STABILITY 4/8 



sketched in Figure 4/5/3. The other lines of the field could be 

 added by considering what would happen after other disturbances 

 (lines starting from points other than A). Although having dif- 

 ferent initial states, all the lines would converge towards 0. 



4/6. In some of our examples, for instance that of the cube, the 

 lines of behaviour terminate in a point at which all movement 

 ceases. In other examples the movement does not wholly cease ; 

 many a thermostat settles down, when close to its resting state, 

 to a regular small oscillation. We shall be little interested in the 

 details of what happens at the exact centre. 



4/7. More important is the underlying theme that in all cases 

 the stable system is characterised by the fact that after a displace- 

 ment we can assign some limit to the subsequent movement of the 

 representative point, whereas in the unstable system such limita- 

 tion is either impossible or depends on facts outside the subject of 

 discussion. Thus, if a thermostat is set at 37° C. and displaced 

 to 40°, we can predict that in the future it will not go outside 

 specified limits, which might be in one apparatus 36° and 40°. 

 On the other hand, if the thermostat has been assembled with a 

 component reversed so that it is unstable (S. 4/12) and if it is 

 displaced to 40°, then we can give no limits to its subsequent 

 temperatures ; unless we introduce such new topics as the melting- 

 point of its solder. 



4/8. These considerations bring us to the definition which will 

 be used. Given an absolute system and a region within its field, 

 a line of behaviour from a point within the region is stable if it 

 never leaves the region. Within one absolute system a change of 

 the region or of the line of behaviour may change the result of 

 the criterion. 



Thus, in Figure 4/3/1 the stability around A can be decided 

 thus : make a mark on each side oi" A so as to define the region ; 

 then as the line of behaviour from any point within this region 

 never leaves it, the line of behaviour is stable. On the other 

 hand, no region can be found around B which gives a stable line 

 of behaviour. Again, consider Figure 4/5/2 : a boundary line 

 is first drawn to enclose A, and B, in order to define which 

 part of the field is being discussed. The line of behaviour from 



47 



