Elastic Solids to Metrology. 553 



In case (ii.) the tube holds fully 2 milligrams more of mer- 

 cury than if it were absolutely non-elastic. 



As a second example, take the case of a glass tube whose 

 height is 12' 7 cm., internal diameter 10 cm., thickness 1*5 mm., 

 for which E and rj have the values assumed above. The 

 internal capacity is approximately 1000 c. c, and the elastic 

 increments when the tube is full of (i.) water, (ii.) mercury, 

 are approximately 



(i.) fy. = 62xl0- 5 c.c, 

 (ii.) Stv = 84xl0- 4 c.c. 



This tube thus holds about 0*11 gramme more mercury than 

 if it were wholly inelastic. 



Spherical Shell containing and surrounded by Liquid. 



§ 14. Another instance in which we can find the changes 

 of the volumes enclosed by the outer and inner surfaces of a 

 hollow vessel, containing and surrounded by liquid, is that of 

 a spherical shell ; but only when it is in equilibrium under its 

 own weight and the pressure systems. Thus external pressure 

 is necessarily existent, though internal pressure is not. 



Take the origin at the common centre, the outer and inner 

 surfaces being respectively of radii a and a'. Draw the 

 axis of z vertically downwards, and suppose the pressure to 

 bep + gp'z outside and p' -\-gp"z inside, so that p f , p" represent 

 the densities of the liquids. In order that equilibrium may 

 exist, the density p of the shell must be connected with the 

 densities of the liquids by the relation 



p(a i -a ;z ) = p'a z -p"a Vi (46) 



There is obviously symmetry round the vertical diameter, 

 and the displacements u along, and iv perpendicular to, the 

 radius vector r depend only on r and the inclination of r 

 to the axis of z. The values of the displacements are 



r(pa 3 —pa 13 ) (aa'yr~ 2 (p —p') 



U ~ ~ (3m-n)(« 3 -a' 3 ) ~ " 4n(a 3 -a' 3 ) 



gr'cosO f Q/ .p^-pW .p'a*-p"a' 3 \ 



+ ir, — . — rr^ n 2{m — 2n)'-— 1) — ^ (5m — n) ( -— lt — *-&- > 



6(m + n)(3m — n) I v J a° — a'° v ' a 6 — a 16 ) 



gr- l co^d{p , -p ,f ){aa , f gr~ 3 cos 0(p' -~p")(aa') 5 



3w(a 3 -a /3 ) ~ y(m + n) (a 5 -a' 5 ) ' * * ' * 



10=777 — . wo ; \ [m—ny 3 ,» 2(2m + n y 5 r , 5 > 



b(m + n)(3?n — ?i) I v ; a 6 — a' 6 v J n b — a ]h ) 



9 r- ] smd(m + 2n)(p'-p i, ){aa / ) 3 _ gr-H md{p' -p"){aa f f 

 + 6n(m + n)(a*-a'*) ~~ lS(m + n)(a 6 — a! 6 ) ' " ' 



(47) 



(48) 



