356 On Steady Motion in an Incompressible Viscous Fluid. 

 From equations (34) we have 



t ds ds ds n 

 a* fly " cfe 



the vortex-lines therefore lie on the surface s = const., and also 

 on the sphere r = const. For the particular case when the 

 flow at an infinitely great distance from the sphere is parallel 

 to the axis of x, all of the functions vanish except fa ; then 

 writing 



fa= — \Xy 



we have 



. J ., 3 a 1 o? \ 3X a /, a\ 2 



(53) 



3\ 



— T 



^( 1 - 



« 2 \ 





3\ 



W 1 - 



« 2 \ 





For this case s reduces to 



3 Xa 



and consequently 



?=o, 



3 X« 







r=- 



3 Xa 

 "2 r*"^ 



(54) 



From these last equations we see that the vortex-lines are 

 circles whose centres lie upon the axis of x. The function % 

 was defined by the differential equations 



dx dx \dz dy) '' 



substituting in these the values of f? , rj, f, from (34) we obtain 



® — P = 5 — ^ — , 



which is easily seen to satisfy the equation 



v 2 (©-P)=o. 



Substituting this value of © — P in equation (17), we have 



