BRIDGMAN. — MERCURY UNDER PRESSURE. 407 



proportional to stress, any stress will produce the same additional 

 strain as the same stress applied from a state of ease. The long inter- 

 val of time for which the maximum pressure was exerted, and the 

 large excess of this pressure over any subsequent pressure, make it ex- 

 ceedingly probable that such a state has actually been reached. The 

 residual stress in the cylinder is then of exactly the same nature as the 

 stress in a gun with shrunk-on hoops, and the deformation may be 

 calculated by the ordinary theory. It may be mentioned, in support 

 of this position, that measurements have been made of the external 

 elastic deformation of another such cylinder of the same steel up to 

 10,000 kgm., showing agreement with the accepted theory to at least 

 5 per cent, with no trace of hysteresis. 



The calculation of the actual deformation of the cylinder according 

 to the elastic theory would be impossible because of the indeterminate 

 nature of the boundary conditions. An approximate solution only 

 can be obtained, in which the end effects are neglected altogether. 

 This solution is nevertheless probably accurate enough, because the 

 end effect in other similar cases in which the rigorous solution is 

 known has been shown to be negligible. It is well to remember in 

 considering the validity of these approximations that the maximum 

 value of the correction at the highest pressure is only 1.6 per cent. 



The internal diameter of the cylinder does not expand uniformly 

 under pressure. The expansion may be supposed to have its greatest 

 value at points removed from the position of the piston by at most a 

 few diameters. The expansion throughout the greater length of the 

 cylinder is constant, therefore, and equal to the value it would have in 

 a cylinder infinitely long and of the same radial dimensions. But for a 

 short distance beyond the head of the piston the expansion varies fi-om 

 its maximum value to a value which may be shown to become half this 

 maximum at the piston itself Now if we suppose that the cylinder is 

 so long that as the piston moves at constant pressure (during freezing 

 or melting) the locality of transition in dimensions travels bodily with 

 the piston, then a moment's consideration shows that the volume swept 

 out by the piston is equal to the length of stroke multiplied by the area 

 of the cylinder at its locality of maximum distortion. The assumption 

 does not seem forced in view of the fact that the length of the cylinder 

 is between ten and fifteen diameters. 



The formula for the displacement under these conditions may bo 

 found in any book on elasticity. We have 



_ hP ( a^ 21/ 4/x + 3k \ 

 a2-62V2/* 3k ■ 12/x y 



