32 Professor Kelland's Note on Fluid Motion. 



the functions being subject to the condition MF+N4> + P^ 



■=/('). 



3. If M, N, P are explicit functions of t only, our equations 



(1.) are reduced to 



*M M^ + N^ + pi", 



at dx ay a z 



^ = M^ + N^+P^, 

 at ax ay dz 



d P ^ifdw ^dw , „</?* 



— =M- r - +N r + P-t"- 



dt dx dy dz 



Hence 



-. (P u ^. a"* u p d* u _ 

 dx* " dxdy dxdz" 



-_ d 2 u ^d 2 u t, d* u 

 M -— + N -j-s + P t— r- = 0, 

 dxdy dy z dydz 



-. d- ti ~~ d* u p d 2 m _ 

 d xdz dydz dx? 



from which equations we obtain, by eliminating M, N and P, 



rf 2 M d*u d*u _ d*_u / d*u \ 2 _ d*u / d*u \ a 

 d&dfdz* dx*\dydz) dy*\dxdz) 



{ d*u*( d 9 u \* a d*u d*u d 2 u _ 



d z 2 \d x dy) " dxdy d xdz dydz 



an equation of precisely the same form as that which occurs 

 in the determination of the principal axes of a system, or of 

 the diametral lines of a surface of the second order. 



Similar equations are true in v and to. We conclude that 

 the motion is such as to be symmetrical with respect to the 

 coordinate planes. 



Cor. — If x, y, z enter in such a way into the expressions 



for the velocities that -=- = -7-, &c, the equations are identi- 

 dy dz ^ 



cally true. 



4. If the motion be confined to two dimensions, the equa- 

 tions are reduced to 



d u d v _ 

 dx dy ' 



du dv _ p 



dy dx ~ ' 



