780 
IV. Accelerated or retarded motion. 
When the motion of the body is not uniform, a second system 
of forces must be exerted by the surface of the body on the fluid. 
Outside 5, these forces can only give rise to an irrotational motion 
the potential ~** of which is defined by: 
op** 
On 
and may therefore be calculated by the methods of hydrodynamics 
for ideal fluids '). 
Within o; all velocities increase together with dV and in the 
transition layer a vortex layer is generated of the intensity 
== (dV, (along optie poe 
[we dna x (9 ot — dv) 0 08 es OO 
On a possibly existing structure of this layer nothing can be 
said directly; the impulse must be equal to dt times the resultant 
of all extra forces that have acted on the fluid (both inside 6; and 
in the transition layer). We can partly (perhaps totally) calculate the 
impulse from the total intensity of the layer, which is given by 
(26); this part must agree with that which may be calculated from 
el), this force will give rise to a “vortex double layer”: a positive and a 
negative layer with intensities of the order <—2 per unit of volume at a distance 
of the order e; so that the intensity per unit of surface fran, and the im- 
pulse per unit of surface f rXw*dn are both finite. For the layer formed by fn 
this is generally not the case; this layer is simple and consists of lines circling 
round the body. 
This may be illustrated by the consideration of a disc moving in its own plane 
while its thickness approaches zero. Then the resultant of the pressure forces 
becomes zero, which must therefore also be the case with the impulse of the vortex 
motion generated by fx. The resultant of the frictional forces remains finite and is 
nearly independent of the thickness of the disc. Therefore the impulse of the 
transition layer cannot or can only partly be due to the fact that it consists of 
vortex lines surrounding the disc. It must have its impulse “in itself” viz. it must 
be a ‘double layer”. 
A double layer may be represented by w= ——. 55 = Born € =. (—; ae a the impulse 
+o 
has the value few da =2 AV vz, independent of t. 
e 
— @ 
") See eg. H. Lams, Hydrodynamics, Ch. V and VI. 
