554 Dr. C. Chraa : Applications of 



As rcos0 = z, all terms in w which depend on change 

 sign as we cross the horizontal plane through the centre. 



The change in the volume £v(=4t7T 3 /3) enclosed by any 

 spherical surface concentric with the shell (a > r > a 1 ) is 

 given by 



Sv = f ff 2Wsin0tf0. 



r J 



As the integral of sin cos 6 between and it vanishes, the 

 only terms in u which contribute to 8v r are those containing 

 p or ;/ ; and it will be found that the increments in the 

 volumes within the outer and inner surfaces and in that of 

 the material are respectively given by 



^,/ n .= - { (pa*-p'a*)l(k) + 3(p-p'yi(4n) } - (a 3 - a' 3 ), J- (49) 



or Sv — — 4ir (pa 3 —p'a n ) /3/c. 



These changes are the same as if the outer and inner 

 surfaces were exposed to uniform pressures p and p'. They 

 do not depend directly on p, p', or p", and at first sight one 

 might be tempted to assume that the expressions were 

 perfectly general. This, however, is not the case. If, for 

 instance, the vessel were wholly or partially supported on a 

 horizontal plane, we should obtain from the general formula 

 (2) for the change in the volume v of the material of the 

 shell 



Sv=-47r(pa»-//a'*)lC31c)-Zal3k; . . (50) 



Z = {±/3)7r 9 {p(a*-a'*)-(p<a*-p''a r6 )} . (51) 



is the total upward thrust exerted by the supporting plane. 



This expression for Bv accords with the previous one only 

 when Z vanishes, i. e. when no part of the weight is borne 

 by the plane. 



Another limitation to the generality of the results (49) is 

 that the pressures p-\-gp'z and p' -\-gp"z are assumed to hold 

 over the whole of their respective surfaces. Thus the shell 

 must be wholly immersed, and it must be either quite full or 

 quite empty of liquid. 



The results would doubtless be very approximately true 

 for a wholly immersed flask consisting of a spherical body 

 and narrow cylindrical neck, the contained liquid, if any, 

 extending into the neck. If necessary an allowance might 

 be made, from the data in § 12, for the change in volume of 

 Ihe neck. 



where 



