\\S HYDRAULIC INVESTIGATIONS. 



pared with that of a body descending or ascending freely 

 along an inclined plane. 

 Demonstration. 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 6 by the pressure of the co- 

 lumn c, the tube being so constituted!, that the tension may 

 vary as the force. Then the relative force of the column c 

 is represented by b <*, since its effieacy increases, according 

 to the laws of hydrostatics, in the ratio of the diameter of 

 the tube; and this force must be equal, in a state of equili- 

 brium, 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 

 6 



corresponding pressure required to distend it will be ex- 



* pressed by a height of the column equal to ( 1 V- 



h-^ a x^—a s a \ b c 



— : ( 1 — — } . Now if the dia- 



v \ x J b — « 



since — j — : c : 

 b 



meter be enlarged in such a degree, that the length of a cer- 

 tain portion of its contents may be contracted in the ratio 

 1 : l— «r, r being very small, then the enlargement will b% 



7" T X 



in the ratio 1:1 + -— > *h at ls > x ' w ^ ^> e ~* but tne incre- 



Q x' be 



ment of the force, or of the height, is — . , which will 



xx b — a 



become — . . Now in a tube filled with an elastic 



c 2x b — a 



fluid, the height being h, the force in similar circumstances 



would be r h, and if we make hzz — . , the velocity 



2x b-^-a J 



of the propagation of an impulse will be the same in both 



cases, and will be equal to the velocity of a body which has 



fallen through the height f k. Supposing x infinite, the 



height capable of producing the necessary pressure becomes 



b c 



, which may be called g, and for every other value of 



b-~a 



x this height is T 1 — — — j g, or g>— < ~, or, since h becomes 



ag 



