Feb. 14, 1878] 



NATURE 



>o3 



establiih that their whole effect in altering the energy of motion 

 of ihe two bodies at any instant is divisible into two parts, that 

 which the forces, removed to the centre of mass of the pair, and 

 actinsj there on their joint mass, will have on the joint mass in 

 absolute space, and that which is represented by the sum of the 

 bodies' changes of actual energy reckoned in a space which has 

 this centre of the masses for its origin. If we call the latter changes 

 their local changes of energy, and professing ourselves entirely 

 ignorant of motion and position in absolute space, confine our 

 attention to describing the motions of the bodies in the specified 

 or local space, the abstract laws of dynamics again tell us that in 

 this local space the motion of the bodies is what arises from an 

 equal and opposite action and reaction exerted mutually between 

 them. Suppose this to be of the permanent kind above 

 described (which occurs frequently in natural actions, as already 

 mentioned), then as regards the local motion and its forces (now 

 equal and opposite, and quite distinct from what they were 

 abstractly), the above proposition may be predicated of them 

 which asserts that the local energy of motion and local potential 

 energy together have a constant sum. In our circumscribed 

 sphere of observation the energy of motion is entirely known, or 

 in other words, if we follow the bodies along any course from 

 one point to another, not only all the changes and the sum of 

 the changes of their actual energies, but also their energies at 

 first, and therefore their energies at last, are known by a successive 

 process of integration. We know from the permanency of the 

 energy-gradients along the line of centres that the sum of the 

 energy changes between the two given points is independent of 

 the course or lapse of time in which the final point is reached. 

 Instead, therefore, of making a new successive integration for every 

 course, one such integration for all expresses the total change of 

 energy between the points, and as this is possible for all points 

 or configurations which the bodies can reach from their first 

 configuration, if a scale of such energy changes reckoned from 

 some starting one is made out for all the different distances from 

 each other at which the bodies can be, the scale value will be 

 nothing at the starting distance, and will have determined values 

 at all other distances. We would use the scale by saying that 

 the actual energy at any distance only differs from the scale value 

 by the starting- energy to be super-added ; or the excess of the 

 actual energy above the scale value is everywhere constant, and 

 everywhere equal to the actual energy at the initial point. This 

 concise description of the motion, as far as the actual energy 

 at any moment is concerned, accords with the mathematical 

 usage of collecting variable quantities thus simply related 

 to each other and to constant quantities on one side, and 

 constant quantities on the other side of an equality ; but a further 

 simplification of its expression is effected if those scale values 

 which mean increase of energy from the starting-point are called 

 "negative," and those denoting loss or decrease of actual energy 

 are called " positive " ; for having constructed a new scale on this 

 convention (which we. may call the negative scale), to use it we 

 must first change the sign of any value in it before applying the 

 last proposition. As that expression tells us that the remainder, 

 on subtracting the former scale value from the actual energy at 

 any point, is constant, this operation of subtraction, after altering 

 the sign of the new scale value, is simply equivalent to adding 

 the new scale value without altering its sign. With this con- 

 vention, therefore, that an increase of actual energy is a nega- 

 tive increase, or, in other words, a decrease of the negative scale 

 value, we may put the sentence describing the actual energy in 

 every part of the motion in these much simpler words. The 

 sum of the actual energy and of the negative scale value is every- 

 where constant and equal to the actual energy at the starting- 

 point of the scale, which we may call the initial actual energy. 

 When increase of actual energy coincides with decrease of " nega- 

 tive scale value" (as we have just seen), and also as it is usual to 

 express it with "work done by a force," increase of negative 



purpose of mechanics). If we continue this process until all the bodies of the 

 material universe are brought, with a knowledge of their masses, under our 

 observation, we reach that abstract field of force, or force-space, which is 

 contemplated in Newton's enunciations. This space may be idenlified with 

 absolute space, because the centre of mass of the universe by which it is 

 defined is as perfectly abstract and metaphysical an idea as any that we can 

 form of absolute space, on the simple ground that we have no reason to 

 attribute to matter a less boundless and limitless extent in the universe than 

 we ascribe to space itself. To define one metaphysical idea by another is 

 not unscientific, nor is the description of force which Newton gives more 

 repugnant to the eyes of common sense than the_ ideas which we form, 

 though quite indefinite, of the extent of the material universe, and of the 

 boundless realms of space. A special office, it may also be suggested as 

 very probable, may be assigned to force, to avoid the occurrence of super- 

 position and mingling of matter in the same points of space, or to give 

 matter impenetrability. 



scale value represents work done against a force as it is expressed 

 in the new phraseology of the science of energy, or with " potential 

 work." The actual energy of the material couplet is everywhere 

 fixed and determinate (when it is once started), but if we speak 

 of the negative scale value as "potential energy" the amount 

 of this at various distances depends upon the distance chosen as 

 the initial one, when it is zero. Thus if we reckon the potential 

 energy of a swinging pendulum, drawn by gravitation towards 

 the centre of the earth (whose motions of rotation and of oscil- 

 lation relatively to the common centre of the globe and of the 

 pendulum-bob may be disregarded, so that, with the exception 

 of gravity, only a force perpendicular to its motion guides the 

 bob in a space, referred to the common centre as origin, which 

 we may identify with the place of the experiment) from the top 

 of the arc, where the actual energy of the bob is zero, this must 

 be the sum of the values of the actual and potential energies 

 throughout the motion, and consequently at the highest point 

 the potential energy is zero, and everywhere else it is negative, 

 while at the lowest point of the arc, where the actual energy is 

 a maximum, the potential energy reaches its greatest negative 

 value. If, on the contrary, we select the lowest point of the arc 

 as the starting-point, and call the potential energy at this point 

 zero, making the sum of the two in all parts of the motion 

 thereby equal to the greatest value which the actual energy can 

 have, the potential energy must elsewhere supply the deficiency as 

 the actual energy abates, or have positive value in all other posi- 

 tions of the bob, and at the highest points of its swing, when the 

 actual energy entirely disappears, it will reach its greatest posi- 

 tive value, equal to the greatest value of the actual energy at its 

 lowest point. By one such system, therefore, the motion is as 

 perfectly described as by the other, and by a different choice of 

 zero-points the individual amount of the potential energy is thus 

 evidently disposable at pleasure, while its difference between two 

 points yet always remains the same. But by taking the zero 

 point where the actual energy has its greatest value, the advan- 

 tage is obtained, as in the last arrangement for a pendulum, that 

 the potential, like its partner, actual energy, will never be less 

 than nothing, and its values will always be positive. With its 

 zero point so taken, and with a special choice of mass in the 

 moving body attracted or repelled, whose course is followed, the 

 series of negative scale values or of potential energies ji'.st de- 

 scribed is termed " the potential" or the " potential function" of 

 the force upon it ; but its definition for any permanent force-pair 

 supposes the total absence of all such constraining forces as the se 

 of the pendulum string, and the bodies must be left per.fecily 

 free to approach or recede from each other to the centre, or to 

 the furthest imaginable distance unimpeded by any forces foreign 

 to the pair. In such material couplets it is also sometimes cu£" 

 tomary to reckon their combined energies actual and potential in 

 a space having for its origin one of the bodies themselves instead 

 of the centre of their mass. The motion of the standard body 

 then disappears, and that of the other body becomes the relative 

 motion of the two, while at the same time a certain mean mass 

 must be supposed centred in the moving body, so that when the 

 product of this, multiplied by its new acceleration, is taken, its 

 impulse relatively to the stationary body (which is now the rate 

 of change of energy of the pair with the distance between them) 

 may not undergo any alteration by the change of origin. Reckoned 

 in this way, either of the bodies may be said to have energies of 

 motion and configuration in the space relative to the other body, 

 whose sum is constant. , 



Newcastle-on-Tyne A. S. Herschkl 



{^^Tohe continued. ) 



Aid of the Sun in Relation to Evolution 



It' is not often that it will fall to the lot of the physicist to 

 harmonise such important theories as those of evolution and the 

 nebular hypothesis, and much credit is due to the boldness and 

 the originality of Dr. CroU's attempt to do this. At the present 

 time the great majority of scientific men hold the trtith of both 

 of these hypotheses in spite of the fact that serious difhculties 

 exist in them which admit of only doubtful explanation, so that 

 it is certain they would be considerably strengthened if it were 

 found possible to dovetail them one to the other without unduly 

 straining the conditions of either. That Dr. CroU has effected 

 this important service is, I think, very questionable, although I 

 fully believe it is attainable. 



In advocating his own views in Nature (vol. xvii. pp. 206, 

 et seq.), and in his other publications Dr. Croll has anticipated 



