Professor Kelland's Note on Fluid Motion. 



31 



or 



W7/ = 



M 



du 

 dy 



du 

 dz 



dv p (Ijw 

 dx da? 



T^rdv -ndw 



xt dv 



dy' 



dw^ 

 d~z ; 



(2.) 



where 



dz 



xt div 

 dx 



dm 

 d? 



du 

 Tz' 



dv 



(S*.) 



p=:= *» 



f/j/ dx' 



1. One way of satisfying all the equations is by supposing 

 M = 0, N =b, P = ; in which case the equations (3.) in- 

 dicate that udx + vdy -\- wdz is a complete differential. 



2. Another way is to suppose M, N and P all absolutely 

 constant; in which case the velocities u, v, w will be deter- 

 mined by the same equation, viz. by either of the equations 

 (1.). Hence it, v, w all have the same form. 



Also the equations (2.) give 



M^ + N* 



dx dx dx 



dv -ndw 



0, 



&c. &c, orMa+ Nu+ Pwisa quantity whose partial dif- 

 ferential coefficients, with respect to each of the coordinates, 

 is zero. This quantity is therefore either zero, or a function 

 of t only. 



a. If it be zero, udx + vdy + ivdz is integrable by a 

 factor, for the equation M?i + N»+Pro = is the well- 

 known equation of condition that this may be the case. 



b. IfMtt+Nu + Ptt> —f{t), udx + vdy + wdz is not 

 a complete differential after being multiplied by a factor. 



The equations are nevertheless integrable in this case, and 

 give as their result, 



u= F(Mz-P#, N*-Py, /), 

 t> = <j>(Mz -P#, Ns-Pj/, t), 



* See ray Memoir on the Theory of Waves, Trans. Roy. Soc. Edin., 

 vol. xv. p. 116. 



