21. The laws of central forces require that provision should be made for radial oscillations, tending towards the time-limit of iso-radial circular oscillations (32 and 8); for tangential velocities, varying inversely as the times and directly as the fourth root of central iso-radial tendencies; for centres of oscillation in lines of force, and for oscillations between systematic and locally dominant centres. We have already seen (15, 17, 18,) how closely the relative masses of Sun and Jupiter provide for the last requirement; if they provide also for the others, the centre of oscillation for Sun's possible atmosphere should be at 1047.875-(√32)1 - 23.875 solar radii. The corresponding height of possible atmosphere, or the height of equality between the velocities of rotation and revolution, is of 23.875

=

= 35.813 r. This would give, for the time of solar rotation, 365.256 ÷ (214.86 ÷ 35.813) — 24.856, differing less than of one per cent. from the vis viva estimate (20).

22. The ratio 1 : 322 is also simply connected with the mass and distance of Jupiter's companion planet, Saturn, and so with the centre of planetary inertia. For 1024 + 1025 + .536 = 2049.54 (3,17), Saturn's mean radius vector being 2049.51 solar radii; 1024 + (35.813 ÷ √.4) = 1080.625 is the limit, of which Jupiter represents a centre of explosive oscillation, and the inertial moment at the limit gives the mass of Sun Saturn; for 1080.625 X (})2 = 3501.2, Bessel's value being 3501.6.

23. Among the many harmonies of planetary mass which manifest a dependence on nebular influences, the following are, perhaps, indicative of some of the earliest forms of activity.

a. The masses of Jupiter and Earth are nearly proportionate to the squares of their periodic times the velocities due to internucleal vis viva; 5.20283

5.2028

321.2; 321.2 X 1047.875 336201,

Neptune (8: π): π2 × 336201 ÷ 8

=

b. The influence of spherical gyration on Venus (11), seems to be further shown by its ratio to Earth, which is the square of the ratio of Uranus to 415289. Hill's estimate of the mass of Sun ÷ Venus is 408134. If the internucleal vis viva of Jupiter were taken at secular perihelion, the resulting theoretical mass-denominators would be Earth, 326222; Venus 402460.

24. The masses of the principal planets, therefore, seem to have been primitively determined by the following influences:-Neptune, by the proportion between the time of direct fall to the centre of planetary inertia, and the time of circular revolution; Uranus by the time of describing the same proportional part of a circle, in the circular orbit; Saturn, by equality of nebular vis viva with Jupiter, when the two centres of condensation were in opposite parts of the nebular belt and on opposite sides of the sun; Jupiter, by the ratio of variability, between incipient fall to a centre of linear atmospheric oscillation and circular revolution; Earth, by the combined action of vis viva and time of revolution; Venus, by the action,

Chase.]

[April 21,

in a spherical mass, of the same force as determined Uranus in a circular disc. The mass-denominators are,

25. These masses, with Hansen and Olufsen's mass for Mars, and Encke's for Mercury, give the following ratios for the extra and intra asteroidal

Here is a further approximation, in the inner system, to the square of the outer ratio, accompanied by suggestive indications of the influence of the ratio between aggregating and dissociating velocities (1 : √2), and of the ratio between the oscillatory and kinetic radii (1.4232 : 2 :: 1; 1.405). The outer is about 220 × the inner, or nearly as Earth's radius-vector is to Sun's radius.

26. To the primitive influences others were subsequently added, depending upon mutual actions and reactions, some of which have already been pointed out, and others are obscurely intimated by harmonies which can hardly be regarded as accidental. To this latter class the following may be added:

a. Jupiter's mass is to Neptune's mass, as TX Neptune's radius vector is to Jupiter's radius vector; Neptune's mass is to Earth's mass, as X Jupiter's radius vector is to Earth's radius vector.

b. The mass of the intra-asteroidal planets: Sun's mass square of Jupiter's secular aphelion square of light-modulus.

c. The limit of possible solar atmosphere: 62 × Sun's radius : : Earth's polar: Earth's equatorial diameter. (62 × 3 = product of number of gravity nodal divisions by number of oscillatory nodal divisions in a linear pendulum; Cfr. Jupiter's mass= = 6 x Neptune's mass).

27. Struve estimated Sun's proper motion at 1.623 × Earth's radius vector per annum, which is .258 times Earth's orbital motion, the motions being, therefore, in the ratio of their densities.

28. If we add the orbital motions of Earth and Jupiter, to Sun's proper

1876.]

[Chase.

motion, we obtain, in units of Earth's mean radius vector, Earth, 1.258; Jupiter, .6964. Jupiter's proper motion is, therefore, that which is due the "centre of explosive oscillation,” (§), of which Earth is a limit.

These accordances furnish new data for approximating to the primitive parabolic abscissas of interstellar action. For if we take the kinetic radius

(2 πε τ π2 +4

: 1.4232 r = radius of mean velocity of radial, synchronous with

with circular, oscillation and the solar modulus limit of possible atmosphere, (35.813 × radius) as determinants, the three known abscissas (; +7+3;5 +27 7+729) give the following logarithms for planetary

log. + 10 + 100 = 1.678467 1.680602 1.795038 1.885522

66

+11 7+1211.910966 1.858650 1.890463 1.920003

+12 7+144

5 + 13 7 + 169

+

147196

§ + 157 +225

+167 +256

+17 7 + 289

= 2.958507 2.941956 2.969211 2.994855

3.249779 3.206444 3.244704 3.279865 = 3.552807 3.521682 3.557093 3.589694 +18 7+3243.867589 3.861415 3.867758 3.873951 +27 7+729

7.229595

7.229595

7.686009

35.813 M+ a Centaurit +28 7+7845=7.661925 7.657096 We have seen (5) that the limits of the planetary system (Mercury and Neptune), were fixed by the radius of internal rupture and the nucleal radius which correspond to the modulus of light. The values,, 4, &c., represent the points at which particles falling from 1r, 2r, 3r, &c., would acquire the velocity of central disintegration (/2gr). The like co-efficients of Mercury and Mars (3) seem to indicate satellite relations to Earth and Venus; all the abscissas between Mercury and Neptune, are within the range of orbital eccentricities, the greatest deviations of the theoretical from the mean values being those of Venus and Mars, which are less than five * The nucleal radius varying as the 34 power of the atmospheric radius. Compare Alexander's ratio for exterior half-planets (Smithsonion Contributions, 280, p. 5.) + Modulus, or the double height of virtual fall which would give the velocity of light, bearing the same ratio to the modular atmosphere, as solar radius bears to solar atmosphere.

The least value is given by Norton; the greatest by Denison. There is no reason for doubting that the true value is between the two.

Britton.]

[Dec. 3,

The co-efficient-products

×}=}=nodal division

2 rkinetic radius;

percent. Jupiter's deviation is about 2.6 per cent.; Saturn's, 1.2 per cent.; Uranus's, 1 per cent.; Earth's, of 1 per cent. in the three principal planetary pairs are,, of linear pendulum; 2 h, X =, nearly 6 4, 7 × 4 = 1. It seems probable that the actual determining orbit was an ellipse, the two foci being within the masses of Sun and a Centauri, but the difference between parabolic and elliptical co-ordinates in our planetary system, is not recognizable so near the focus.

Improved Mounted Burettes for Volumetric Analysis.

BY J. BLODGET BRITTON.

(Abstract of remarks made before the American Philosophical Society, December 3d, 1875.)

More than a year ago (meeting of November 20th, 1874, Vol. XIV, No. 93, p. 189, of the published proceedings) I exhibited several burettes mounted on stands, and delicately graduated for close, volumetric analysis, differing from each other, but each having the screw and spring valve adjustment, and also one of the rod-stoppered kind, having the screw and top spring lever adjustment.

I now exhibit and shall explain three others also mounted, and delicately graduated, yet still more improved.

The shortest one (Fig. 1 in the annexed plate), has a capacity of 100 cc, with its scale divided into tenths, or one thousand degrees. The glass rod inside of the tube is operated by means of a thumb-screw (A) which works against a horizontal lever (B) under the stand. The lower end of the light vertical iron rod (C) inside of the wooden upright of the stand, rests upon the front of the lever, and its upper end encircles the glass rod directly under its knob. By turning the thumb-screw, the iron rod, and with it the glass one, may be raised or lowered, and the vent of the burette opened or closed at pleasure. The flow may thus be completely controlled or instantly stopped.

The second instrument (Fig. 2 in the plate) is between four and five feet long, and has a capacity of 240 cc, with its scale divided into tenths or two thousand four hundred degrees. It is operated with the thumb-screw (A) like the shorter instrument from which it differs in construction only in the position of the screw and lever and shape of the latter. The screw is placed