Hence, when Q, the quantity of redundant fluid originally introduced into the sphere is given, the values of V and of the density p are likewise given. In fact, by writing in the preceding equation for

T (2), and sin (n = 27),

their values, we thence immediately deduce

The foregoing formulæ present no difficulties where n>2, but when n <2, the value of p, if extended to the surface of the sphere A, would require an infinite quantity of fluid of one name to have been originally introduced into its interior, and therefore, agreeably to a preceding observation, could not be strictly realized. In order then to determine the modification which in this case ought to be introduced, let us in the first place make n>2, and conceive an inner sphere B whose radius is a-da, in which the density of the fluid is still defined by the first of the equations (12); then, supposing afterwards the rest of the fluid in the exterior shell to be considered on A's surface, the portion so condensed, if we neglect quantities of the order da, compared with those retained, will be

and since, in the transfer of the fluid to A's surface, its particles move over spaces of the order da only, the alteration which will thence be produced in V will evidently be of the order

and consequently the value of V will become

v = 2 r (" + 1) r ( 1 = ′′ ) a 2¬~Q + k. da3 ;

'n
2

k being a quantity which remains finite when da vanishes.

In establishing the preceding results, n has been supposed greater than 2, but p the density of the fluid within B and the quantity of it condensed on A's surface being still determined by the same formulæ, the foregoing value of V ought to hold good in virtue of the generality of analysis whatever n may be, and therefore when n is a positive quantity and Sa is exceedingly small, we shall have without sensible errors

Conceiving now P' to represent the density of the fluid condensed on A's surface, 4πa P' will be the total quantity thereon contained, which being equated to the value before given, there results

Moreover as

represents the total quantity of redundant finid

in the entire sphere A, the quantity contained in B is

If now when n is supposed less than 2, we adopt an hypothesis similar to Dufay's, and conceive that the quantities of fluid of opposite denominations in the interior of A are exceedingly great when this body is in a natural state, then after having introduced the quantity Q of redundant fluid, we may always by means of the expression just given, determine the value of da, so that the whole of the fluid of contrary name to Q, may be contained in the inner sphere B, the density in every part of it being determined by the first of the equations (12). If afterwards the whole of the fluid of the same name as Qis condensed upon A's surface, the value of V in the interior of B as before determined will evidently be constant, provided we neglect indefinitely small quantities of the order Sa3. Hence all the fluid contained in B will be in equilibrium, and as the shell included between the two concentric spheres A and B is entirely void of fluid, it follows that the whole system must be in equilibrium.

From what has preceded, we see that the first of the formulæ (12) which served to give the density p within the sphere A when n is greater than 2, is still sensibly correct when n represents any positive quantity less than 2, provided we do not extend it to the immediate vicinity of A's surface. But as the foregoing solution is only approximative, and supposes the quantities of the two fluids which originally neutralized each other to be exceedingly great, we shall in the following article endeavour to exhibit a rigorous solution of the problem, in case n>2, which will be totally independent of this supposition.

8. Let us here in the first place conceive a spherical surface whose radius is a, covered with fluid of the uniform density P', and suppose it is required to determine the value of the density p in the interior of a concentric conducting sphere, the radius of which is taken for the unit of space, so that the fluid therein

contained, may be in equilibrium in virtue of the joint action of that contained in the sphere itself, and on the exterior spherical surface.

If now V represents the value of V due to the exterior surface, it is clear from what Laplace has shown (Mec. Cel. Liv. II. No. 12) that

do being an element of this surface, and g' being the distance of this element from the point p to which V' is supposed to belong.

If afterwards we conceive that the function V is due to the fluid within the sphere itself, it is easy to prove as in the last article, that in consequence of the equilibrium we must have

But and consequently Vis of the form Y, therefore by employing the method before explained (Art. 4), we get

(0)

f

(0)

13

ƒ (x', y', z') = ƒ'"'"= ƒ“+ƒ,".r"3+ƒ». r'"+&c.=B2+B ̧r'"+B2r'+&c.;

where, as in the present case, f, f, f, etc. are all constant quantities, they have for the sake of simplicity been replaced by

B1, B1, B1, &c.

Hitherto the exponent 8 has remained quite arbitrary, but

4-n.6-n... 2t-2ť'+2-n 4 .6...2t2t'+2

n-2.n-1... n + 2t - 2ť-3

4-n.6-n...2t-2t+2-nn-2.n-1...n+2t'-3

2

Giving now to t the successive values 0, 1, 2, 3, &c. and taking the sum of the functions thence resulting, there arises

V=V®=V ® + V®+V+V+ etc. = 8. V

(0) 1

(0)

where the sign S is referred to the variable t and Σ to t.

Again, by substituting for V and V' their values in the equation V'+V= const. and expanding the function V' we obtain

4-n.6-n...2t-2t'+2-nn-2.n−1...n+2t'+3 X

4.6...26-21+2

2.3.4...2t+1

which by equating separately the coefficients of the various powers of the indeterminate quantity r, and reducing, gives generally

Then by assigning to tits successive values 1, 2, 3, &c., there results for the determination of the quantities B., B1, B2, &c.,

the following system of equations,