Prof. Challis on a new Equation in Hydrodynamics. 283 



The validity of this equation was confirmed, first, by show- 

 ing that it holds good at the same time that udx + vdy + wdz 

 is integrable per se, if N be a function of t only, which for this 

 case it manifestly must be; and then by employing it to obtain 

 from equation (2.) the equation 



,^d V V^ 



/W-P=/^-l''^ + T (*•) 



m which V is the velocity at any point of a line s drawn at a 

 given instant in the direction of the motion of the particles 

 through which it passes. 



The equation (4.) is readily shown to be true when udx 

 + vdy + 'wdj^ is assumed to be an exact differential. That it 

 is true without any limitation, as the reasoning to which I am 

 referring demonstrates by means of equation (3.), may be 

 concisely shown as follows. If F be the sum of the resolved 

 parts in the direction of an arbitrary line s of the forces im- 

 pressed on a mass of fluid, then in case of equilibrium tZP 

 = Fds; and hence, by D'Alembert's principle, when there 

 is motion and the effective accelerative force in the same di- 

 rection is^ dP = {F —f) ds. If, therefore, there be no im- 

 pressed force, d P +J'd s — 0: and supposing the arbitrary 

 line to be taken in the direction of the motion of the particles 



through which it passes, / = ( -77 ) • But on this supposition 



/dY\ _dy ^^_1Z V — 



\dt ) ~ dt ds' dt'' dt ' ds' 



dV dV 



Hence — dF = -j— £?s + V . -5— d s, which gives by integra- 

 tion equation (4.). 



I proceed now to adduce another argument in confirmation 

 of equation (3.), by employing this equation in making a de- 

 duction from equation (1.), the truth of which may be esta- 

 blished by independent considerations. For this purpose it 

 will be necessary to make use of the formula for the sum of 

 the reciprocals of the principal radii of curvature of a curve 

 surface whose equation is = 0, expressed in partial differ- 

 ential coefficients of 0. To those conversant with the pro- 

 cesses of analytical geometry, there will be no difficulty in 

 proving that, if r and ;•' be the radii of curvature, this for- 

 mula is 



\r '^ r") Vdx'^'^ dy'^'^ d z^J " dx^' dx'^ "^ dy^ ' Jf 

 U2 



