438 Prof. Challis on the general Differential 



plane. But clearly in these cases the condition that the lines of 

 motion are normals to continuous surfaces is satisfied, and the 

 principle above enunciated as the foundation of a third general 

 equation is consequently involved. For the solution of other 

 problems, the differential function udx 4- vdy + ivdz is equated to 

 (d(f))j the differential with respect to coordinates of a new variable 



d>. so that w=- -y-, v= -=-£, and w = -—-. I am not aware that 

 T dx dy dz 



any problems have been attempted in which that supposition 

 has not been actually or virtually made. But whence arises the 

 necessity for a new variable, and what does the variable itself 

 signify ? Respecting the meaning of the variable, a very explicit 

 an swer can be given. For since in the expression udx -f- vdy -f wdz 

 the differentials dx, dy, dz are independent and arbitrary, we may 

 assume them to be such that that expression is equal to zero. It 

 will then be seen that (ckf>) = is the differential equation of a 

 surface which is everywhere cut at right angles by the directions 

 of the lines of motion in the elements through which it passes. 

 It is evident that there will be an unlimited number of such 

 surfaces, the function <j> being applicable at all times to all parts 

 of the fluid. Thus the introduction of this variable is really a 

 recognition of the principle that the lines of motion are subject 

 to the above geometrical condition. The further step that I have 

 taken is to regard this principle as necessary and fundamental, 

 and to reason from it. According to this view, the substitution 

 of (dej)) for udx + vdy + wdz would be a consequence of that prin- 

 ciple. The following considerations will, however, show that 

 this substitution is not sufficiently general, and would unduly 

 restrict the investigation of the laws of the motion of fluids. 



5. It is known from analytical geometry that udx + vdy + wdz = 

 would equally be the differential equation of a surface cutting at 

 right angles the directions of the motion, if u, v, and w y instead 

 of being equal, were respectively proportional to the partial 

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

 of x y y, z, and t, that is, X and yjr being both unknown func- 

 tions of x 3 y, z, and /, if 



„ d^lr % d'dr % d^Jr 



u=\~-, v=\-t-. w=\-r-. 

 dx dy dz 



and consequently 



X(<^r) = udx + vdy + wdz. 

 It is admitted that the right-hand side of the last equation is 

 not an exact differential in every case of the motion of fluids ; 

 so that, although by substituting (d<fi) for it a resulting differ- 

 ential equation involving only x, y, z, and t, with cf> as the prin- 

 cipal variable, might be found, this equation would not possess. 



