188 Mr CHALLIS's RESEARCHES IN THE THEORY 



f^k'sec'6 VTf 

 direction gives j— — , which = — — = velocity at Z. Hence the 



vertical velocity is the same at all points of any horizontal plane, and 

 the fluid will consequently descend in parallel slices.* Let us now 

 determine the pressure at any point on the particular supposition that 

 V is uniform. Then if 



Vk"&eee ^, , .. , „r dw Vsec'd ^,dk 2F"Asec'0 



w = :: the velocity at W, -7- = ; — x 2«-7-- = y, . 



p' ■'at p^ at p- 



And 



Idt^' ^-Jdt'^P-J 7 = -p + ^ 



» 



Hence 



„ 2r'kse&9 r'k'sec'e 

 p = C-g. + . __. 



And as when z = h, p — 0, and p cos Q = h, it follows that 



The above solution I do not consider to be of any value, except as 

 illustrating the process to be followed in determining mathematically 

 the way in which the interior of a mass of fluid is affected as to 

 velocity and pressure, in consequence of given conditions at the 

 boundaries. This part of the theory of fluid motion is very 

 defective. 



* I obtained this result in the number of the Phil. Mag. and Annals of Philosophy 

 for .Jan. 1831, but omitted to shew that it is entirely dependent on the arbitrary condition 

 that the inferior -surface of the fluid is bounded by a horizontal plane. Qji any other 

 supposition the problem would be one of much greater difficulty. This omission has not 

 without reason caused a misapprehension as to the application of the solution, on the part of 

 Berzelius in a notice taken of it in his Annual Review. {Jahres-Bericht, 1833.) 



