Manchester Alemozrs, Vol. xlvi. (1902), No. 1. 9 



die and thus also flattening the slope of the hole, or, if 

 the latter is objectionable, using the same slope and 

 backing off the edges slightly at each face. At the same 

 time weight might be saved and liability to failure reduced 

 by decreasing the value of //. There are practical objections 

 to the latter course, however, as the dies could not then be 

 rebored for use with larger tubes after wear had taken 

 place. 



In Section II. there is given the method of obtaining 

 equation (i) from the general equations, as the case of a 

 thick plate supported in a circular ring and loaded over 

 an inner circular aperture may have a wider application 

 than that siven above. 



II. Stress in an annular die supported at the outer edge 

 and loaded over the inner Jace. 



In Fig. I. let the direction of motion be taken as the 

 positive direction of x, and let r be the radial distance of 

 any point from ox. 

 Let 7"= total pull in direction x. 

 d= thickness of die = 2e. 



2^ = reduction in diameter of tube. 



z<! = radial displacements 



. , J . , ^ 1 of any point xr. 



w = axial displacement J ^ ^ 



/ = radial pressure assumed increasing outwards. 

 ^= longitudinal pressure increasing in direction of ox. 

 / = tangential pressure, normal to any radial plane. 

 J- = shearing stress in direction <?.ar over planes perpen- 

 dicular to r. 



General Equations. 



The general equations of internal equilibrium, neglect- 

 ing gravitational forces, became 



