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 2s' in the place of s, and as Q is always a rational and entire function of cos e', sin cos a', and sine sin a', the remaining integrations may immediately be effected.

Having thus the part of T', due to any term Ar' 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 TTT, &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,

V-T-T-TT- &e. = V' + U';

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 In this way the left side gives

zero.

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

Hitherto the value of the exponent B 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 r2.

For this purpose, we may remark that

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

n

If therefore we make += any whole number positive

2

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

n

2

+B+t+3-s' is equal to zero or any negative 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 r2.

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

p = (1 − r22)3. ƒ {(r'2),

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

p = (1 − r'3)3 ƒ (x', y', z') = (1 − x" — y' — z"2)® ƒ (x', y', z'),

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

s

the second to a similar function of x, y, z and the rectangular co-ordinates of p.

To prove this, we may remark that the corresponding value V will become

V = fradr'd€ d=' sin 8′ (1 − r'")3ƒ (x', y', x') 9TM;

the integral being conceived to comprehend the whole volume of the sphere.

Let now the function ƒ be divided into two parts, so that

ƒ (x', y', z') = ƒ, (x', y', x') +ƒ, (x', y', z);

f containing all the terms of the function f, in which the sum of the exponents of x, y, z' is an odd number; and f, the remaining terms, or those where the same sum is an even number. In this way we get

V = V1 + V1;

the functions V, and V, corresponding to ƒ, and ƒ,, being

1

V1=fr"dr'de 'sin @′ (1 — r'3)aƒ, (x', y', x') 9TM*,

V1=fr"dr'de d☛' sin ′ (1 — r')3 ƒ, (x', y', x') g'".

We will in the first place endeavour to determine the value V1; and for this purpose, by writing for é, y', ' their values before given in r', 0, ', we g..t

ƒ1 ( x', y', x') = ?' yr (2');

the coefficients of the various povers of r" in ↓ (r) being evidently rational and entire functions of cos e, sin cos ☎', and sin sin. Thus

V1 = fr''dr'de' d' sin ′ (1-2) ′ (2^^) g1";

this integral, like the foregoing, comprehending the whole volume of the sphere.

Now as the density corresponding to the function V, is

P1 = (1-x" — y'2 — 2'*)°ƒ, (x', y', '),

it is clear that it may be expanded in an ascending series of the entire powers of x, y, z, and the various products of these powers consequently, as was before remarked (Art. 1), V, ad

mits of an analogous expansion in entire powers and products of x, y, z. Moreover, as the density p, retains the same numerical value, and merely changes its sign when we pass from the element do to a point diametrically opposite, where the co-ordinates x, y, z' are replaced by x', - y', -z': it is easy to see.. that the function V, depending upon p1, possesses a similar property, and merely changes its sign when x, y, z, the co-ordinates of p, are changed into - x, y, z. Hence the nature of the function V, is such that it can contain none but the odd powers of r, when we substitute for the rectangular coordinates x, y, z, their values in the polar co-ordinates r, 0, w.

Having premised these remarks, let us now suppose V, is divided into two parts, one V due to the sphere B which passes through the particle p, and the other V" due to the exterior shell S. Then it is evident by proceeding, as in the case where p = (1 − r13)3 ƒ (r3), that V will be of the form

V1 = p3TM* {A+Br2 + Cr* + &c.};

the coefficients A, B, C, &c. being quantities independent of the variable r.

In like manner we have also

V"fr"dr'do'da' sin €′ (1 — r'")". r'f (r'") g1" ;

=

the integrals being taken from rr to r1, from 0 to π, and from 0 to '=2#.

By substituting now the second expansion of g1 before used (Art. 1), the last expression will become

V1" = T+T,+T,+T,+ &c.

of which series the general term is

T‚= fd0'du̸′ sin 0′ Q. fr'"dr' (1 — '4'2) ® 1717. y^ (r'2).

Moreover, the general term of the function (r) being represented by A, the portion of T, due to this term will be

r2 [de da' sin O'Q‚A' fr'1¬n+2 ̄* dr′ (1 − p′2) ♬ .

the limits of the integrals being the same as before.

(a);

If now we effect the integrations relative to r' by means of the formula (3), Art. 1, and reject as before those powers of the variable r, in which it is affected, with the exponent n, since these ought not to enter into the function V, the last formula will become

1

--

2

and as V, ought to contain none but the odd powers of r, we may make s=28' + 1, and disregard all those terms in which a is an even number, since they will necessarily vanish after all the operations have been effected. Thus the only remaining terms will be of the form

where, as A, and Q+ are both rational and entire functions of cos, sin & cos', sin e' sin a', the remaining integrations from Ø = 0 to π, and '= 0 to 2π, may easily be effected in the ordinary way.

If now we follow the process employed in the preceding article, and suppose T, T, T, &c. are what T., T1, T,, &c. become when we use the truncated formula (a') instead of the complete one (a), we shall readily get

In like manner, from the value of V, before given, we get

V1" = fr2dr'de do sin 0′ (1 − r22)ß $ (r'2) g1";

the integrals being taken from rr to r=1, from 0 to

=