﻿Prof. J. Milne — On the Flotation of Icebergs. 67 



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 can 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, Tc the radius of gyration of that plane, 

 and li the height or distance between the centre of gravity of the 

 floating hodij and that of the displaced fluid, for stable equilibrium 

 we must have 



A¥>YJi 

 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, the diameter of the base cannot he 

 less than two-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 he less than twice the height. 



]N"oTE. — Case I.^ Moment of Inertia of a circular lamina about a 



ttR- 



diameter= -^, but this = A^^ 



or 7rE-F= -, - /. ^' =x •*• ^ —'^ 



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* 



/.radius of plane of flotation is rp^ 



3 



.'. radius of gyration ^ — 2 (^) 



The Area of the plane of flotation ^ 



A^irr^p'^ (2) 



Let G be the centre of Gravity of the Cone and E that of the dis- 

 placed water, 



GG=^ and EC=| a />* 



/.GE or h=j (1— p*) (3) 



Y the immersed volume = the volume of the Ice Cone multiplied by 



por-^p (4) 



1^0 w substituting in AF> Yh 



irrY • ^> - ^p-j{ l—p') 



or r > aV-^ — 1 



P' 

 1 See Woodcut, Fig. 1, p. 69. 



