192 Mr CHALLIS's RESEARCHES IN THE THEORY 



This result shews that the weight of fluid displaced is greater than 

 the weight of the sphere, and consequently that the centre O is lower 

 than it would be in a state of rest. 



Suppose a portion of the sphere to be cut off" by a horizontal 

 section at the distance of b from the centre ; and let 7 become 7', the 

 centre being still above the surface of the water. Then if we suppose 

 the motion to be always in the direction of the radii*, and the horizontal 

 bottom to have no effect in impressing motion, the equation for this 

 case will be. 



W=w- 



TrF'a' 



ttTV 

 = w : — 





The difference between W and w is here less than before on account 



of both the factors —; and 1 — -yr ; for -?- is greater than - . This 



a* b* b ^ a 



seems to shew that curved bottoins tend to depress the vessel when it 



begins to move, and consequently to increase the resistance. 



As the term —-— ff sm^6eos26 cos uidOdu) is positive from 0=:sin"' — 

 to = 45°, and from = 135° to 6 = the supplement of sin"' — , let us 



Cv 



integrate for the portion of the surface corresponding to these limits, 

 or what amounts to the same, take the double of the integral between 

 the first limits, those of w remaining the same as before. In order to 

 abstract from the consideration of the portion of the surface not taken 

 into account in this integration, we may suppose the portions for 

 which we integrate to be connected by a cylindrical surface, the radius 

 of which = a sin 45°. The length of this cylindrical part may be any 

 we please : the vertical pressure against it will be only equal to the 



* This again cannot be true in the direction of the radii which pass through the lower 

 circular boundary of the surface. 



