Let us therefore suppose that the body A is a sphere, whose centre is at the origin O of the co-ordinates, the radius being 1; and p is such a function of x', y', ', that where we substitute for x', y', ' their values in polar co-ordinates

x=r' cose, y=r' sine cos', 'r' sin e' sin w',

it shall reduce itself to the form

f being the characteristic of any rational and entire function whatever: which is in fact equivalent to supposing

p = (1 − x1—y" — z')3 . ƒ (x2 + y2 + z′′).

Now, when as in the present case, p can be expanded in a series of the entire powers of the quantities x', y', z', and of the various products of these powers, the function V will always admit of a similar expansion in the entire powers and products of the quantities x, y, z, provided the point p continues within the body 4*, and as moreover V evidently depends on the distance Opr and is independent of 0 and w, the two other polar co-ordinates of p, it is easy to see that the quantity V, when we substitute for x, y, z these values

x=r cos 0, y=r sine cos, z= r sin 0 sin ☎,

will become a function of r, only containing none but the even powers of this variable.

and p= (1-').ƒ(r'3),

· [ pdv = fr" dr' dơ dw' sin€ (1 − r'1)® ƒ (r'3), g1* ;

f

* The truth of this assertion will become tolerably clear, if we recollect that V may be regarded as the sum of every element pdv of the body's mass divided by the (n − 1)th power of the distance of each element from the point p, supposing the density of the body A to be expressed by p, a continuous function of x, y, z. For then the quantity V is represented by a continuous function, so long as p remains within A; but there is in general a violation of the law of continuity whenever the point p passes from the interior to the exterior space. This truth, however, as enunciated in the text, is demonstrable, but since the present paper is a long one, I have suppressed the demonstrations to save room.

=

the integrals being taken from 0 to '2′′, from 0′=0 to e', and from r'=0 to r' = 1.

Now may be considered as composed of two parts, one V due to the sphere B whose centre is at the origin 0, and surface passes through the point p, and another V" due to the shell S exterior to B. In order to obtain the first part, we must expand the quantity g1 in an ascending series of the powers of

In this way we get

g1TM” = [ra3 — 2rr' {cos ̧0 cos ✪ + sin & sin e' cos (~' — ~)} +r3]' ̄ ̄”

r

If then we substitute this series for g1 in the value of V', and after having expanded the quantity (1), we effect the integrations relative to r', ', and ', we shall have a result of the form

V' = p1” {A+Br2 + Cr*+ &c.}

seeing that in obtaining the part of V before represented by V, the integral relative to ought to be taken from r'=0 to r'=r

only.

To obtain the value of V", we must expand the quantity

91 in an ascending series of the powers of, and we shall thus have

g1TM" = (r3 — 2rr′ [cos 0 cos & + sin @

sin ✔ cos (~—')] +=

the coefficients Qo, Q1, Q,, &c. being the same as before.

The expansion here given being substituted in V", there will arise a series of the form

V" = T2+ T2+ T2+ T2+&c.

of which the general term T, is

T‚= fd0' dw' sin 0' Q. fr'a dr' „7-x (1 − r'")3. ƒ (r'") ;

=

the integrals being taken from r'=r to r'=1, from 0′ =0 to , and from 0 to 2π. This will be evident by recollecting that the triple integral by which the value of V" is expressed, is the same as the one before given for V, except that the integration relative to r', instead of extending from r' = 1, ought only to extend from r' = r to r = 1.

=

0 to

But the general term in the function f(r") being represented by Ar*, the part of T, dependent on this term will evidently be

12148-8-n

Ar fde' da' sin e'. Q. fr'2443—8—” dr' (1 — y'2)3.................................. (2) ;

the limits of the integrals being the same as before.

...

We thus see that the value of T, and consequently of " would immediately be obtained, provided we had the value of the general integral

but since the first line of this expression is the well-known

when n=2.p=b+1 and q = 2 (B+1) we have ultimately,

By means of the result here obtained, we shall readily find the value of the expression (2), which will evidently contain one term multiplied by and an infinite number of others, in all of

which the quantity r is affected with the exponent n. But, as in the case under consideration, n may represent any number whatever, fractionary or irrational, it is clear that none of the terms last mentioned can enter into V, seeing that it ought to contain the even powers of r only, thence the terms of this kind entering into V" must necessarily be destroyed by corresponding ones in V'. By rejecting them, therefore, the formula (2) will become

But as Vought to contain the even powers of r only, those terms in which the exponent s is an odd number, will vanish of themselves after all the integrations have been effected, and consequently the only terms which can appear in V, are of the form

where, since s is an even number, we have written 28' in the place of s, and as Q is always a rational and entire function of cos e', sine cos', and sine sin a', the remaining integrations may immediately be effected.

Having thus the part of T', due to any term Ap's of the function f(r'") we have immediately the value of T'", and consequently of V", since

V" = U' + T,' + T',' + T',' + T',' + &c.;

U' representing the sum of all the terms in V" which have been rejected on account of their form, and T T T the value of T, TT,, &c. obtained by employing the truncated formula (2) in the place of the complete one (2).

But - V=V' + V" = V' + U' + T' + T',' + T',' + T' + &c. or by transposition,

and as in this equation, the function on the left side contains none but the even powers of the indeterminate quantity r, whilst that on the right does not contain any of the even powers of r, it is clear that each of its sides ought to be equated separately to zero. In this In this way the left side gives

V=T' + T,' + T,' + T'' + &c..........(5).

Hitherto the value of the exponent 8 has remained quite arbitrary, but the known properties of the function I will enable us so to determine B, that the series just given shall contain a finite number of terms only. We shall thus greatly simplify the value of V, and reduce it in fact to a rational and entire function of r3.

For this purpose, we may remark that

I (0) = ∞, I (− 1) = ∞, г (− 2) = ∞, in infinitum.

If therefore we make

n

2

+B= any whole number positive

or negative, the denominator of the function (4) will become infinite, and consequently the function itself will vanish when s' is

so great that − +B+t+3 — s' is equal to zero or any negative

2

number, and as the value of t never exceeds a certain number, seeing that f(r) is a rational and entire function, it is clear that the series (4) will terminate of itself, and V become a rational and entire function of r3.

2. The method that has been employed in the preceding article, where the function by which the density is 'expressed is of the particular form

may, by means of a very slight modification, be applied to the far more general value

p = (1 — r'2)3 ƒ' (x', y', z') − (1 − x22 — y′′ — z'o)3 ƒ (x', y', z'′),

=

where f is the characteristic of any rational and entire function whatever and the same value of which reduces V to a rational and entire function of 2 in the first case, reduces it in