130 Mr. Challis on the General Equations 



M. Poisson supposes the vessel to be such that the variation 

 of z is small compared to the corresponding variation of x. 

 Hence if ^ and p' differ by a very small quantity, 



^=g(^-y)-^-^.-- — (--^-;,)very nearly, 



and passing from differences to differentials, 



dp __ J ,dxdu t^u^jl 



which is the equation in the Traite de Mec. torn. ii. p. 449. 



Upon the same principles as those by which the preceding 

 problem was solved, it will be possible to find the velocity of 

 the fluid issuing from a vessel of any shape, whatever be the 

 nature and position of the orifice. For let fluid be compelled 

 to move through any canal, continuous or not, and lying in 

 one or several planes, and at the same time let it be acted 

 upon by gravity. Suppose the transverse section at every 

 point to be a square. Then every small portion of it may be 

 considered a frustum of a pyramid ; and if the pyramid, the 

 frustum of which is terminated at a given point be completed, 

 and r be the distance of its vertex from this point, by what has 

 been proved, 



*=/W + ^, ^=-^ = vel. 



Now let the transverse sections of the tube be indefinitely 

 diminished, their proportions remaining the same. In this 

 case the motions of the particles perpendicularly to the axis 

 of the canal, will be indefinitely small, and no error will be 

 induced by making r infinite in the preceding expressions, 

 that is in supposing each very small portion of the tube pris- 

 matic. Hence since F (t) is arbitrary, we may put — = p(^ {t). 



Again, if 5 be the distance measured along the axis, of the 

 point under considex'ation, from a fixed point in the axis, 



— = -r-, because the line s touches all the lines r. 



ds dr' 



equations exactly like those for motion in space of one dimen- 

 sion. 



and ^ = ^.^_;^'(0 5-|'-/'(0, 



X being 



