[135] 

 ON THE STABILITY OF FLOATING BODIES. 



BY JAMES LOUDON, M.A., 



Mathematical Tutor and Dean, University College, Toi onto. 



The following direct method of determining the nature of the equi- 

 librium of a floating body was devised by the writer in January, 1870. 

 The particular case of the fluid being of constant density is taken ; 

 and the displacement is supposed to take place round a certain line in 

 the plane of floatation. 



Let Ox, Oy, Oz be axes fixed in space, and Ox', Oi/', Oz' axes 

 fixed in the body ; p the density of the fluid, p' the density of the 

 body; V the volume of the fluid displaced, V the volume of the 

 body J X, f, i the centre of gravity of the body, x', y,z' the centre of 

 gravity of the fluid displaced ; p, p' the pressures at a point of the 

 element d S of the surface of the body before and after displacement, 

 respectively. 



Then, before displacement, for equilibrium we have 



— gp'V'.x + g pffz (xdx — ydij) dy == 0, 



and pV = pV, 



.•. p' V'x = p Vx' = p^^ z (xdx — ydy) dy (1) 



After displacement through an angle 8 around Oy, x, y, z become 

 X -\- z8 6, y, z — xd d, respectively, and the sum of the moments of 

 the fluid pressures about Oy 



= — gp' F'(S + z ^&)-\-gpf/{^ i^^^^—y^y) dy—de (x^dx—zxdz) dy] 



= — 9P' V'z do — gp 80 ffix'dx — zxdz) dy, by (1) 



= gp80 I — Vz — ff^dxdy -\- ff zxdzdy\ 



= gp80 { Viz' - z) -ffx^dxdy], (2) 



if Oy passes through the centre of gravity of the plane of floatation 

 •.• Fs' ^ ff zxdzdy, by the properties of the centre of gravity; 

 and equilibrium is stable or unstable according as (2) is negative or 

 positive, i. e., with the usual notation, according as V. HO 5 Mk^. 



Note. — In tating the moment of p^ about Oy, p' is resolved into forces 

 parallel to Ox', Oy', Oz' ; so that the moment of p'dS, dS being projected on 

 yz', z'z', z'y, =p' (xdzdy — zdzdy) ^ gp [z — xdd) {xdx — zdz) dy. 



Kovember 12, 18*70. 



