384 Mr. Challis on the general Equations 



the accelerative forces impressed at the point ; and this substitu- 

 tion is legitimate, because, as the forces of nature are directed 

 to fixed or moveable centres, Xdx + Ydy + Zdz will be generally 

 a complete differential with respect to x, y, and z, of a function 

 of X, y, z, and t: ^ is a function of x, y, z, and t, which was 

 introduced in the course of the investigation of the preceding 

 equations, by substituting d(p for udx + vdy + wdz; for it appeared 

 that these equations do not admit of a simple fonn, unless udx 

 +vd^ + wdz be a complete differential of a function of x, y, and z, 

 which may also contain t, but is not differentiated with respect 

 to this variable. Consequently the equations (i), (2), (3) cannot 

 be made use of, except in cases in which we are assured that 

 this condition is satisfied. 



It has occurred to me that the analytical fact, that the equa- 

 tions of the motion admit of simplification when udx + vdy + wdz 

 is a complete differential of a function of x, y, z, has reference 

 to the manner of action of the parts of the fluid on each 

 other. If this be such that the motion in every elementary 

 portion is directed to a fixed or moveable centre, udx+vdy+wdz 

 will be a complete differential of a function of x, y, z, for the 

 same reason that Xdx+ Ydy-\-Zdz is a complete differential of the 

 same variables. And it is possible to obtain an integral of (2), 

 which will accord with this character of the motion. For suppose 

 to be a function of t and /, r- being equal to .r +_y- + z". Then 



by substituting in (2) we obtain " f = 0, an equation which is 



not contradictory to the supposition. In consequence of the 

 supposed nature of the function (p, 



_d(p X _d<p y _d(p z 



dr r dr r dr r 



values, which prove that the velocity is directed to or from the 



