July 24, 1890] 



NATURE 



305 



When the thickness of the cylinder is considerable, compared 

 with the bore, this solution of Barlow will give a very fair 

 indication of the true result. 



(5) But Rankine showed (" Applied Mechanics," § 273) that, 

 by superposing the state of hydrostatic stress produced by equal 

 internal and external pressures, we obtain the algebraical solu- 

 tion of the most general case where the internal and external 

 applied pressures are arbitrary. 



For if we suppose the state of stress in the cylinder is a hydro- 

 static stress, composed of a radial pressure /, and an equal 

 circumferential pressure -t, then equation (2) becomes— 



/ pdr = pr - pii-i ; 



and differentiating with respect to r, 



p = d{pr)ldr, or dpidr = o ; 

 so that 



P = f>, a constant ; and then t = - b . . . (4). 



(6) The superposition of this state of stress on Barlow's s^ate 

 of stress gives — 



values which will be found to verify equation (2) ; and now the 

 constants a and /> are determined for arbitrarily applied internal 

 and external pressures /, and p^ by the equations 



so that 



(A- - A) r,W . 



ro^ - r? 



poU 



.2 _ 



r, — ■* - To — 

 f, - Porj" - PiTi "^ 



These results were first obtained by Lame and Hart (the late 

 Sir Andrew Searle Hart, of Dublin), but in a much more com - 

 plicated manner. Lame's solution was given in his " Le9ons 

 sur !a theorie mathematique de I'elasticite des corps solides " ; 

 while Hart's treatment of the question will be found in Note W 

 to Robert Mallet's " Physical Conditions involved in the Con- 

 struction of Artillery" (1856). An investigation of the same 

 problem by Maxwell, when about eighteen years old, in the 

 Trans. R. S. Edin., vol. xx. 1850, has been generally over- 

 looked. 



Rankine's treatment analyzes the mechanical signification 

 of the separate terms of the solution, and obtains them by 

 simple reasoning from the state of stress, without an appeal ta 

 the laws of elasticity and the consequent state of strain. 



(7) Putting /.r," = poVo' makes b —o, and gives the particular 

 case considered first by Barlow ; and putting /, = /„ makes 

 a — o, and gives the additional particular case of uniform 

 hydrostatic stress invented by Rankine. 



But, in the general case, a and b may have any values, posi- 

 tive or negative, according to the relations between/,- and A. ^i 

 and To. 



Thus, as in Fig. I, with/, = o, we find — 



n 



and then 



r,-3 _ r 



PiTo- 



-3-2 



p = ar 



t =/,li 



n 



'• = ".: 



NO. 1082 



Vi-^ - To-- 



'-^^P^rzl^^. ,« 



--l^._^p.rijyLfy. ....,8, 



, VOL. 42] 



(8) Now using t to denote the average y?i\\i& of the circum- 

 ferential tension, so that 



then 



t = /.•;',/(;- 

 + r^' ti - 



'-.). 



rAro + n) 



rp r p - T j 

 n To + ri 



(10) 



thus showing that the maximum tension /,• may exceed the 

 average tension /■ by a considerable amount ; and it is this 

 maximum tension ti which must be carefully watched and kept 

 down below a certain working value ; so that, with given (■„ the 

 maximum allowable pressure in the tube is given by 



ro» -I- ri' 



This is the formula now used in the design of a hydraulic 

 press, or of a thick tube, of bore 2>-j, to stand an internal 

 pressure pi ; ti being fixed by the strength of the material, and 

 then Tg being calculated. 



We notice that /< is always less than /<, so that a tube, how- 



