4/3 



STABILITY 



The result is that if any transient disturbance cools or overheats 

 the main object, the thermostat brings its temperature back to 

 the usual value. By this return the system demonstrates its 

 stability. 



4/2. An important feature of stability is that it does not refer 

 to a material body or ' machine ' but only to some aspect of it. 

 This statement may be proved most simply by an example showing 

 that a single material body can be in two different equilibrial 

 conditions at the same time. Consider a square card balanced 

 exactly on one edge; to displacements at right angles to this edge 

 the card is unstable; to displacements exactly parallel to this 

 edge it is, theoretically at least, stable. 



The example supports the thesis that we do not, in general, 

 study physical bodies but only entities carefully abstracted from 

 them. The matter will become clearer when we conform to the 

 requirements of S. 2/10 and define stability in terms of the results 

 of primary operations. This may be done as follows. 



4/3. Consider a corrugated surface, laid horizontally, with a ball 

 rolling from a ridge down towards a trough. A photograph taken 

 in the middle of its roll would look like 

 Figure 4/3/1. We might think of the 

 ball as being unstable because it has 

 rolled away from the ridge, until we 

 realise that we can also think of it as 

 stable because it is rolling towards the 

 trough. The duality shows we are 

 approaching the concept in the wrong 

 way. The situation can be made clearer 

 if we remove the ball and consider only Figure 4/3/1 



the surface. The top of the ridge, as 

 it would affect the roll of a ball, is now recognised as a position 

 of unstable equilibrium, and the bottom of the trough as a position 

 of stability. We now see that, if friction is sufficiently marked 

 for us to be able to neglect momentum, the system composed of 

 the single variable ' distance of the ball laterally ' is state-deter- 

 mined, and has a definite, permanent field, which is sketched in 

 the Figure. 



From B the lines of behaviour diverge, but to A they converge. 



45 



