Prof. Challis on the Principles of Hydrodynamics. 235 



It may be remarked, tliat this equation rests on the principle 

 of constancy of mass, and on the axiom that the directions of 

 motion are normals to continuous surfaces ; but on account of 

 the disappearance of u and /3 in the investigation, it does not 

 involve the same consideration of continuity as that which con- 

 ducted to equation (3.). 



The other mode of investigating equation (7.) is given by 

 Professor Tardy in the Number of the Philosophical Magazine 

 for March 1850, p. 173. It will only be necessary to exhibit 

 here so much of the process as will convey an idea of the prin- 

 ciple of the reasoning. The equation 



« = X^ 

 ax 



gives by differentiating with respect to x, 



du d^yjr dX «?i|r 



dx dx^ dx dx ' 

 Putting, for the sake of brevity, 



dr_,dr,dr_Ti, 



dx^ "^ rf/ ■*■ dz^ ~ ' 

 we have 



and thence by differentiating, 



dV d^d^r df^ djf^d^^\r 



dx __ dx _Y dx dx^ dy dy^ dz dz^ 



dx~ '^ R3~ • 



Hence, substituting for X and -y in the above expression for -r-* 

 and observing that ~f- = -^, the result is 



du u dY 



dx ~ Y 



dx^ dx^ dx^ 

 dx dy dxdy dx~Sz dxdz 



By obtaining analogous expressions for -j- and -7-, adding the 

 three together, and having regard to the known expression for 

 — + — in terms of partial differential coefficients of 1^, and to 

 the equality 



ds dx'Y '^ dy'y dz ' \' 



