(1) This can be proved in much finer detail, but the details are hardly required here. Its derivation from (b) is almost direct.

(2) A configuration of the system corresponds to a point in the space. The behavior of the system in time corresponds to the point moving along a path. All the paths make the field and this defines the organization of the system. “Experimental control” is equivalent to the ability to start the point where we like in the field (by using the n arbitrary constants in the solution of equation 1).

(3) We do not need here to know anything of the physiological mechanism underlying such change.

(4) To preserve the form of equations (1) we may use any continuous approximation to step-function form such as ; where q is positive and large. The function θ determines those x and h combinations where h' changes to h”; for where θ(x ₁, ... , x _n; h) is positive, h becomes h', while where it is negative h changes to h”.

(5) The whole x-field remains constant until h changes; then the x-field changes, in general to something quite different. Consequently, if the x-point, started from always followed a particular path, then after change of h it can follow a different path even though started from the same (x) point.

(6) (α) The hypothesis (b) demands absoluteness. (β) The “reorganization” hypothesis demands step-functions, as we have shown above. (γ) As we are not concerned with the possibility that the nervous system is limited in its variability of behavior, we may assume this to be infinite in the absence of any reason to the contrary.

(7) Since a commutive system has an infinite number of x-fields, it clearly cannot be handled with the explicitness of equations (1). Assuming we have in front of us a real, physical example of a commutive system, we can clearly no longer know (a) the number of h’s, (b) their values, (c) the forms of the f’s in f _i(x ₁, ... ; h ₁, ...), (d) the x-fields, since these depond on the f’s. But, and this is sufficient for the rest of the paper, (a) we can observe the values and changes of the x’s, (b) we can control the x’s starting point, and (c) we can test for constancy of the h’s by seeing whether the x’s path in the x-space repeats itself.

(8) The disturbances are needed to test for, and demonstrate, equilibrium.

(9) It is not necessary, for there are other ways of getting h-constancy, depending on some relation between the distribution of θ-points and the direction of paths in the fields. We have no right to say that these combinations cannot occur.

(10) Some minor postulate that the disturbance is to return the x-point to A, B, ... by some definite route devoid of θ-points is required here.

(11) Firstly, by selecting special examples it is easy to show that “equilibrium” belongs strictly to a single path and not to a field. Other special cases show that the path is to be confined to a region and not necessarily to terminate at a point. It may easily be shown th~ the common examples of equilibrium are all special cases, or limits, of.the definition given. Nothing less general seems to be adequate.

(12) The x-point, is first to be fixed, but p is not to depend on where it is fixed.

(13) “At least” because, as shown in Note 9, other ways of getting h-constancy are possible, though rare and of little importance p does not include these, so their presence may result in an increase of the proportion which have become h-constant.

(14) This does not apply to a single field, i.e., the “commutive” part is necessary; for in a single field, as t → ∞, P does not → 1 but → p. If p is small the effect is quite different.

(15) The hypotheses about p and m are really only that (1) p ≠ 0 and (2) m → ∞ as t → ∞. These merely exclude peculiar cases.

(16) If the θ-points are distributed systematically, the paths of equilibrium will automatically have systematic properties imposed on them. This leads to interesting and important developments which ha~e already been explored by the author, but they are outside the scope of this paper.

(17) The above discussion on deals with one system where the x-point either is, or not, on an equilibrium path, and the equilibrium is therefore “all or none.” But there are several ways in which we can get independent equilibria, and then, following the same principles as before, the number of equilibria must tend to increase by accumlation. But this leads beyond the scope of the present paper.

(18) We have supposed the environment to remain constant as far as its organization is concerned (though not the values of its variables). This means that the enviromment, can interact freely with the nervous system in the above theory. A single change of the envirommental organization would, however, wreck an established equilibrium. But regular chanes between a finite set of enviromment can be shown to tend to Equilibrium. If the enviromment should change its organization irregularly, the whole paper becomes inapplicable since postulate (b) of § 1 is no longer true.

(19) Ashby, W. R. J. ment, Sci., 1940, 86, 478.