4G2 



HYDRODYNAMICS. 



um nnd 



i-'tability of 



Kinating 



I 1 * It 



'= , and OP= , consequently GO= -j = 



tt A ' 



Bodies, h ; and since 

 fluent KH 3 X 



fluent KH 3 X</ z 3*-r 4 



making 



12V 



= h, we shall obtain the limits be- 



tween the stable and instable equilibriums. Thus 



r 8 

 or n* r= r. But since 2r 



I In 



.12 5T l-"/ 



= 6, or the diameter of the base, we have n' 2 n = 



Hence, if the diameter of the base bears a greater 

 proportion to the length of the axis than that of A/ 2 

 to 1, there is no value of the specific gravity n, which 

 will cause the solid to float in the equilibrium of indif- 

 ference : It follows, therefore, from the preceding inves- 

 tigations, that in this case it will always float perma- 

 nently with its axis in a vertical line. 



If the diameter of the base bears a less proportion to 

 the length of the axis than V 2 to 1, then there are al- 

 ways two values of n, which will be the limits of sta- 

 bility and instability. In order to determine the ratio 

 between the length and diameter of the cylinder which 

 limits the case of stability and instability when the spe- 

 cific gravity is given, we obtain from the equation n n 2 



fit (, ___ 



= - , the equation -j =. V 8 n 8 n 2 , from which it 



8 1* I 



follows, that since n is given, the diameter of the base 

 should be to the length of the axis of the cylinder in a 

 greater proportion than that of A/8 8 " to 1, in 

 order that the solid may float permanently with its axis 

 upwards ; but if the diameter of the base should be 

 to the length of the axis in a less proportion to that, 

 the solid will overset. For example, if n = \, then 

 VsiT 8 n- = */\ = T?2247 ; that is, the diameter of 

 the base should be in a greater proportion to the length 

 of the axis than 1.2247 to 1, in order that it may float 

 permanently. If the proportion is less than this, it 

 will overset. 



Application We shall now conclude this Section by following Mr 

 ot' the pte- Atwood in his application of the preceding principles 

 cccUngprin- to the stability of ships. We have already seen that 



rtabuiw of the force of stabilit y (>f a shl P or an y ther kdy > 



Miips. represented by WxGZ.W being the weight of the 



vessel and its lading. When the angle of inclination 

 is so small as to be considered evanescent, we have 



fluent of AB 3 x<*2X* 

 seen that GZ = -- ^ ---- h s ; but since 



the first member of this equation is equal to ET, and 

 since /i=OG=EG, it follows that - ~ =ES, 





fluent A B" d z 



h = GS, which is an invariable 



quantity, whatever be the inclination of the floating 

 body, provided it is very small ; that is, the point S is 

 immoveable with respect to G. This point S is called 

 the Mctacenlre or centre of equilibrium ; for if the cen- 

 tre of gravity G coincides with the point S, the stabili- 

 ty, or GZ x W=W x SG x s, will be =0, or the solid 

 will float in all positions alike, without any effort to 

 restore itself if it is inclined, or to incline itself farther. 

 If the centre of gravity G is situated beneath the me- 



and 



Bodies. 



tacentre S, the solid will always float with stability, 

 as the measure of that stability" W x SG X* tends al- 

 ways to turn the body in <i direction contrary to that 

 in which it is inclined. If the centre of gravity is pla- 

 ced above the metacentrc, the force W x SG x * ha- _ y 

 ving passed through 0, tends to turn the vessel in the 

 same direction as that in which it is inclined, and it 

 will therefore float with an instable equilibrium. 



When the angles of inclination, however, are larpr, 

 the stability of the vessel will, as has already bevn 



b A 

 shewn, be measured by WxGZrr^ -- dixW- In 



the application of this formula to practice, h A is the 

 only quantity which requires to be determined ; for all 

 the other values can be easily ascertained from tlm na- 

 ture of the case. In order to find b A, the following 

 observations must be attended to. If a line parallel to 

 the horizon passes from the head to the stern of the 

 vessel when the ship floats uprightly, this line is called 

 the longer axis, to distinguish it from the shorter axis. 

 which passes through the same centre, but in a direc- 

 tion perpendicular to the former. If we conceive a 

 vertical plane to pass through the longer axis when the 

 ship floats uprightly, it will divide the vessel into two 

 parts perfectly similar and equal. A ship in equili- 

 brium, may also be conceived to be divided into two 

 parts by the horizontal plane which passes through the 

 surface of the water, and this section is called the prin- 

 cipal section of the mater, represented in section by AB, 

 Fig. I, which will be transferred to IN when the vessel 

 is made to heel or revolve through the angle SGK. 

 The real section of the water will now be AB, which 

 may be called the secondary section of the water. These 

 two planes inclined at the angle of heeling SGK, inter- 

 sect each other in X, and this line of intersection will 

 obviously be parallel to the longer axis. 



The position of the point X clearly depends on the 

 shape of the sides of the vessel. In a parallelepiped, 

 with two plane angles immersed, as in Fig. 5, the point 

 X bisects the lines ZR, FQ, corresponding to AB, IN 

 in Fig. 1 ; but, when the same solid floats with only 

 one plane angle immersed, as in Fig. 10, the point X 

 no longer bisects these lines, but is removed towards 

 the parts immersed by the inclination. As the breadth 

 of vessels, therefore, has no regular proportion from 

 the head to the stern, the position of X, which is ne- PLATE 

 cessary to the determination of b A, must obviously CCCXV. 

 be determined practically by approximation. We must *'S- 25> 

 therefore conceive the equal volumes NXP, LXW, 

 Fig. 1 and 25, one of which is immersed, and the other 

 raised by the heeling of the ship, to be divided into 

 segments by vertical lines, perpendicular to the longer 

 axis, and at distances of two or three feet. These seg- 

 ments will therefore have the form of wedges, as shewn 

 in Fig. 25, NXP being the inclination of the planes on 

 the faces of the wedges. 



The solid contents of the immersed wedges NXD 

 must now be found by approximation ; and making X I 

 =r AB NX, and XW = AB PX, the solid contents of 

 all the wedges, 1XW raised by heeling, must also be ob- 

 tained. If the size of the immersed wedges is not equal 

 to the size of the elevated wedges, the position of the 

 point X must be altered, till this equality is obtained. 

 To find b A, therefore, let the area PXNTP, and its cen- 

 tre of gravity f, be determined by approximation. Draw 

 dc perpendicular to'PX, and X c will be the distance of 

 the centre of gravity from the point X, estimated in the 

 horizontal direction PX; and ex being found in a si- 



"' 







