Mr EARNSHAW, ON FLUID MOTION. 205 



2. If the expression udx + vchj + tvdz be integrable either imme- 

 diately or by a multiplier, the integral of the equation 



udx + vdy + wda = 0, 

 will be of the form 



fix, y, X, f) ^ 0, 



which will furnish the surfaces alluded to in last article. But if the 

 above expression should neither be integrable at once nor by a mul- 

 tiplier, the integral of the above equation will be of the form 



/{x, I/, z,f) = 0\ 

 xl^ix, y, x) = i ' 

 and will denote, not a series of surfaces, but a series of curve lines. 



3. It appears that all the particles through which the surface 

 passes whose equation is 



udx + vdy + wdz = 0, 



are connected by the common property proved in Art. 1; and as we 

 have no other idea of a tvave-surface than that it is the locus of 

 particles in a similar state of disturbance, we may be permitted to 

 take the above equation as the expression of that similarity which 

 constitutes a wave-surface; or in other words, we may assume the 

 equation 



udx + vdy + wdz = 0, 



as the mathematical definition of a wave-surface, ox of a wave-line, as the 

 case may be. 



By the assistance of this definition we may enunciate the propo- 

 sition of Art. 1 in these terms ; — 



The motion of every jxirticle of the fluid is perpendicular to the 

 wave-surface in which it is situated. 



4. It is proved by Pont^coulant in his "Theorie Analytique du 

 Syst^me du Monde," Tom. I. p. 163, and by most other writers on 

 Hydrodynamics, that if udx + vdy + wdz be at any one instant a complete 



D D 2 



