428 The Rev. Prof. Challis on the Necessity of Three 



in all cases of motion, unless it may be regarded as an arbi- 

 trary or discontinuous function. The same remark applies 

 to the functions that u, v, w are of the coordinates and the 

 time. To represent the velocities at all points and at all times, 

 whatever motion be given to the fluid, they must be arbitrary 

 or discontinuous functions. As, however, in the investigation 

 of the equations (1.) and (2.) u, v, w were assumed to be func- 

 tions of constant form for at least indefinitely small variations 

 of the coordinates and the time (for they would not otherwise 

 be subject to mathematical reasoning), the same supposition 

 must be made with respect to the function vp. Hence So;, 83/, 

 8z and ^t, being indefinitely small variations, we have for a 

 given form of the function, 



4/(^,7/, 2,^) = 0, 



and ^^{x-{-^x,i/ + ^t/, z + ^Syt + Bt) =0. 



Expanding the second equation to first powers of the small 

 quantities, and putting for shortness' sake, ^ for 4; (.r, j/, z, t), 

 the result is, by reason of the first equation, 



at ax ay dz 



It is plain that so far this equation is nothing more than an 

 expression of geometrical continuity, and that the variables 

 8^, 83/, Iz are in no way related to each other. We may, there- 

 fore, suppose that lx = ulti ^y = vlts^nA lz = *(solt\ and the 

 equation then becomes 



rf\I/ fZvI/ d"^ d^ „ ,„ . 



dt dx dy dz ^ ' 



It now expresses that the normal surfaces to the directions of 

 the motion of the same elementary particle in successive in- 

 stants are geometrically continuous, and consequently may be 

 called the equation of continuity of the motion. 



We have now arrived at the third general equation, and 

 might proceed to test the truth of it by employing it with 

 equation (1.) in the investigation of equation (A.). As I have 

 already given the mathematical process for this purpose in the 

 Cambridge Philosophical Transactions (vol. vii. part iii. p. 

 385) and in the Philosophical Magazine (S. 3. vol. xx. April 

 1842), it will be unnecessary to adduce it here. 



The foregoing argument establishes the truth of equation 

 (3.). Being true, it may be presumed to be as necessary for 

 the general consideration of fluid motion as equations (1.) 

 and (2.). This, however, will more distinctly appear by 

 showing, as I proceed to do, the defective state of analytical 

 hydrodynamics when only equations (1.) and (2.) are made 

 use of. 



