47s The Rev. J. Challis on the Principles of the 



are varying. If u, v, w be the parts of the velocity at a 

 point whose coordinates are x, j/, z, resolved in the direc- 

 tions of the axes of rectangular coordinates, the equation thus 

 obtained for an incompi'essible fluid is, 



du dv d-iX) ^ ... 



-T— + -7- +-7- =0 (A. 



dx dy dz 



Under this form it is not proper for application to proposed 

 instances of motion, for which purpose it must admit of exact 

 or approximate integration, and therefore consist of partial 

 differential coefficients of a principal variable. The trans- 

 formation into the required form is made by assuming u d x 

 + vdy 4 ISO d z to he an exact differential of a function of 

 a; y and ^. For if <f> be the function, evidently 

 d <i> d <t> dit> 



dx dy dz 



and T-| + -, J + -^ = . . . . (B.) 



d x^ dy dz^ 



No general rule has hitherto been given for determining 

 when it is allowable to assume u d x -\- v d y ■\- w d z iohe 

 an exact differential, nor has it been ascertained to what par- 

 ticular circumstance of the motion this analytical condition 

 refers. This must be considered a defect in the mathematical 

 theory of hydrodynamics. 



The equation (A.) may also be arrived at by the following 

 considerations. In whatever manner a mass of fluid is in mo- 

 tion, we may conceive an unlimited number of surfaces de- 

 scribed in it, so that each is perpendicular to the direction of 

 the motion at a given time of the particles through which it 

 passes. These surfaces are not necessarily continuous, but 

 must be supposed to consist of continuous parts possessing 

 the general properties of curve surfaces. Hence if, in two of 

 the surfaces separated by an indefinitely small interval, two 

 indefinitely small rectangular areas be taken opposite to each 

 other, so that the lines joining the corresponding angular 

 points are normals to the surfaces, then the directions of mo- 

 tion in the fluid element lying between them will, by a known 

 property of curve surfaces, all pass through ivio focal lines per- 

 pendicular to the directions of the normals, and situated in 

 planes at right angles to each other. The normal velocity 

 will in general vary in passing from one point to another of 

 the intersecting surface, in a manner depending on the arbi- 

 trary disturbance of the fluid ; but taking it to be uniform 

 through the extent of each of the indefinitely small areas, the 

 continuity of the fluid will be maintained if the same quantity 



