Stability of Viscous Fluid Motion. 615 



Kelvin's equation (6) with JJ = /3y has been given by Oseen*. 

 In this case the initial value of f is concentrated at one point 

 (f, t]), and the problem may naturally be regarded as an 

 extension of one of Fourier relating to the conduction of 

 heat. Oseen finds 



Hfi 4:vt(l + T \8W) 4vt 



«**f>- °\ M1+ U — ■ • ( 12 > 



where 



c 



jj S(lv,0)d£d v ; ..... (13) 



and the result may be verified by substitution. 



"The curves f= const, constitute a system of coaxal and 

 similar ellipses, whose centre at t = coincides with the 

 point f, T), and then moves with the velocity J3rj parallel to 

 the ,?>axis. For very small values of t the eccentricity of the 

 ellipse is very small and the angle which the major axis 

 makes with the ,r-axis is about 45°. With increasing t this 

 angle becomes smaller. At the same time the eccentricity 

 becomes larger. For infinitely great values of t, the angle 

 becomes infinitely small and the eccentricity infinitely 

 great." 



When /3 — in (12), we fall back on Fourier's solution. 

 Without loss of generality we may suppose f = 0, ?? = 0, and 

 then (r 2 = .r 2 + ?/ 2 ) 



K*'»'0-nb-« ( 14 > 



representing the diffusion of heat, or vorticity, in two 

 dimensions. It may be worth while to notice the corre- 

 sponding tangential velocity in the hydrodynamical problem. 

 If yfr be the stream-function, 



9 y- (p r . ' /- "> __ j d ( r d *\ 

 **~da*+dtf -r'dr\ df)* 



so that (IaIt C /a ,.,,,. ,--, 



r -/ =z-(l — e-'- 4 " f ) (15) 



dr 7T v / v 



the constant of integration being determined from the known 

 value of dyfr/dr when ?* = co . When r is small (15) gives 



j?*ss (lb) 



becoming finite when r = so soon as t is finite. 



* Arkiv for Matematik, Astronomi och Fysilc, Upsala. Bd. vii. No. 15, 

 1911. 



