534 Dr. C. Chree : Applications of 



mean for all the vertical "fibres" constituting the bar), we 

 have by (1) 



whence SZ/Z=^Z/2E (3) 



Similarly from (2) 



Sv= (l/3k)^gpz dx dy dz, 

 whence Bv/v=gpl/6k (4) 



Here $1 and Bv represent how much the length and volume 

 are greater than they would be in the absence of gravity. 



If the bar instead of being suspended were supported in 

 the same position, whether on a smooth plane or on a series 

 of points lying in one horizontal plane and exercising no 

 friction, we should find in place of (3) and (4) 



S7/Z=-^Z/2E, (5) 



8v/v=—gpl/6k (6) 



By taking the arithmetic mean of the lengths or volumes 

 of the bar when suspended and when supported, we clearly 

 eliminate the action of gravity. 



Even when the bar is kept in a uniform position, its length 

 and volume are affected by any change in g such as follows 

 change o£ place on the earth's surface. Again, as the elastic 

 constants E and k will vary in general with the temperature, 

 the changes in length and volume which accompany changes 

 in temperature are in part — generally of course only in small 

 part — of elastic, not of direct thermal origin. 



Pressure of Surrounding Medium. 



§3. The expressions (3) to (6) neglect the pressure of the 

 liquid or gaseous medium surrounding the solid. As this 

 may vary, it is desirable to estimate its effect. So far as 

 change of volume is concerned this is easily done as follows, 

 for a body of any shape. The densities p, p' of the solid and 

 surrounding medium are supposed uniform. Take the case 

 when the solid is suspended, and let p denote the pressure in 

 the medium at the level of the centre of gravity of the solid. 

 Take the origin of coordinates at the centre of gravity, and 

 suppose that the axis of z is drawn vertically upwards. We 

 know by elementary hydrostatics that the point of suspension 



