46 Professor Osborne Reynolds [March 28, 



Such a theory would stand in the same relation to the movements 

 of troops as that of hydrodynamics does to the movements of water. 

 For although only the disciplined motion is recognised in military 

 tactics, troops have another manner of motion when anything disturbs 

 their order. And this is precisely how it is with water : it will move 

 in a perfectly direct disciplined manner under some circumstances, 

 while under others it becomes a mass of eddies and cross streams 

 which may be well likened to the motion of a whirling, struggling mob 

 where each individual particle is obstructing the others. 



Nor does the analogy end here : the circumstances which deter- 

 mine whether the motion of troops shall be a march or a scramble, 

 are closely analogous to those which determine whether the motion of 

 water shall be direct or sinuous. 



In both cases there is a certain influence necessary for order : 

 with troops it is discipline ; with water it is viscosity or treacliness. 



The better the discipline of the troops, or the more treacly the 

 fluid, the less likely is steady motion to be disturbed under any 

 circumstances. On the other hand, speed and size are in both cases 

 influences conducive to unsteadiness. The larger the army, and the 

 more rapid the evolutions, the greater the chance of disorder ; so with 

 fluid the larger the channel, and the greater the velocity, the more 

 chance of eddies. 



With troops some evolutions are much more difficult to effect with 

 steadiness than others, and some evolutions which would be perfectly 

 safe on parade, would be sheer madness in the presence of an enemy. 

 So it is with water. 



One of my chief objects in introducing this analogy of the troops 

 is to emphasise the fact, that even while executing manoeuvres in a 

 steady manner there may be a fundamental difference in the condition 

 of the fluid. This is easily realised in the case of troops. Difficult and 

 easy manoeuvres may be executed in equally steady manners if all 

 goes well, but the conditions of the moving troops are essentially 

 different. For while in the one case any slight disarrangement would 

 be easily rectified, in the other it would inevitably lead to a scramble. 

 The source of such a change in the manner of motion under such 

 circumstances, may be ascribed either to the delicacy of the manoeuvre, 

 or to the upsetting disturbance, but as a matter of fact, both of these 

 causes are necessary. In the case of extreme delicacy an indefinitely 

 small disturbance, such as is always to be counted on, will eftect the 

 change. 



Under these circumstances we may well describe the condition of 

 the troops in the simple manoeuvre as stable, while that in the 

 delicate manoeuvre is unstable, i. e. will break down on the smallest 

 disarrangement. The small disarrangement is the immediate source 

 of the break-down in the same sense as the sound of a voice is 

 sometimes the cause of an avalanche ; but if wc regard such dis- 

 arrangement as certain to occur, then the source of the disturbance is 

 a condition of instability. 



