If we now conceive the point p to move from a given initial position (x, y, z) to a given final position (x,, Y1, z1) the work done by the force during the motion will be represented by

F(x,y1, z1) - F(x。, Yo, zo).

If then we suppose F(x, y, z) to be such that it has only a single value for any one point in space, the work will be completely determined by the original and final positions; and it follows that the work done by the force is always the same, whatever path may have been followed by the point in passing from one position to the other.

If the three conditions (1) are not satisfied, the integration cannot be effected in the same general manner. If, however, the path be known in which the motion takes place, the integration becomes thereby possible. If in this case two points are given as the original and final positions, and various curves are conceived as drawn between these points, along any of which the point p may move, then for each of these paths we may obtain a determinate value for the work done; but the values corresponding to these different paths need not be equal, as in the first case, but on the contrary are in general different.

§ 6. On the Ergal.

In those cases in which equation (12) holds, or the work done can be simply expressed as a function of the co-ordinates, this function plays a very important part in our calculations. Hamilton gave to it the special name of "force function"; a name applicable also to the more general case where, instead of a single moving point, any number of such points are considered, and where the condition is fulfilled that the work done depends only on the position of the points. In the later and more extended investigations with regard to the quantities which are expressed by this function, it has become needful to introduce a special name for the negative value of the function, or in other words for that quantity, the subtraction of which gives the work performed; and Rankine proposed for this the term 'potential energy.' This name sets forth very clearly the character of the quantity; but it

is somewhat long, and the author has ventured to propose in its place the term "Ergal."

Among the cases in which the force acting on a point has an Ergal, the most prominent is that in which the force. originates in attractions or repulsions, exerted on the moving point from fixed points, and the value of which depends only on the distance; in other words the case in which the force may be classed as a central force. Let us take as centre of force a fixed point π, with co-ordinates §, n, , and let P be its distance from the moving point p, so that

ρ · √ (§ − ∞)2 + (n − y)* + (5 − 2)3 .................................. (13).

=

Let us express the force which π exerts on p by p' (p), in which a positive value of the function expresses attraction, and a negative value repulsion; we then have for the components of the force the expressions

X = '(p)

But by (13) larly for the

dp

dx

that

х

§-*; Y=4(p)"="; Z=$(p)2==.

P

__§-*; hence X=

==

P

:

— $'(p)de, and simi

other two axes. If (p) be a function such

But, since in the expression for p given in equation (13) the quantities x, y, z are the only variables, and (p) may therefore be treated as a function of those three quantities, the

expression in brackets forms a perfect differential, and we may write :

·dė

Xdr+ Ydy+Zdz == dò (p) ............(16).

The element of work is thus given by the negative differential of (p); whence it follows that (p) is in this case the Ergal.

Again, instead of a single fixed point, we may have any number of fixed points π1, π2, π,, &c., the distances of which from p are P1, P2, P3, &c., and which exert on it forces I'′(P1), P' (P2), Ó' (P), &c. Then if, as in equation (14), we assume (p), 2(p), (p), &c. to be the integrals of the above functions, we obtain, exactly as in equation (15),

x _ _ dp,(p1) __ do,(p12) __ do,(p2)

1

dx

2

da

dx

[P1 (P1) + $2 (P2) + $s(Pg) + .....],

d

d

dy

dz

Σφ (ρ) (17α),

Similarly Y=-(p), Z=- p.)............

whence

Xd+Ydy+Zdz=- Σφ (ρ) ......... (18).

Thus the sum Σ (p) is here the Ergal.

§ 7. General Extension of the foregoing.

Hitherto we have only considered a single moving point; we will now extend the method to comprise a system composed of any number of moving points, which are in part acted on by external forces, and in part act mutually on each other.

If this whole system makes an indefinitely small movement, the forces acting on any one point, which forces we may conceive as combined into a single resultant, will perform a quantity of work which may be represented by the expression (Xdx + Ydy + Zdz). Hence the sum of all the

work done by all the forces acting in the system may be represented by an expression of the form

Σ(Xdx + Ydy + Zdz),

in which the summation extends to all the moving points. This complex expression, like the simpler one treated above, may have under certain circumstances the important peculiarity that it is the complete differential of some function of the co-ordinates of all the moving points; in which case we call this function, taken negatively, the Ergal of the whole system. It follows from this that in a finite movement of the system the total work done is simply equal to the difference between the initial and final values of the Ergal; and therefore (assuming that the function which represents the Ergal is such as to have only one value for one position of the points) the work done is completely determined by the initial and final positions of the points, without its being needful to know the paths, by which these have moved from one position to the other.

This state of things, which, it is obvious, simplifies greatly the determination of the work done, occurs when all the forces acting in the system are central forces, which either act upon the moving points from fixed points, or are actions between the moving points themselves.

First, as regards central forces acting from fixed points, we have already discussed their effect for a single moving point; and this discussion will extend also to the motion of the whole system of points, since the quantity of work done in the motion of a number of points is simply equal to the sum of the quantities of work done in the motion of each several point. We can therefore express the part of the Ergal relating to the action of the fixed points, as before, by

(p), if we only give such an extension to the summation, that it shall comprise not only as many terms as there are fixed points, but as many terms as there are combinations of one fixed and one moving point.

Next as regards the forces acting between the moving points themselves, we will for the present consider only two points p and p', with co-ordinates x, y, z, and x', y', z',

respectively. If r be the distance between these points, we have

r = 2)2. √(x' — x)* + (y' − y)2 + (z' — z)* .........(19). We may denote the force which the points exert on each other by f'(r), a positive value being used for attraction, and a negative for repulsion.

Then the components of the force which the point p exerts in this mutual action are

and the components of the opposite force exerted by p' are

so that the components of force in the direction of x may also be written

dr

dr

-ƒ' (r) dx2 -ƒ' (r) dx'

and if f(r) be a function such that

Similarly the components in the direction of y may be

and those in the direction of z

−df (r); −df (r).

dz

dz