DESIGN FOR A BRAIN 4/4 



We conclude tentatively that the concept of ' stability ' belongs 

 not to a material body but to a field. It is shown by a field if 

 the lines of behaviour converge. (An exact definition is given in 



S. 4/8.) 



4/4. The points A and B are such that the ball, if released on 

 either of them, and mathematically perfect, will stay there. 

 Given a field, a state of equilibrium is one from which the repre- 

 sentative point does not move. When the primary operation is 

 applied, the transition from that state can be described as ' to 

 itself '. 



(Notice that this definition, while saying what happens at the 

 equilibrial state, does not restrict how the lines of behaviour may 

 run around it. They may converge in to it, or diverge from it, 

 or behave in other ways.) 



Although the variables do not change value when the system 

 is at a state of equilibrium, this invariance does not imply that 

 the ' machine ' is inactive. Thus, a motionless Watt's governor 

 is compatible with the engine working at a non-zero rate. (The 

 matter has been treated more fully in /. to C, S. 11/15.) 



4/5. To illustrate that the concept of stability belongs to a 

 field, let us examine the fields of the previous examples. 



The cube resting on one face yields a state-determined system 

 which has two variables: 



(x) the angle at which the face makes with the horizontal, and 

 (y) the rate at which this angle changes. 



(This system allows for the momentum of the cube.) If the cube 

 does not bounce when the face meets the table, the field is similar 

 to that sketched in Figure 4/5/1. The stability of the cube 

 when resting on a face corresponds in the field to the convergence 

 of the lines of behaviour to the centre. 



The square card balanced on its edge can be represented approxi- 

 mately by two variables which measure displacements at right 

 angles (a?) and parallel (y) to the lower edge. The field will 

 resemble that sketched in Figure 4/5/2. Displacement from the 

 origin to A is followed by a return of the representative point 

 to -O, and this return corresponds to the stability. Displacement 

 from O to B is followed by a departure from the region under 



46 



