544 Dr. C. Chree : Applications' of 



It is unnecessary to reproduce the calculation, which pre- 

 sents few difficulties. The final result may be put in the 

 form 



8h/h= (1/E) {g(p-p'){ih cos cc- V a sin a) - (l-2 v )p}, (28) 



where p is the pressure in the liquid at the depth of the centre 

 of gravity of the cylinder. An alternative form, see (27), is 



Bh/h=(ljE){g(p-p') { jj!$y~% -a-2v)p} • • (28') 



The direct influence of the support on the length thus 

 vanishes if 



V = {hfta)\ 



On the theory of uniconstant isotropy rj = 1/4 ; in a material 

 for which this held, the above relation would become 



/i = a, 



or the height would bo equal to half the diameter. 



If in (28') we suppose a/h vanishingly small, we obtain the 

 correct result for the elastic stretching of a cylinder whose 

 axis is vertical ; while if we suppose h/a negligible, we get the 

 mean elastic reduction in thickness of a thin disk supported 

 from a point in its rim with its plane vertical. 



Solid Cylinder or Prism with axis vertical ; Complete Solution. 



§ 10. As already stated, the case of a right cylinder or 

 prism with the axis vertical can be solved completely. 



Take the origin at the C.G., the axis of z being drawn 

 vertical^ upwards, and let h denote the length of the axis or 

 vertical edges. If p' be the density of the liquid, and p the 

 pressure in the liquid at the level of the C.G., then the pres- 

 sure at a height z is p— gpz. 



The body stress equations are 



dxx ' { dxy dxz _dxy ^dyy [ dyz _ Q 

 dx dy dz dx dy dz ' 



~ L d ~ V ■ ■ ■ (29) 



dxz dyz zz j 



On the curved surface, or vertical sides, the direction co- 

 sines of the normal being as before X, fi, 0, we must have 

 (7^x; + ^)/\=(Xx^+fi^)/fi=-(p--gp f z). . . (30) 



The conditions over the flat ends vary according to the 

 special circumstances. 



