Elastic Solids to Metrology. 543 



relative increment hp in its density in the latter case would 

 be given by 



tp*=ffppf\(i-v 3 i->iv)m,)+w}. ■ • ( 26 ) 



Here the density of the air is neglected, the axis of z is sup- 

 posed vertical, J is the depth of the C.G. of the solid below 

 the point of suspension, and £' its depth below the surface of 

 the liquid. 



Cylinder suspended from the rim. 



§ 9. The formulae (8) and (9) apply whatever be the shape 

 of the body ; thus the change in volume of a suspended or 

 supported solid is always ascertainable, neglecting variability 

 in the surrounding medium. Changes of linear dimensions 

 are, however, less manageable. As we shall see presently, we 

 can obtain a complete solution when a right cylinder has its 

 axis or a rectangular prism one of its sides vertical ; so that 

 in that case we can do better than use the formula (1). In 

 many cases probably it would be difficult to make much even 

 of (1) . The following, however, is a case where it applies 

 satisfactorily. 



An isotropic right circular cylinder of density p, radius a, 

 and height h, is suspended by a point in its rim, in a liquid 

 of density p\ to find the elastic change in its volume. 



Take the origin of coordinates at the centre of the upper 

 face, the axis of z along the axis of the cylinder, and the 

 plane of xz to contain the point of suspension ( — a, 0, 0). 

 As the C.G. must lie on the vertical through the point of 

 suspension, the axis of the cylinder must be inclined to the 

 vertical at the angle a. where 



«=tan- 1 (2a/A) (27) 



The data required as to the components of the fluid pres- 

 sures are as follows, II denoting the pressure in the liquid at 

 the centre of the upper face : — 



Surface. Values of Specified Components. 



Upper plane face, z=0, F = G=0, 



Lower „ „ z = 7t, „ , H= — {U+gp'(h cos a + #sin a)}, 



Cylindrical surface H = 0, F/x=G/y= — a~ 1 \U-\-gp'(z cos «+# Bin a)}. 



Throughout the volume X=#/osina, Y = 0, Z=gp cosa. 



At the point of suspension £ = 0, «.!•= — a, and the com- 

 ponent of the tension of the supporting string parallel to x is 

 given by 



\\ JVS= ~7r(/3 — p')d 2 hg sin a. 



