1097 
evident that neither this choice of v nor the method of OsreN gives 
the true convection of the vorticity along the surface, as the con- 
vection velocity must sink to zero there, which is not the case with 
the value of v, taken above. 
Boussinesq has shown that for the stationary two dimensional 
flow equation (20) takes a simple form, when we take as coordinates 
the stream function and the potential of the flow v, (here the relative 
flow image is used). When as in the notation of Boussinesq Ua is 
written for the stream-function, Up for the potential, then the equation 
0—p (Ee de _ at (= aig =) (20%) 
Ont Ov dx dydy) = 
becomes: 
8 dw 0?w Ow 
= oa ae) Bag 
or: 
Ow pv (O?w dw 
Ta (53 : =) GND oO 
A difficulty in the solution of this equation is that the boundary 
conditions are expressed in v and not in w. The limiting case 
however treated by Boussinesq himself in the problem of heat 
transport ') is simple: »/U is very small (this involves a very great 
R), so that the vortex motion is confined to a very thin boundary 
layer and the derivative of a quantity with respect to a (viz. in a 
direction perpendicular to the boundary layer) will be much greater 
than the derivative with respect to 8 (in the direction of the boun- 
dary layer). 
Then we may assume: 
re (22) 
and also: 
ge 28 
v= — De . = 5 5 5 A ° . ° ( ) 
where vs is the (true) velocity in the boundary layer parallel to 
the surface. 
By means of these formulae we can calculate the distribution of 
the vorticity and the current in the boundary layer, when we suppose 
the velocity outside the boundary layer to be known. In analogy 
1) J. Boussinesg, le. p. 295/296. — Boussinesg also treats the problem for a 
solid of rotation (p. 305). In thecalculation of the heat transport Kine uses the 
complete equation (21) (}.c.) 
71 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
