334 



On the Statical Stability of a Ship. 



[Jan. 24, 



the radius of curvature of the locus of the centres of buoyancy, corre- 

 sponding to the selected inclination. 



The conic is now implicitly determined. It remains to show what use 

 is to be made of these data. 



Let be the radius of curvature, corresponding to the angle 6, made 



between the normal and axis of a conic ; then 

 (1— e sm^ 0_)2 



From these we obtain 



2 . . ' («) 



p^ sm" 



Poi— jo tcos2 



1-^'- a ' (J) 



- Pof>/sin2 

 f^o^~P/cos^^ 



«e'=-^ — 2 — ^; W 



and these afford the means of calculating all the elements of the conic. 

 Now, let us take any other inclination ^ : we may calculate p from the 



foregoing value of by means of the formula 



p = {e) 



(l-e2sin^# 



Now, if X be the distance of the centre of gravity of the ship below the 

 metacentre of the upright position, and p the perpendicular from the 

 centre of gravity on the normal of the conic in the inclined position, we 

 shall have 



^^^^^Mzit, (/) 



sin^ ]Oo^+p/cos0 

 and ^ X D is the moment of stability, D being the displacement. 



Strictly, it is only necessary to use the formulae («), (e), (/) in actual 

 work. Formula (/) shows clearly how an alteration in the position of the 

 weights affects the stability. If \ be altered, the altered value of p is 

 obtained (geometrically) by a very obvious construction. 



In Mr. Scott Russell's treatise on ' Naval Architecture,' p. 604, it is 

 shown how the stability may be obtained by geometrical construction when 

 the conic is known. 



It is worth while to remark that the condition that the conic should be 



