460 



HYDRODYNAMICS. 



Uquilibri- Sin. n RX X t r 



um and sin. nXRrr , and substituting in place ot 



stability of V4-H* 



Floating 1 



Bodies, sin. RX its value - , we obtain sin. n XR 



N Y ' Vl+/ 



=r, and cos. XR= 



and since X rf= - "*- '- , we have X c = 



3 



a 2+t- _, v , 

 =X-i^. But X4 = 



by similar triangles, and therefore b c r= 2 X c = 

 =l.. Now the immersed part ACDB= 



The preceding proposition is applicable only to the case Equilibri. 

 where the surface of the water intersects the parallel sides s . a l .",\'. t an - f 

 YH.VVV. In order to obtain the angle of inclination from no .,/ ng 

 a position of equilibrium, with the flat surface horizon- 

 tal, and the specific gravity of the solid, when the fluid 

 surface passes through one extremity of the base, let 

 AECD, Fig. 6, be a vertical section of the square pa- pV g 

 rallelopiped, and let the water line IK pass through D. 

 Then putting CD=rt, and *=tangent of KDC the an- 



u 2 t 

 gle required, we have KC=a /, and area KCJ : -^ ; 



but area KCD : area AECD=n: 1, we have n=-~ Sub- 

 stitutingthis valueof n in the formula, or value of* 2 in the 



/ i rt in n 



_ 

 preceding propo3ition,we obtain s- = g- 



, 



t 



2 a c n= V, and the volume QXR = = A. Substitu- 

 ting, therefore, these values in the equation GZ = 



consequently = 



or 6P- 



iA . ,, 2X2+/* 

 _ hs, we have GZ - X 



!2 = G< 3/* 2-f-G/ 3 3t* 2t*, or 4l'=6t 2, 

 which gives <r=|rri, that is *=^,and 1=1. The first 



^ .. of these values corresponds to an angle of 26 33' 51", 



3 x vl +/2 an d the second to an angle of 45, as shewn in Fig. 7. F 'g- "> 



In the first of these cases, KCD : ABCD, 1 .- 4, and 

 therefore n=^, and the equilibrium is that of stability ; 

 in the second case n = J , and the position of equilibri- 

 um is also one of stability. 



ccn i 

 ;GZ=- 



2 a 



. By substituting for t 1 its equal 



the formula becomes GZ = - , ; 



. As it may be more convenient to make a 



express the whole breadth AB or PQ, instead of the 

 lialf breadth, the equation will, by this change, become 



PROP. IX. 



To find the position of equilibrium as in Prop. VIII. 

 when the fluid surface intersects one of the extremities 

 of the upper side, as shewn in Fig. 8. Fig. 8. 



GZ=- 



: , ' --. By making this 



B 



Putting ABK = t, we have areas 



=, and 





value =0, we obtain g =i 2 ^Vl'lg c''ii~+ a'' OT 

 2 _12c 2 n 12 c 2 2 2 a* 

 ~ 12c* 12 c' n 1 fl 1 " 

 In the case of a square parallelepiped, we have a=c, 



lo n 12 n 2 2 



and therefore s 2 =-^ --,. In order, therc- 



KCDB= , consequently n-=^' , which, being 



substituted for n in the equation of Prop. VIII. gives 



JZ fi t ^ I* a 

 - = _ - , the same as formerly. Hence 



a ( 



_ i 



12ra 12 it- l' 

 fore, to ascertain from this equation the angle through 

 which the solid revolves.let us take wzr.24, which being 

 between .211 and .789, will place the solid with a flat 

 surface upward and horizontal, and in an instable equi- 



.1888 /.1888 



hbnum, consequently * = __, and^W ^^= 



the sine of 23 29', the angle of revolution, after which 

 the solid will settle in a position of stable equilibrium. 



The preceding equation determines also the specific 

 gravity n, which will make the solid float at the angle 

 ; for, by resolving that equation, we obtain n = 



-^zt:- -, and apply ing'this to the particular angle 



since /=: Jrt:^, we have H= =-f, or n 



PROP. X. 



To determine the position in which the parallelepi- 

 ped will float permanently with a plane angle obliquely 

 upward, when the specific gravity is between T 8 T and 

 JU. or between 34 and " 



*.-*-. 



of 23 29', we have ?j=0.5 + 0.26 = 0.7G and 0.24, 

 the two specific gravities, which will cause it to float 

 in stable equilibrium at the angle of 23 29'. 



PROP. VIII. 



To ascertain the position of equilibrium, &c. as in 

 Prop. VII. when the surface of the fluid passes through 

 one of the extremities of the base of the floating solid. 



It follows from Prop. VI. that when n is between 

 T V and -5%, or between f ^ and $%, the solid will float 

 permanently with the diagonal inclined to a vertical 

 line. In order to find the angle, let 1VCF represent pj g . 

 the square parallelepiped floating with its angle I placed 

 obliquely, and let its inclination to a vertical line be 

 OGT. Let DE be the surface of the fluid, and taking 

 CB a mean proportional between EC and CD, draw BA 

 parallel to FV, and cutting 1C in H, CH will be the 

 depth to which the solid sinks when IV is vertical, and 

 therefore area BXE = area XDA. Make CO = | CH, 

 and O will be the centre of gravity of the volume 

 ABC. Bisect EB in K and AD in R, and draw KX, RX, 

 and take XM = } XR and XL = |. XK, and M, L will 

 respectively be the centres of gravity of the triangles 

 X AD, BXE. Let fall the perpendiculars M P, QL upon 



