of the Motion of Incompressible Fluids. 125 



sphere, the velocity of each will vary inversely as the square 

 of the distance from the centre. It is this law of their motions 



that the equation q = 5- points out. If a disturbance act 



at the origin of coordinates, the motions consequent upon it 

 will be the same at the same distance from the origin in all 

 dii'ections at a given instant, provided F {t) be the same for 

 all directions ; that is, if the disturbance be similarly related 

 to all the parts of the surrounding space, as in the instance 

 just adduced. But, generally speaking, the form of the func- 

 tion F {t) will be different for every different direction from 

 the point of distui'bance, and will vary from one direction to 

 another, either according to a law, or in a manner regulated 

 by no law of continuity. All this flows from the arbitrary 

 nature of the functions introduced by the integration of a par- 

 tial differential equation. We may however conclude, that 

 if a pyramid, the vertical angle of which is very small, be 

 placed with its vertex at the point towards which, or from 

 which, the motion tends ; for all or a given portion of the 

 particles included within its faces, F {t) will be ultimately the 

 same at a given instant, and consequently the velocities at 

 that instant will at different distances from the vertex vary 

 inversely as the square of the distances. An instance of mo- 

 tion in obedience to the law here considered, is afforded by 

 fluid issuing from a small orifice. The portion of fluid in- 

 cluded between the orifice and the vena contractu^ constitutes 

 a frustum of a cone, and at a given instant the velocities of 

 the particles vary inversely as the square of their distances 

 from its vertex. 



If we now choose an origin of coordinates and axes, the 

 positions of which are fixed in space, if .i', j/, z, be the coordi- 

 nates of a point at which the velocity is </, and «, /3, y, be 

 those of the point towards which, or from which the motion 

 tends, 



. = zli!) 



and a = fU) + ^ ^*^ , 



for the eciuation -r4 + -r4 + -7^ = is satisfied by this value 



* a I* rfy a r* 



of <p. «, /3, y, vary in general with the time, and tiepend, as 

 well as the form of F, on the given conditions of the problem 

 to be solved. Also the diflercntial cqualioiJ is satisfied by 



^ = /Y/) J. F.(0 , JMO 



^ -^^ ''"'" ^(,._<.').-f:(^y_,3')'+(-.-y)' A/^.--«"r+(y->T+(--/')' 

 + &c. to as many terms as wc please. This c(]uatiou is ajjpli- 



cablc 



