67 
Prof. J. Milne — On the Flotation of Icebergs. 
Being thus convinced, from our own sense of reason, that there 
are cones of ice which can float with their base downwards, and 
also that there are others which ean float with their base upwards, 
the question then is to define these cones. 
1. Adopting from Thomson and Tait, Natural Philosophy, § 767, 
that where V is the volume of a body immersed in a fluid, A the 
area of its plane of flotation, h the radius of gyration of that plane, 
and /< the height or distance between the centre of gravity of the 
floating body and that of the displaced fluid, for stable equilibrium 
we must have 
A Jfc 2 > V h 
We shall find for a cone of ice to float with its vertex downwards 
in sea-water, the radius of the base of the cone must be greater than 
196 times its height, — or, roughly, tlie diameter of the base cannot be 
less than tico-fifths of the height. 
2. Again adopting the same method for a cone of ice floating in sea- 
water with its base downwards and horizontal, we shall find that the 
radius of the cone must be greater than 1-05 times its height, — 
or roughly the diameter of the base cannot be less than twice the height. 
Note. — Case I. 1 iloment of Inertia of a circular lamina about a 
diameter = 7r ^ L , hut this = AP 
. AF= 
irR 2 
*B* . 7 „ R 2 
or tRI- = - - . — . 
k = * 
' A 14 
Let r be the radius of the base of the cone and a its height. Also let 
the density of the floating cone compared with the liquid be p, then — 
AC:BC=i:p* i 
.'.radius of plane of flotation is rp 3 
1 ' 
O 3 
.’. radius of gyration h —-sr (1) 
The Area of the plane of flotation 2 
A=7 rr*p 3 (2) 
Let G be the centre of Gravity of the Cone and E that of the dis- 
placed water, 
GC=J and EC=! a p* 
.'.GE or h=j (1— p*) 
(3) 
V the immersed volume=the volume of the Ice Cone multiplied by 
p or ~r p ( 4 ) 
Now substituting in AP > Xh 
, 4 r 3 p 2 irr 1 /! 3 a ,, i, 
' ~r> -rP'T (!- p *) 
or 
r > aV -\- — i 
1 See Woodcut, Fig. 1, p. 69. 
