548 Dr. C. Chree : Applications of 



When cavities exist complications arise. It is always possible 

 to find the change in the volume of the material itself; but in 

 the case, for example, of a flask containing liquid, what is of 

 most interest is not the change in volume of the material 

 but the change in the volume contained by the inner surface. 

 The change in volume of the material of the flask may be 

 obtained, at least very approximately, as follows : — Take the 

 origin in the inner surface of the base of the flask, whose 

 bottom is supposed a plane surface z=— t; take the axis of 

 z vertically upwards, and let h be the height of the liquid 

 surface above £ = 0. The material being supposed of density 

 p, the change in its volume is given by (1). Throughout the 

 volume 



X=Y=0, Z=-gp; 



and supposing p' the density of the liquid, we have, over the 

 inner surface S, 



FA=G/ /i =H/v= w '(A- 2 ), 



where X, //,, v are direction-cosines of the normal (above 

 the level of the liquid F = G=H = 0, neglecting the atmo- 

 spheric pressure). Over the outer surface no force is supposed 

 to act, except over the lower surface S' of the base where H 

 alone exists. Thus we have 



M8v=-$gpzdx dy dz+§gp'(h-z)(\x + /j,y + vz)d$ +JJ(-^)H^S'. 



The values of the several integrals are as follows : — 



— jjj gpzdx dy dz = -gpzv, 



jj gp'Qi - z) (\x + p,y + vz)dB =gp f v / (Sh - 4?) , 



Here v is the volume of the material of the flask, z the height 

 of its C.G. above the upper surface of its bottom ; v' similarly 

 is the volume of the liquid, z 1 the height of its C.G. 

 In all we have 



8v=-gpv(z + t)l{3k)+g P 'v , (M-fz , -t)l3k. . . (37) 



This expression becomes neater if instead of z, z\ and h we 

 use £, ?', and H, the heights repectively of the two C.G.'s 

 and the liquid surface above the lower surface of the bottom 

 of the flask (i. e. ^=z + t, &c). We then get 



a»^-^fwt/(3*)+^V(3H-4n/3*, • • (37') 

 or 



8t?=-W?/(3^)-r-W / (3H-4?)/3^ « . . (37"] 



