138 
Proceedings of the Royal Society of Edinburgh. [Sess. 
where A r is a number, and equating the dimensions of each term on the 
right to [k], it is easily found that 
r = s=l -t ; u = 0. 
where r is indeterminable so far. The quantities A r and r, in fact, must 
depend on the shape of the boundaries. 
.-. K=vf 1 (miv) (i) 
or 
TJZ/k = F(UZ/v) (2) 
where the form of the functions depends on the geometry of the problem. 
From (1) it follows that for a given value of JJl/v the strength of the 
vorticity is directly proportional to the kinematic viscosity. Equation (2) 
indicates that the discussion of the stability of flow in a viscous fluid to 
a given disturbance, and all the circumstances of the motion generally, 
may equally well be centred round the non-dimensional group JJI/k ; that 
in fact we may imagine a given disturbance in vorticity, specified by k, 
applied to the fluid, and seek to determine the value of JJI/k, which is 
critical in the sense that it separates the region of values of this non- 
dimensional grouping for which the motion is stable from those for which 
the motion is unstable. Having determined this critical value of JJI/k, 
how to determine the exact relation (2) from which to evaluate the critical 
quantity JJl/v is clearly the next step, and if this can be completely 
effected the problem will definitely be solved. For the present I propose 
to restrict myself to determining whether such a critical value of JJI/k 
exists at all, and if so to evaluating it. It need scarcely be said that the 
discussion so far is not limited to the question of flow in a channel, 
although that is the case ,we have had in view. 
§ 4. It will be presumed that a fluid is moving between and parallel 
to the walls of a uniform channel, with the parabolic distribution in 
velocity corresponding to the steady motion of a viscous fluid in such a 
case, so that in the usual notation 
u = JJ(1 - y 2 /a 2 ) ; v = 0, 
where 2a is the breadth of the channel. 
Let it be disturbed in such a manner as to give rise to two vortices of 
strengths —k and +/c situated at the points (x = 0, y — h), (x = 0, y=—h) 
respectively. These may be imagined to have been produced by the 
sudden even insertion of a plate of breadth 2 h stemming the fibw 
symmetrically about the axis of x. Experiment shows that two vortices 
