an Incompressible Viscous Fluid. 

 we have also the analogous equation for £ , 77, f, 



dec dy dz 



345 



From equations (5) we obtain 

 \ dx \dz dy) ) I 



cW 



-(2-2)} 



+ 10 



■/*?_■ 



L cfo ' V<iy dx) ) ~ 



I <i# ' \dz dy) ) \dy * \dx dz) j 

 \ dz * \dy dx) ) " 



> • (6) 



0. 



If we have a surface @ = const, defined by the differential 

 equations 



d®_dP (<^__d%\ 



dx ' \dz dy)' 



dx 



dy dy \dx dzj' 



dy dy 



dz dz \dy 



these equations may be written 



(7) 



drj\ 



dx)' 



cm d® , d® 



«j- + v-j- +W-J-: 

 dx dy dz 



p d®, d® 

 dx dy 



+«? 



(8) 



and we have, as a result, that there exist in the fluid an infi- 

 nite number of surfaces © = constant, each of which is covered 

 by a network of stream-lines and vortex-lines. 

 The expression 



is not an exact differential unless V 2 f=0, \/ 2 r} = 0, V 2 ?=0; 

 and consequently equations (7) do not always hold. Equa- 

 tions (6), however, must always hold, as they are obtained in- 

 dependently of the supposition contained in (7). Equations 

 (6) may be written, for brevity, in the form 



^ 1 + 7 7 ^ 2 + ?<D 8 =OJ 



(9) 



