bounding curve consists of detached ovals, or possesses multiple points. This theorem seems to have been first given by Stokes (Smith's Prize Exam. 1854), in the form

It solves the problem suggested by the result of § 472 above. 478.] If o represent the vector force acting on a particle of matter at p, S.adp represents the work done while the particle is displaced along dp, so that the single integral

SS.odp

of last section, taken with a negative sign, represents the work done during a complete cycle. When this integral vanishes it is evident that, if the path be divided into any two parts, the work spent during the particle's motion through one part is equal to that gained in the other. Hence the system of forces must be conservative, i. e., must do the same amount of work for all paths having the same extremities.

But the equivalent double integral must also vanish. Hence a conservative system is such that

Sfds S. Vo Uv = 0,

whatever be the form of the finite portion of surface of which ds is an element. Hence, as Vo has a fixed value at each point of space, while Uv may be altered at will, we must have

If we call X, Y, Z the component forces parallel to rectangular axes, this extremely simple equation is equivalent to the well-known

Returning to the quaternion form, as far less complex, we see that

that is, P is the potential of a distribution of matter, magnetism, or statical electricity, of volume-density r.

T

Hence, for a non-closed path, under conservative forces

depending solely on the values of P at the extremities of the path. 479.] A vector theorem, which is of great use, and which corresponds to the Scalar theorem of § 473, may easily be obtained. Thus, with the notation already employed,

Now

and

V (V.VV.rdr)σ = −S (TV) V.dr—S (dTV) VTσo,

d (S(TV) Vo1T) = S(TV) V.odr+S(drv) Vo ̧TM. Subtracting, and omitting the term which is the same at both limits, we have SV.odr-V. (V.UvV) ods.

Extended as above to any closed curve, this takes at once the form SV.odp=ffdsV.(V.UvV)o.

Of course, in many cases of the attempted representation of a quaternion surface-integral by another taken round its bounding curve, we are met by ambiguities as in the case of the spaceintegral, § 474 but their origin, both analytically and physically, is in general obvious.

480.] If P be any scalar function of p, we have (by the process of § 477, above)

SPdr =ƒ(P。−S (TV) Po)dī

=-SS.TVP.dr.

V.VV.Tdr dr S.TV -TS.dTV,

=

But

and

These give

SPdr =

(TSTV - V.VrdTV) P。 = dsV.UvvP。,

Hence, for a closed curve of any form, we have

Pdp=ffdsV.UvvP,

from which the theorems of §§ 477, 479 may easily be deduced. 481.] Commencing afresh with the fundamental integral

put

and we have

SSSSVods = SSS.o Uvds,

σ = κβ,

SSSSB uds = ffu S. ẞUvds;
Ꮪ Ᏸ Ꮩ

from which at once Vuds = ffu Uvds,

or

SSSVrds = SUv.rds...

Putting ur for 7, and taking the scalar, we have

(1)

(2)

(3)

482.] As one example of the important results derived from these simple formulae, take the following, viz. :—

SSV.(VoUv)rds = ffo S + Uvds -ƒ ƒ Uv Sords,

where by (3) and (1) we see that the right-hand member may be written = SSS(S(TV)o+oSVT-V Sor)ds

==

- Sƒƒ V.V (Vo)τds...

(4)

This, and similar formulae, are easily applied to find the potential and vector-force due to various distributions of magnetism. To shew how this is introduced, we briefly sketch the mode of expressing the potential of a distribution.

483.] Leto be the vector expressing the direction and intensity of magnetisation, per unit of volume, at the element ds. Then if the magnet be placed in a field of magnetic force whose potential is u, we have for its potential energy

This shews at once that the magnetism may be resolved into a volume-density S(Vo), and a surface-density - So Uv. Hence, for a solenoidal distribution, S.Vo = 0.

What Thomson has called a lamellar distribution (Phil. Trans. 1852), obviously requires that

A complex lamellar distribution requires that the same expression be integrable by the aid of a factor. If this be u, we have at once

V≤ (uo) = 0,

or

S.. Vo = 0.

With these preliminaries we see at once that (4) may be written SSV.(Vo Uv)rds = − SSSV.TV Vods-ƒƒƒ V.ords +ƒƒƒ So▼.rds. Now, if

T

= √(}),

where r is the distance between any external point and the element

ds, the last term on the right is the vector-force exerted by the magnet on a unit-pole placed at the point. The second term on the right vanishes by Laplace's equation, and the first vanishes as above if the distribution of magnetism be lamellar, thus giving Thomson's result in the form of a surface integral.

484.] An application may be made of similar transformations to Ampère's Directrice de l'action électrodynamique, which, § 432 above, is the vector-integral Vpdp Tp3

where dp is an element of a closed circuit, and the integration extends round the circuit. This may be written

Of this the last term vanishes, unless the origin is in, or infinitely near to, the surface over which the double integration extends. The value of the first term is seen (by what precedes) to be the vector-force due to uniform normal magnetisation of the same surface.

whence, by differentiation, or by putting p+ a for P, and expanding in ascending powers of Ta (both of which tacitly assume that the origin is external to the space integrated through, i. e., that Tp nowhere vanishes), we have

- 2 f f f ds Up = f f V. Up V. Uv Up ds = 2

Трг

and this, again, involves

Τρ

;

Τρ

Upds.

486.] The interpretation of these, and of more complex formulae of a similar kind, leads to many curious theorems in attraction and in potentials. Thus, from (1) of § 481, we have

which gives the attraction of a mass of density t in terms of the potentials of volume distributions and surface distributions. Putting σ = it1 + j tą + k tz,

this becomes

Tp2

By putting σ = p, and taking the scalar, we recover a formula given above; and by taking the vector we have

This may be easily verified from the formula

SPdp=ffUv.VPds,

σ = tUp,

Again if, in the fundamental integral, we put

487.] As another application, let us consider briefly the Stressfunction in an elastic solid.

At any point of a strained body let à be the vector stress per unit of area perpendicular to i, μ and v the same for planes perpendicular to j and k respectively.

Then, by considering an indefinitely small tetrahedron, we have for the stress per unit of area perpendicular to a unit-vector ∞ the expression

==

λSiw+uSjw+v Skw 4w,

so that the stress across any plane is represented by a linear and vector function of the unit normal to the plane.

But if we consider the equilibrium, as regards rotation, of an infinitely small parallelepiped whose edges are parallel to i, j, k respectively, we have (supposing there are no molecular couples) V (iλ+jμ+kv) = 0,

This shews (173) that in this case is self-conjugate, or, in other words, involves not nine distinct constants but only six.

488.] Consider next the equilibrium, as regards translation, of any portion of the solid filling a simply-connected closed space. Let u be the potential of the external forces. Then the condition is obviously ££$ (Uv) ds +ƒfƒds▾u = 0,

where is the normal vector of the element of surface ds. Here