130 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION 



Then, agreeably to the law of the resistances which we have 

 established above, we have 



:: ^ : (r + x) 2 ' 



this result expresses the strain at the point x, or the resistance of the 

 material whose thickness is an ; and the fluxion of this quantity as 

 referred to the variable thickness x, is 



consequently, the fluent, or the sum of all the strains, is 



/fr*x 

 - - 4- C, and this when x t becomes 

 (r + xf 



Therefore, if the strain or resistance /, were to act uniformly on the 

 thickness expressed by ^ , it would produce the same effect, as if 



all the variable strains were to act on the whole thickness t. 



The above law being admitted, let us suppose that the interior 

 radius of the cylinder, and the pressure per square inch on the surface 

 are given, and let it be required to determine the thickness such, that 

 the strain and resistance may be in equilibrio. 



Here it is manifest, that the greatest strain the thickness 



CT t 



can resist, is - , and the strain to which it is actually exposed, is 

 nr; consequently, when these are equal, we have 



crt 



***%? 



from which, by expunging the common factor r, we get 



ct 



-7+? (105). 



If this value of n be compared with its respective values, as indi- 

 cated in the equations (104) preceding, we shall have the following 

 expressions, for the thickness of metal in the cylinder to resist any 

 pressure, while the elastic power of the material remains perfect, viz. 



Pr _ pr . _ wr 



'* 



.7854CD 9 P .7854cd 8 ' .78540? w 



