in Reply to Professor P. Tardy. 297 



sequence of the adoption of the principle just enunciated, so 

 that there is no occasion for the application of the criterion 

 of integrability by a factor. Now, as is well known, one fun- 

 damental hydrodynamical equation rests on the principle that 

 the mass of a given element in motion continues from one in- 

 stant to the next to be the same. That is, if p dx dy dz be 

 the mass of the element, this equation of continuity is 



Z.pdxdydz=Oi 



the symbols of operation S and d being independent of each 

 other, and the former having reference to time as well as space. 

 This equation is equivalent to (3.). In an analogous manner, 

 another fundamental equation rests on the principle that a cer- 

 tain geometrical condition, viz. that the directions of motion 

 in a given element are normals to a continuous surface, con- 

 tinues from one instant to the next. That condition is ana- 

 lytically expressed by the equation 



and the second equation of continuity consequently is 



8,(#)=0, (4.) 



the symbols I and d being used in the manner stated above. 

 Now 



And 



By adding to this the analogous expressions for Z,-j- dy and 

 ^ , -J- dz, the result by reason of equation (4.) is 



or 



Since by hypothesis the reasoning applies to a given element, 

 ^x=udt, By=vdt, dz—wdt, 

 Phil. Mag. S. 3. Vol. 36. No. 243. April 1 850. X 



