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through the points P, P„ then will the value of V at any point p, on L, he expressed by

X being the distance Pp, measured along the line L, considered as increasing in the direction PP,, and Vt, the given value of Fat P. For it is very easy to see that the value of Vfurnished by this construction, satisfies the partial differential equation (a), and is its general integral; moreover the system of lines L, L', L", &c. belonging to the points P, P', P", &c. on 8, are evidently those along which the electric fluid tends to move, and will move during the following instant.

Let now V+ D V represent what V becomes at the end of the time t + dt; substituting this for V in (a) we obtain

Q_dV dDV dV dDV dV dDV

dx' dx dy' dy dz' dz ""••'"\/•

Then, if we designate by PI V, the augmentation of the potential function, arising from the change which takes place in the exterior forces during the element of time dt,

DV-0V

will be the increment of the potential function, due to the corresponding alterations Dp and Dp in the densities of the electric fluid at the surface of A and within it, which may be determined from DV—D'Vby Art 7. But, by the known theory of partial differential equations, the most general value of DV satisfying (J), will be constant along every one of the lines L, L', L", &c., and may vary arbitrarily in passing from one of them to another: as it is also along these lines the electric fluid moves during the instant dt, it is clear the total quantity of fluid in any infinitely thin needle, formed by them, and terminating in the opposite surfaces of A, will undergo no alteration during this instant. Hence therefore

O^jDp'dv+Dpde + Dpfo,* (c);

dv being an element of the volume of the needle, and da, da,, the two elements of A'a surface by which it is terminated. This condition, combined with the equation (b), will completely determine the value of D V, and we shall thus have the value of the potential function V+ D V, at the instant of time t + dt, when its value V, at the time t, is known.

As an application of this general solution; suppose the body A is a solid of revolution, whose axis is that of the co-ordinate *, and let the two other axes Xy Y, situate in its equator, be fixed in space. If now the exterior electric forces are such that they may be reduced to two, one equal to c, acting parallel to z, the other equal to b, directed parallel to a line in the plane (xy), making the variable angle $ with X; the value of the potential function arising from the exterior forces, will be

zc xb cos <Byb sin <f>;

where b and c are constant quantities, and § varies with the time so as to be constantly increasing. When the time is equal to ty suppose the value of Tto be

V= fi (x cos w + y sin v):

then the system of lines L, L', L" will make the angle -a with

the plane (xz), and be perpendicular to another plane whose

equation is

0 = ajcosw+ysinsr.

If during the instant of time dt, <f> becomes <j> + D<f>, the augmentation of the potential function due to the elementary change in the exterior forces, will be

D' V= (x sin <f> y cos <j>) bD<f>;

moreover the equation (b) becomes

dDVi . dDV ,,„

0 = cosw. —?—hsinw . —=— ...... (o)

and therefore the general value of D V is

DV = DF(ycosvr xsinv; z);

DF being the characteristic of an infinitely small arbitrary function. But, it has been before remarked that the value of D V will be completely determined, by satisfying the equation (5) and the condition (c). Let us then assume

DF{y cos v x sin -a; z) = hD<ff > (y cos or — * sin sr);

A being a quantity independent of z, y, «, and see if it be possible to determine A so a3 to satisfy the condition (c). Now on this supposition

.D V—17.V— KD<f> (y cos «r — xsin tr)~(a! sin £ — y cos £) fefy = 2ty {y (A cos w + i cos <ff>) — a; (A sin a + b cos <f>)J.

The value of Dp' corresponding to this potential function is (Art. 7)

2y=o,

and on account of the parallelism of the lines L, L, &c. to each other, and to A'a equator da = dcr». The condition (c) thus becomes

Q = Dp + DPl («'):

Dp and Dp, being the elementary densities on .4's surface at opposite ends of any of the lines L, L, &c. corresponding to the potential function DV—HV. But it is easy to see from the form of this function, that these elementary densities at opposite ends of any line perpendicular to a plane whose equation is

0 = y (A cos sr + b cos <p) x (A sin «r +. b sin <p),

are equal and of contrary signs, and therefore the condition (c) will be satisfied by making this plane coincide with that perpendicular to L, 11, &c., whose equation, as before remarked, is

O=sajcos« r + ysin«r;

that is the condition (c) will be satisfied, if A be determined by the equation

A cos «r + b cos § A sin -a + b sin <f>
siner costfr'

which by reduction becomes

0 = A + b cos {<f> - w),

and consequently

V+DV**f$ (xcosv + ysinw) +hD<j> (y cos w - x s'm «r) = fix jcos «r + -* sin w cos (^ — w) Ztyj

+ £y | sin o — g cos wcos (<j> - «r) Ztyl-,

= fix cos •! «• — -?. cos (£ — «r) ityy

+ /9y sin \tr — 3 cos (£ — w) Ztyj'.

When therefore <£ is augmented by the infinitely small angle

D$, vt receives the corresponding increment — -5 Cob (<f> — «-) D$,

and the form of V remains unaltered; the preceding reasoning is consequently applicable to every instant, and the general relation between <f> and -a expressed by

0 = Dv + -5 cos (4> - v) D<f>: a common differential equation, which by integration gives

II being an arbitrary constant, and 7, as in the former part of this article, the smallest root of

0 = Jsin7—ft.

Let «t, and <£„ be the initial values of w and $; then the total potential function at the next instant, if the electric fluid remained fixed, would be

Vt = fi (x cos «r0 + y sin «x0) 4- (x sin £, - 7 cos <f>t) bd<f>,

and the whole force to move a particle p, whose co-ordinates are x, y, z,

which, in order that oar solution may be applicable, roust not be less than /3, and consequently the angle ^>, — «70 most be between 0 and ir: when this is the case, w is immediately determined from <f> by what has preceded. In fact, by finding the value of H from the initial values «j, and <£,, and making t« J»+i7+Jw—\$, we obtain

. „_ tang,

"" 5 ~&=*«*y + tan y tan £ {e»-*s «*T _ l j '•

£ being the initial value of f.

We have, in the latter part of this article, considered the body A at rest, and the line X', parallel to the direction of b, as revolving round it: but if, as in the former, we now suppose this line immovable and the body to turn the contrary way, so that the relative motion of X' to X may remain unaltered, the electric state of the body referred to the axes X, Y, Z, evidently depending on this relative motion only, will consequently remain the same as before. In order to determine it on the supposition just made, let X be the axis of x\ one of the co-ordinates of p, referred to the rectangular axes X', V, Z, also y, a, the other two; the direction X' Y, being that in which A revolves. Then, if w' be the angle the system of lines L, L\ &c. forms with the plane (x, z), we shall have

<p, as before stated, being the angle included by the axes X, X'. Moreover the general values of V and £ will be

P-=/9(x'cos«r' + y,sin«0 and t=lir + \i-±v',

and the initial condition, in order that our solution may be applicable, will evidently become <f>0 vjt = «r'0 = a quantity betwixt 0 and It.

As an example, let tan y = , since we know by experiment

that 7 is generally very small; then taking the most unfavourable case, viz. where v\ = 0, and supposing the body to make one revolution only, the value of J, determined from its initial one, £ ■ far + J7 — Jw'0> will be found extremely small and only

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