212 Prof. F. El gar. The Variation of 



gravity will therefore appear in fig. 3 as a straight line parallel to 

 the axis OX. Let GG' be this line. The equilibrium of the floating 

 body will best able in the upright position for those depths of flotation 

 at which the ordinates to the curve of metacentres LMN are greater 

 than OG, and unstable when they are less. At the points where the 

 curve LMN intersects GG' the equilibrium will be neutral. 



Fio. 7. 



Triangular section. 



Section whose upper half is rectangular and lower 

 half elliptical, as in fig. 6. 



Elliptical section. 



In actual ships the locus of the centre of gravity is not a straight 

 line such as GG', any more than the curve of metacentres is a hyper- 

 bola like LMN ; and the fundamental difference exists between them in 

 practice, that whereas the curve of metacentres is constant for a ship, 

 the locus of the centre of gravity is very variable in its character. 

 The diagram in fig. 3 may serve, nevertheless, to illustrate the nature 

 of the problem that has to be dealt with in investigating the stability 

 of ships. The curve of metacentres for a ship can be readily con- 

 structed by applying Bougner's theorem, viz., height of metacentre 



above centre of gravity of displaced volume =3- - y -, where y is the 



half-ordinate of the plane of flotation and V the volume of displace- 

 ment. The integration of y*dx is effected by one of Simpson's rules ; 

 %nd the volume of displacement and corresponding height of centre of 

 gravity of displaced volume are computed by the same means. The 

 curve of metacentres when once constructed is the same for all con- 

 ditions of the ship, as it can only be altered by changes in her 

 dimensions or form. In this important respect it differs entirely 

 from the locus of the centre of gravity. 



The locus of the centre of gravity of a ship is usually very irregular, 

 and is neither fixed in character nor position. It varies with dif- 

 ferent weights and descriptions of loading ; and, unlike the curve of 

 metacentres, its ordinates cannot always be expressed in terms of the 



