298 Prof. Challis on a iieiio Equation in Hydrodyjiamics, 



Hence substituting in the above equation and integrating, we 

 obtain 



^4' d-l/ d^ d^ 

 ''=*-+^"+^''+i'" + xW- • • (5-) 



Lastly, by taking account of the equalities, 



, d-^ d-l/ d-h 



da; df dz* 



and supposing the arbitrary function of the time to be included 

 in \I/, we have the equation (2.) which it was required to deduce. 

 It may here be remarked, that the equation (5.) may be 

 put under the form 



Whence, by integration, 



^+xi(0+c=o, 



an equation which applies to a given element. Hence since 

 C is a quantity altogether arbitrary, it cannot be affirmed that 

 ^ has necessarily the same value at the same time for different 

 elements. Again, if Ix, 8j/, 82; be the variations of the co-ordi- 

 nates at a given time from the point xyz to any point indefi- 

 nitely near, we have 



Hence integrating along an arbitrary line, 



4' = \I/(a:, y, !s, t) +'^-4/(a, b, c, t\ 



"^ being the value of -^ at the arbitrary point ahc at the given 

 time. This equation gives the value of \|/ at the point ocyz i 

 but as this value contains the arbitrary quantity '^—\J/(a,i,c, if), 

 it cannot be affirmed that rj' has necessarily the same value at 

 the same time for any other point of space. From these two 

 results we may conclude that the value of 4/ is in no respect 

 limited by the investigation which has conducted to the equa- 

 tion (2.). 



I am quite at a loss to understand how any statement made 

 by me in the Supplementary Number of the Philosophical 

 Magazine for last June, should have led Professor Tardy to 

 suppose that, according to ray views, " the particles of the 

 fluid which are on the surface vI/ = remain on it during suc- 

 cessive instants." I distinctly affirmed, what the preceding 

 investigation has perhaps made more apparent, that the new 

 equation *' expresses the condition that the directions of the 



