1895.] 



of a Viscotis Incompressible Fluid. 



319 



and, combining it with the equation 



dt 

 we obtain 





dr 



dz 



\d7'' ^ r dr ^ dz' 



-9&) 



rrJdi TrOO) T^a> V { fd(0 



U— +F^ = Z7-+-j — 

 dr dz r to [\dr 



or 



I.e. 



9ft) 



d^ 



„ V d(o) fd(o (o 

 0) dz] \dr 7 



y-(i)]-v=. 



ft) \dr r j) 



d /ft) 

 dz\ 



"){^--a^^°^^ + l-(")|^-''a7^^^^=^- 



Adopting the same notation as before, the above equation 

 becomes 



X 



d{r, z) 



= 0. 



Thus we have ft)/r a function of ')(^ and t, being therefore at any 

 instant constant along any of the curves % = const., as in theabove 

 special case of steady motion. 



5. In the three-dimensional case, the equations of motion 

 may be written in the form 



|^,,K_„f),2.(|_|)=0, 



dy 



.(17). 



In addition to these we have 



d^u dv 



2f = 



dy 



_ du dw 



dz' '^^dz~dx ' 



9.}'—— — — 



^~dx dy' 



du dv 9^ _ p. 

 dx dy dz 



Suppose that we seek to determine three quantities /, m, n 

 such that 



OX dy dz 



.(18). 



