128 THEORY OF CONSTRUCTION AND SCIENTIFIC DESCRIPTION 



posing a certain uniform pressure to be exerted all round the interior 

 boundary; it will readily appear, from the 

 theory of resistance, that each successive 

 circular lamina, estimated from the interior 

 towards the exterior circumference, offers a 

 less and less resistance to the straining force. 

 But it is obvious from the very nature of 

 the subject, that by reason of the internal 

 pressure or strain, the metal must undergo a 

 certain degree of extension, and since the resistance of the outer 

 boundary is less than that of the inner one, it follows, that the exten- 

 sion must also be less ; this is manifest, for the resistance which any 

 body offers to the force by which it is strained, is proportional to the 

 extension which it undergoes, divided by its length ; now, since the 

 resistances of the several laminee, decrease as they recede from the 

 interior boundary towards the exterior, while at the same time, the 

 corresponding circumferences increase; it is manifest, that the exten- 

 sion for the several laminse decreases to the last or exterior boundary, 

 where it is the least of all : It is therefore the law of the decreasing 

 resistance, that the present enquiry is instituted to determine. 



Put d ~ ab, the interior diameter of the cylinder before the pressure 



is applied, 



e m the increase of d occasioned by the pressure, 

 d' A B, the exterior diameter in its original state, 

 e the increase induced by pressure. 



Then (d -\- e) and d' -f e'), are respectively the interior and exterior 

 diameters of the cylinder as affected by extension. 



By the principles of mensuration, the area of the annulus, or cir- 

 cular ring contained between the interior and exterior boundaries : 



Is equal to the difference of the squares of the diameters, 

 drawn into the constant fraction 0.7854 ; or it is proportional 

 to the sum of the diameters, drawn into their difference. 



But according to the nature of the present enquiry, the area of the 

 ring is the same, both before and after the extension takes place ; 

 consequently, we have 



(d' + c') f (<* + *)* = <** d*; 



therefore, by expanding the terms on the left hand side, we get 

 e'* d"' 2de e ? d 3 d* ; 



