clustering power noticed by Herschel to be in progress throughout the sidereal universe. Such an investigation of the central forces governing the motions in clusters is very desirable, because it might be expected to throw light on the mode of evolution of clusters as the highest type of the perfect sidereal system. If it can be shown that a clustering power is really at work, and is of such a nature as to produce these globular masses of stars, it will be less important to consider the details of those systems which have not yet reached a state of symmetry and full maturity; for the governing principle being established for the most perfect types, it must be held to be the same in all.

II. GENERAL EXPRESSIONS FOR THE POTENTIAL OF AN ATTRACTING

MASS.

If we have a mass M' of any figure whatever, in which the law of density is o' = f(x', y', z′), where (x', y', z′) are the coördinates of the element dm' of the attracting mass, and this element attracts a unit mass whose coördinates are (x, y, z); then the element of the attracting mass is

And the expressions for the forces acting on the unit mass when resolved along the coördinate axes become

In spherical coördinates we may take the angle for the longitude, for the latitude, and r for the radius of the sphere; and then the required expressions become

The element of mass dm' defined in (1) has the equivalent form

o'dr'dy'dz'o'dr.rde-r sin edo.

The element of the potential due to this differential element is

(4)

(5)

(6)

(7)

If we make use of the equations (1), (4), (5) in equation (2) we may obtain the corresponding expressions for the forces resolved along the coördinate axes:

These expressions will hold rigorously true for any law of density whatever, so long as it is finite and continuous. In the physical universe these conditions always are fulfilled; and hence if these several integrals can be evaluated, they will give the potentials and forces exerted on a unit mass by an attracting body such as a cluster of stars, or the spherical shell surrounding the nucleus of a cluster.

But before considering the attraction of a cluster in detail, we

shall first examine the cumulative effect of central forces on the law of density. The problem is intricate and must be treated by methods of great generality, but as it will elucidate the subsequent procedure for determining the attraction of such a mass upon a neighboring point, we shall give the analysis with enough detail to establish clearly the secular effect of close appulses of individual stars upon the figure and internal arrangement of these wonderful masses of stars.

III. THE CUMULATIVE EFFECT OF THE CENTRAL FORCES UPON THE FIGURE AND COMPRESSION OF A GLOBULAR CLUSTER OF STARS.

Suppose a globular cluster of stars to be in a moderate state of compression, with density increasing towards the center. Imagine the whole of the mass at the epoch to to be divided into two parts by a spherical surface of radius r, drawn about the center of gravity of the entire system; and let the external boundary of the cluster be R, so chosen that no star, from the motions existing at the initial epoch, will cross the border r=R. The stars in the outer shell, between the surfaces r and R, with coördinates (r', y', '), will give rise to a potential U. Those of the nucleus or series of internal shells, between r=0, and r=r, with coördinates (x, y, z), will give rise to a potential V. Accordingly we have

U=

o'dx'dy' dz'

= S S S √ (x − x )2 + (j' − y)2 + (5′ — 2)2′

V=

-SSS

-

odxdydz

√(x' — x)2 + (y' − y)2 + (≈′ — ≈)2

And the forces resolved along the coördinate axes are

=2=

да

= S S S =

o'(a'-x)da'dy'dz'

V′(x' − x)2 + (y' − y)2 + (~' — ≈)2'

o' (y' — y)dx'dy' dz'

(9)

[(x' — x)2 + (y' − y)2 + (≈' — 2)2]" (10)

o' (≈' — z)dx' dy' dz'

= S S S [(x2 = x)2 + (12 = 2)2 + (~' — ∞)2]? ;

(y'

with similar expressions for

av av av

dx' dy' მი

The integration for the mutual potential energy of the stars in the outer shell relative to those in the central sphere of radius r leads to a sextuple integral

ΩΞ

(11)

And the total of the mutual attractive forces resolved along the coördinate axes are

Now it is easy to prove (cf. Thomson and Tait's "Natural Philosophy," §§ 547-548) that the sextuple integral (11) can be put into the form

= SS S ¤ Udxdyd2 = S S S o' Vdx' dy' dz'.

ΩΞ

By actual derivation of the expressions (9) we easily find that

+

+

(13)

дх дх ду ду Os dxdydz= 4′′N, (14)

dz dz

4 being introduced owing to the integration over the closed sphere surface (cf. Williamson's "Integral Calculus," edition of 1896, p. 330; Bertrand, "Calcul Integral," p. 480).

As the right members of (13) give the mutual potential energy

of the bodies of the system, it suffices for us to deal with the integral of (14). This triple integral admits of transformation by Green's theorem ("Essay on the Application of Mathematics to Electricity and Magnetism," Nottingham, 1828). If U and V be functions of x, y, z, the rectangular coördinates of a point; then provided U and V are finite and continuous for all points within a given closed surface S, it is easy to show (cf. Williamson's "Integral Calculus,” 7th edition, 1896, p. 328; Riemann, "Schwere, Electricität und Magnetismus," p. 73; Thomson and Tait's "Natural Philosophy," Part I., Vol., I., p. 167; Bertrand, "Calcul Integral," p. 480):

SSS (ex or + oy oy + ou or) dxdyds

ду ду

дп

U

მე

Ꮩ V

·SSS (+2+) dxdyd: (15)

The case in which one of the functions, U for example, becomes infinite within the surface S was also considered by Green, and is of prime importance in the present investigation of the theory of globular clusters. To simplify the treatment, suppose U to become infinite at one point P only; then infinitely near this point U may be taken as sensibly equal to 1/r, where r is the distance from P. Imagine an infinitely small sphere, of radius a, described about P as a center. Equation (15) obviously is applicable to all points exterior to this little sphere. Moreover, since

it is clear that the triple integral of the right members of (15) may be supposed to extend through the entire enclosed space S, since the part arising from the points within this little sphere is a small quantity of the same order as a2, and therefore of the second order of small quantities.

Moreover, since near P the function U is sensibly equal to 1/r,