388 Mr. Challis on the general Equations 



sequently applicable to fluid contained in any irregular vessel 

 and moving- in any manner. The form and value of F{t), and 

 the values of a, fi, y, for each point, at a given instant, must be 

 determined from the shape of the vessel, the velocity and direc- 

 tion of the velocity at the given instant at parts of the fluid 

 which are free, and from the law just proved of the communi- 

 cation of velocity according to the inverse square of the distance. 

 For on these data alone depend the quantity and direction of 

 the velocity at any point in the interior of the fluid mass. And 

 as, when motion takes place, there must, at least, be two parts 

 of the fluid which are free, and where the pressure is known, 

 the equation {J) is proper for determining the motion at these 

 parts at any instant, when the motion is given at a given 

 instant. This equation is to be made use of, according to the 

 following principles. As it has been shewn that ^ is generally 

 a function of r and t, which retains its form, at a given instant, 

 when r varies in an indefinitely small degree, 



let ^ = X {r, t) ; then f = x (r\ t) ; 



<P' - 'P = xi'^' (f-r) = a>(r'-r), 



suppo.sing the velocity, which is -~ , to be represented by a>. 



,T d(b' d<b dot , , 



"^"" -Jtdt-lTt^'-'^- 



Here r' — r may be considered the increment of a line s, drawn 

 from a point at which the pressure and the direction of the velocity 

 are known, continually in the direction of the motion of the 

 particles through which it pas.ses. 



Consequently, f = V-f^ i,-"- -f(l) (D). 



