween and 18 26' 9*. 



Atwood has collected into the following abstract, 

 the varion* poiitions which the square parallelopipcd 

 assumes a* depending upon it* ipecific gravity. 



1 . If H is between nd A \/\ iT(* shewn in 



r" Til ll<-' " "* _., . ... 



< float permane 



nit lx>t 



between and U.21 1 , the 

 with a flat surface upward*, and 



IS, 



jjf .. 

 Fi*> l. 



reen .211 and 25, (as aliewn in Fig*. 

 12, IS, 14), it will fljat permanently with a flat sur- 

 face upward, but inclined to the horizon at different 

 angles from O 9 corresponding to .21 1 to 26' S4' corre 

 spending to .25. 



.1 between .25 or T ' T and .281 or / (a* 

 15, *hewn in Figs. It, 15, 16), the solid will float with 

 only inimere<l, the diagonal l>eing inclined 



at various angle* from 18 26", corre- 

 sponding to .25 or -,',, and corresponding to .281 



T ' T ',, 



It n is above .281 or y,, (a* in Fig*. 16, 17), the 



solid ill float permanently with it* diagonal vertical, 

 till the specific gravity become* .718 or 



it H it below .718 or !; and ~j or \\, (a* in 

 i-- Figv 1 T, 1 ^), the colid will float with the diagonal in- 

 clined to the vertical at angles varying from 0* corre- 



spending to .718 to 18 9 20 1 coiTCSponding to ,~5, tlircc Hquilibn 

 angles of the solid being immersed. U1 " and 



6. If n i between .7 1 8 or |$ and .789 (as in Figs. 1 9, p,^ 

 20, ai), the solid will float with a flat surface upward H< ^| lcs f 

 and inclined to the horizon at various angles bet WITH -- 

 26' 34' corresponding to .75 and corresponding to PLATE 

 .789. t'CTXV. 



" 7. If 11 is between .789 and 1.000, the solid will float J'j 19 ' "> 

 permanently with a flat surface parallel to the horizon. 



8. When the solid revolves round its larger axis, or 

 axis of motion, so as to complete an entire revolution 

 of 360, it will pass either through 16 or 18 positions 

 of equilibrium. If n is between .211 and .'281, or be- 

 tween .719 and .789, the positions of equilibrium will 

 be tixtetn, eight of which will be positions of stable 

 and the other eight of instable equilibrium, alternating 

 w ith each other. If n is not within these limits, the 

 solid in the course of its revolution will pass only 

 through tight positions, four of which are positions of 

 stable, and the other four of instable, equilibrium. 



I n the preceding propositions, the solid is supposed 

 to have a uniform figure in respect to the axis of mo- 

 tion, o that all iu vertical sections arc equal. But 

 when the solid has such a form that the sections arc un- 

 equal, a different proccs* of investigation, tluuiyli de- 

 pending on the same principles, must be employed fur 

 _' iU position* of equilibrium. We- shall con- 

 tent ouneivn with giving Mr Atwood's application of 

 the preceding principle* to a cylinder. 



PHOP. XI. 



To determine the position* of *Uble and inUble equi- 

 lil>rium, in a cylinder placed on the surface of a fluid, 

 with it* axi* in a vertical i 



. j|. 



Let EFC1> represent the cylinder with it* axi* NP 



vertical, and let it sink to the depth (Jl 1 . M.tke QA=r, 



the distance between the centre* of gravity G and ( >, 



' >=*, and let AIBHSKA represent the circular 



> of the cylinder formed by the surface of 1 1 it- 



fluid. Draw any diameter IS, and another diameter AB, 



perpendicular to IS, and let IS be the direction of the 



XB round which the cylinder i* moveable. Through 



U" draw the double ordinate Kli, and make 



QWa and T=3.14159, the ratio of tlu 



the diameter of a circle. Now, from 

 111. it follow*, that the solid will float perma- 



nently with the axi* vertical, when fluent of KH ^ rf:! 



is greater than A, and that the equilibrium will be in. 

 table if /i i* the greater of the two, and therefore when 

 these two quantities are equal, the equilibrium will 

 be the limit between stability and instability. To 

 apply this to the present case, we must find the 

 K H) v Ai 



fluent of --^. Now, *ince QS=r, QW=t, we 



have, by Euclid, B. III. Prop. 35. 



KH=2X VPH?, and KH 3 i/*=8XrvZFtxrf, th 

 fluent of which quantity, when : increases from to r, is 



wr wr w 



8 x - r -Tf- wl for both *emicuele* 



4 ID 16 



fluent of KH'X<'i=3wr ; and since l'Q=/w, and tin- 

 area of the circle AIBS=r t*, we have V= r" / n. But 





