552 Dr. C. Chree : Applications of 



reduction we obtain from (41) and (42), noticing that 1= h, 



. gpR _ a' 2 p—p' 2 pa 2 — pa' 2 



bv ol v o---M~ a 2 -a' 2 ~n~ U a'-a" 2 ' 



gpR a 2 p-p> __ 2pa 2 -pW 2 



fo >i = -"3iT- -tfZ^—n- U a 2 -a'* > > 



v — v. v 6k \_ jr a 2 — a 12 J ■ 



The elastic changes in the volumes v and v if as well as in the 

 volume v{~v —v^ of the cylindrical walls, are exactly the 

 same as if the pressure over each surface had a constant 

 value equal to its mean value. 



The term gpH/(?>k) depends only on the weight and elastic 

 properties of the tube itself, and so may be left out of account 

 in considering the influence of the liquid pressures. Omitting 

 it, we see that if a circular tube be filled with liquid of 

 density p' to a mark at height /*, the volume within the tube 

 up to the mark is increased to an extent $v { which is given by 



8»>,=iS-A{(« 2 /«) + 2 K73*) }/(« 2 - a 12 ) 



= c,p'h{(l + n )a' + (l-2 v )a'*l/{K(a*-a'r>. . (44) 



If the thickness t of the walls of the tube be small compared 

 to a, this becomes approximately 



SvJr ( =^p'h(a/t)(2- v )fE (45) 



The increase in volume per unit of volume inside a thin- 

 walled tube thus varies directly as the density of the liquid, 

 the height of the murk, and the radius of the tube, while it 

 varies inversely as the thickness of the wall. We see that 

 for exact purposes the cubic capacity of a tube cannot be 

 defined without reference to the density of the liquid which 

 it is to contain ; also all copies of a standard of capacity 

 should be identical in height, diameter, thickness, and 

 elasticity. 



As a numerical illustration, suppose a tube of glass 1 cm. 

 radius, with walls 1 mm. thick, to hold (i.) water, (ii.) mercury, 

 to a height of 16 cm. This gives a content of about 50 c.c, 

 and the increments in the capacity in the two cases are 

 approximately, if we assume 77 = 1/4 and E = 6 x 10 8 grammes 

 weight per sq. cm., 



(i.) Sv,= 12xl0- 6 c.c, 

 (ii.) Sr.=:16xl0- 5 c.c. 



