400 Mr. Mallet on the Physical Conditions 



" If the cylinder be conceived divided by a plane, it is evident that the force which 

 tends to separate its portions is proportional to the area of this plane, and is a maximum 

 where the plane passes through the axis ; and the same is evidently true of each of the 

 cylindrical laminte. 



" Let p be the radius of any of these cylinders, and 2P the corresponding force, the 

 length of the cylinder being unity. Also, let p + 8 be the radius of the same cylinder 

 when extended, then (according to the common theory), 



dp p 



" Again, let two consecutive sides of the cylinder subtend at the axis, the angle 0, the 

 portion of the lamina included between these sides sustains a pressure P9, and its original 

 thickness having been d^, and its thickness under pressure d^ - dS, we will have (according 

 to theory) 



PO = -k'pe~, or P^-lcpf.. (2) 



dQ ap 



" Multiplying the sides of this equation by those of the preceding equation, we get 



PdP=kk'SdS; (3) 



therefore, 



P-- = kk' {S- - A% 



(A being the value of S at the outer surface of the cylinder, where P= 0) ; and eliminating 

 P between the equations 2 and 3, 



dS 



Jk dp 



v^(o--Av \A' p 

 and, by integration, 



8+v/(g^-An Yp V-^' (4) 



(R being the radius of the outer surface of the cylinder). 



" But if r be the radius of the inner surface, and n the pressure of the fluid on the unit 

 of surface, the value of P for the inner surface will be n>' ; and, substituting this value in 

 equation 3, we have 



Wr' = kk'{S''-A^) (5) 



(8' being the value of 8 at the inner surface) ; and eliminating A between equations 4 and 5, 

 we get 



g' = UlL ^- '' + '--' . (G) 



