210 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



we find that 



f*. = 7.8 X 10 5 kgm./cm. 2 , 

 k = 15 X 10 5 kgm./cm. 2 , 

 Ar = 1.4 X 10- 7 X r X P. 



The second component part of the strain is that due to the pressure 

 of the escaping liquid over the curved surface of the piston. Here an 

 approximation must be introduced, for the determination of the strain 

 in a cylinder under a given system of normal stresses on the curved 

 surface seems to be a mathematical problem not yet solved, while in 

 this case the problem is additionally complicated because the stress 

 system is not given but depends in turn on the strain. The approxi- 

 mation made is the assumption that the radial displacement at any 

 point is proportional to the normal pressure at that point, and is the 

 same as that in an infinite cylinder subjected to the same pressure 

 over its entire length. This assumption is probably fairly close to 

 the truth where the extent of the cylinder exposed to the pressure is 

 long compared with the radius, and the pressure varies gradually 

 from point to point, as is the case here. 



The piston then assumes under 

 the external pressure the form of 

 a frustum of cone, as is shown 

 in Figure 4. It will appear in the 

 following that it is absolutely im- 

 material whether the generating 

 lines of the frustrum of the cone 

 into which the piston has been 

 deformed are straight, as drawn 

 in Figure 4, or not. The displace- 

 ment at any point due to the ex- 

 ternal pressure is, on the above 

 assumptions, 



CD 



ABE 



Figure 4. Exaggerated effect of the 

 pressure in distorting the cylinder and 

 piston. 



4/x + 3k 

 Ar = \-! X r X p 



18 fXK 



= —3.5 X 10- 7 X r X p. 



p increases along the piston from 

 its full value, P, at the inner end, 

 E, to zero at the outer end, C. Now by adding these two components 

 of strain, we find that the total radial displacement of the piston consists 

 of a shrinkage of 2.1 X 10 — 7 X r x P at the inner end, and a swelling 

 of 1.4 X 10~ 7 X r X P at the outer end. 



