OF THE MOTIONS OF FLUIDS. 319 



pressible ; and if the fluidity were at the same time so per- 

 fect, that no particle of the fluid should be afl'ected by any 

 pressure not tending directly towards it. A distinguished 

 mathematician of the present day appears indeed to have 

 assumed, that the pressure is transmitted downwards with a 

 velocity determined by the depth, and related to the velo- 

 city of the horizontal transmission, if not identical with it: 

 but it seems sufficiently obvious, that if the canal be sup- 

 posed incompressible, the pressure must descend in it, as 

 it confessedly would do in an organ pipe, with a velocity 

 dependent only on the intimate elasticity of the medium, 

 which in this proposition is supposed infinite. 



Now the difference of the forces on each side of the thin 

 transverse section of the canal, constituting a partial pres- 

 sure, is the immediate cause of the horizontal motion ; and 

 the vertical motion is the effect of the modification of the 

 horizontal motion : and the difference of the pressures is 

 every where to the weight of ihe column or section, or of 

 any of its parts, as the difference of the heights to the thick- 

 ness of the column, or as the fluxion of the height y to that 

 of the horizontal length of the canal x. Hence, if the 

 weight of any particle be called g, the horizontal force act- 

 ing on it will be-r^<;r. Such therefore is the force acting 



horizontally on any elementary column : but the elon- 

 gation or abbreviation of the column depends on the 

 difference of the velocities, with which its two transverse 

 surfaces are made to advance, and this elevation or depres- 

 sion of the upper surface is therefore to the whole height, 

 as the variation of the fluxion of the length, or thickness, 

 produced by the operation of the force, is to the whole 

 fluxion of the length ; that is, Sy is to y as ^dx to da:, or as 

 ^Dx to Bx, But the force which produces the change 



