an Incompressible Viscous Fluid. 347 



Differentiating these for x, y, and z respectively and adding, 

 we have 



V 2 (P-©)=0, (13) 



or 



V 2 P=V 2 ® (14) 



Denoting by 6 the angle between the stream-line and the 

 vortex-line at the point where the normal n is drawn to the 

 surface ©= const., we have, since 



and 



qco sin 6=\/ (v^—wrj) 2 + (ic% — u%) 2 + (utj—v^) 2 , 



dn-^ 03 *™ 6 (l0) 



From this, as in the case of no viscosity, follows that the pro- 

 duct 



qco sin 6 dn 



must be constant over each of the surfaces ©= const., dn de- 

 noting the normal drawn to the consecutive surface. 



For the determination of the pressure p it will be convenient 



dx 

 to resume equations (4). Since u — g-j- y &c, these become 



1 dp dY du - 



- /'= - 7i Q IT + T V \ 



p ax ax as 



1 dp dV dv , __■ , / 1C n 



l± = _d_V dv, 

 p dz dz * ds 



Multiplying these by -=-> ~j -j- respectively and adding these 



results, 



1 dp = rf(P-e) dV 1 d(q 2 ) ^ 

 p ds ds ds 2 ds ' 



from which, since p is constant, 



p=p{(T-&)-Y-iq°\ +C (17) 



As we have integrated along the stream-line \ ds, the quantity 

 C is only constant for this particular line. 



Let us assume that a solid sphere is immersed in the fluid, 

 the latter moving past the sphere in such a manner that the 



