404 Mr. Mallet on the Physical Conditions 



;> = - ^(^ - '') 1737- • - - - ^7j77 • ? 



and uniting this with the value of P, given in equation (13), we have the original condition, 



p=^(i^-,r)-^^ -. (14) 



" IV°. To find the pressure on each shell during the construction of the tube, before 

 the outer shells have been put on, we must suppose the pressure = at the outer surface of 

 the shell last put on, whose radius we may represent by ?/ ; therefore, we must apply at this 

 surface a pressure, such that 



to counteract that given by equation (14), and we must suppose the pressure = 0, as before, 

 when a- = r; then substituting, in equation (12), y for r and r for R, and the above value of 

 P for Fr, we have 



p= - 1 IK-l ) -5F5 • — , -^; ; . -= + 1 — — — , 



' ^ ^'R + r y r--y X (H + r) {r -i- y) x 



and, uniting this with the value given in equation (14), 



p^^R(x-r){y-.v)^ (1.5) 



x{r + y) 



and 



''1 dP ,^ R{x^--ry) 

 ,r dx X- {}• + y) 



(1<5) 



" The practical construction of a tube on these principles is immediately derived from 

 equations (13) and (16). Thus, if it be required to construct a tube capable of sustaining 

 a pressure double of the tenacity due to the material, we must make F= 2T, or R-r = 2i-; 

 that is, the thickness of the tube must be equal to its internal diameter, and, in order to 

 produce the required pressures on the successive shells of which the tube consists (the num- 

 ber of which theoretically should be infinite) it would, perhaps, practically be sufiicient to 

 divide the thickness into four (or a greater number) of equal parts. In this case, if the 



3 2 



inner radius be r, the outer radius of the first shell would be -r, of the second -, of the 



2 r 



third g ?■, and of the fourth 3r = R. Then, to find the mean tension of the second shell, when 



first put into its place, and before it has been enveloped by the third, we must substitute, in 



7 

 equation (16), the values R = 3r, y = 2r, x = r »' ; 



