ON THE THEORY OF HYDRAULICS. 2^7 



We have hitherto considered the motions of fluids as continued pvirtcipally 

 in the same direction ; but they are frequently subjected to alternations of 

 motion, which bear a considerable analogy to the vibrations of pendulums; 

 thus, if a long tube be immersed in a fluid, in a vertical direction, and the 

 surface of the fluid within the tube be elevated a very little, by some external' 

 cause, the whole contents of the fluid will be urged downwards by a force, 

 which decreases in proportion to the elevation of the surface above the 

 general level of the vessel, and when both surfaces have acquired the same 

 level, the motion will be continued by the inertia of the particles of the fluidi 

 until it be destroyed by the difl^erence of pressures, which now tends to retard' 

 it; and this alternation will continue, until the motion be destroyed by fi-ic- 

 tion and by otlier resistances. It is also obvious, that since any two vibra- 

 tions, in which the forces are proportional to the spaces to be described, arc 

 performed in equal times, these alternations will require exactly the sam^' 

 time for their completion, as the vibrations of a pendulum, of which the length- 

 is equal to that of the whole tube; for the relative force in the tube is to 

 the whole force of gravity as the elevation or depression is to the whole 

 length of the tube. Hence it follows, that if two such tubes were united' 

 below, so as to form a single bent tube, the vibrations might take place in 

 the whole' compound tube, in the same manner, and in the same time, as in 

 each of the separate tubes; nor would the effects be materially altered if 

 any part of the middle of the tube were in a horizontal or in an obHque di- 

 rection, provided that the whole length remained unaltered. In such a tube 

 also, all vibrations, even if of considerable extent, would be performed in the 

 same time, and would long remain nearly of the same magnitude; but in ai 

 single tube, open below, the vibrations would continually become less ex-" 

 tensive, and their duration would also be altered as well as their extent; 

 besides the unavoidable resistances, which would in both cases interfere with' 

 the regularity of the effects. 



But it does not appear that the laws of the vibrations of fluids in pipes will 

 at all serve to elucidate the phenomena of waves. Sir Isaac Newton has sup- 

 posed that each wave may be compared with the fluid oscillating in a bent 

 pipe; but the analogy is by far too distant to allow us to found any demon- 

 stration on it. The motions of waves have been investigated in a new and 

 improved manner by Mr. Lagrange; and Ihave given a concise demonstra^ 



