Hydraulic Investigations '. 1 83 



excess of its circumference or diameter above the natural 

 extent, which is nearly the usual constitution of elastic bo- 

 dies, it may be shown that there is a certain finite height 

 which will cause an infinite extension, and that the height 

 of the modulus of elasticity, for each point, is equal to half 

 its height above the base of this imaginary column ; which 

 may therefore be called with propriety the modular column 

 of the pipe: consequently the velocity of an impulse will be 

 at every point equal to half of that which is due to the height 

 of the point above the base ; and the velocity of an im- 

 pulse ascending through the pipe being every where half as 

 great as that of a body falling through the corresponding point 

 in the modular column, the whole time of ascent will he pre- 

 cisely twice as great as that of the descent of the failing 

 body; and in the same manner if the pipe be inclined, the 

 motion of the impulse may be compared with that of a body 

 descending or ascending freely along an inclined plane. 



These propositions may be thus demonstrated : let a be 

 the diameter of the pipe in its most natural state, and let 

 this diameter be increased to b by the pressure of the column 

 c y the tube being so constituted that the tension may vary 

 as the force. Then the relative force of the column c is re- 

 presented by be, since its efficacy increases, according to the 

 laws of hydrostatics, in the ratio of the diameter of the tube j 

 and this force must be equal, in a state of equilibrium, to 

 the tension arising from the change from a to b, that is, to 



b — a; consequently the height c varies as —. , and if the 

 tube be enlarged to any diameter x> the corresponding pres- 

 sure required to distend it will be expressed by a height of 



the column equal to (1 V r— , since ~7 a : c : : 



1 \ x/ b—a b 



: ('l — — J t . Now if the diameter be enlarged 



in such a degree that the length of a certain portion of its 

 contents may be contracted in the ratio I : l — r, r bein* 



very small, then the enlargement will be in the ratio 1:1+-, 



TX 



that is, x' will be — j but the increment of the force, or 



M 4 of 



