MancJicstcr Mem oil's, Vol. xlvi. (1902J, No. 1. 13 



The value of the radial pressure over the inner face 

 will not differ greatly from that obtained by treating r as 

 constant and equal to r^, then 



i)= -\ 2r; - b J (e- - x^)x niT /e-x\ 

 3 (^ ^J 47rre \ f / 



one root is x=i. In order that there shall be no root 

 between x^ +£ and x= — £, we must have 



w>^<o 



i.e., r^ Se'?'^ -Tec ^ 2e' 



/< 7 — V- or >— . 



I4£e + 4cr^ c 



For the 6 cases given, in which / is assumed ="3, 



/ must be 



I. II. III. IV. V. \i. 

 less than -92 ri8 '90 200 I'zy "54 

 or greater than 6-66 12-00. iO'5 900 54 6-5 

 a condition which for the usual values of / would be 

 fulfilled. 



Thus the form of the pressure curve over the inner 

 face will be somewhat as shewn in Fig. II., and seems in 

 agreement with the most natural suppositions. 

 The constant term in the expression for t is 

 a + 1) 1 . 2{a-\- />) 



a 



7)1 T 7-v-^r, 



and since the total outward radial pressure over an axial 



strip of inner face is y^ the above becomes 



^r-c + r^^ 



— p —r, :. 



f'l" — 'V 



where /' is the average radial pressure over the face 

 considering the die as a thick cylinder, and resolving T 

 radially according to the slope of the hole. 



