where n is the coefficient of rigidity,

incompressibility,

V is the potential of attraction,

pw?

2

[x2(b2+c2)+y2(a2+c2) +z2 (a2+b2) — 2xy ab— 2xz ac−2yz bc],

2 is the known Laplacian symbol.

V may now be divided in two parts, the first corresponding to the potential of the unstrained body, the other containing all the terms which arise in consequence of the deformation : so we may write

V=V1+V2.

On the other hand, with the help of the formula

Now if the Z axis of coordinates coincides with the polar axis of the earth corresponding to the unstrained state, the potential

V1+pw2(x2+y2)

exerts no deformation, so that in equations (I.) V reduces to V2 and becomes

[x2a2+ y2f2+z2 (c2-1)+2xy ab+2xz ac+2yz be].

We see that is a solid harmonic of the second degree.
But the disturbing potential is

V2+.

To find it, we proceed after the method of Prof. Darwin*, and obtain readily

W2=V2+$=19n+2gpR $.

"On Bodily Tides," Phil. Trans. for 1879, p. 9. The calculus is "literatim" the same.

Here we have made the supposition that the body is incompressible. [R means the mean radius of the Earth.]

We must now calculate the products and the moments of inertia about old axes after deformation.

As this calculation implies the knowledge of the displacements, we shall take the expressions of the displacements given by Thomson and Tait * for the case of a sphere

Before calculating the products of inertia we remark that the deformation is evidently symmetrical with respect to the plane passing through the axis of z and the axis of rotation. Hence, taking this plane for the plane XZ †, we obtain first, if D, E, F are the products of inertia,

with the help of the formulas II., III., and IV. is very easy, and gives

P

MR2. u2,

19n+2gpR

where M is the mass of the earth,

u the equatorial velocity (u=wR).

We shall also need the difference C-A. This difference, being very little changed by the deformation, may be directly calculated from the known values

*Treat. on Nat. Phil. arts. 837, 838.

In this manner we introduce coordinates moving respectively to the body, but evidently it does not change anything in the results.

Now, by a known theorem, if

D=F=0,

the moments of inertia about the new principal axes are the roots of the equation

(H−A) x2+(H−B) y2+(H−C) ≈2 + 2 Exz=0, (V.) where A, B, C are the given moments about the old axes. In the present case we may put

A=B.

Now, if the angle between the axis of the new greatest moment of inertia and the axis of ≈ is 0, then by the transformation

we may easily find the angle from the formula V. obtain

We

But by definition the angle (POP') is equal to of the angle POR, and

c=cos (POR)

a=sin (POR)

As the angles POR and POP' are very small, their cosines are nearly equal to unity, and their sines are nearly equal to the arcs; but the arc POR is equal to of the arc POP' [the arc ]. Hence, neglecting small quantities of second order, we obtain from the formula VI.

But Prof. Newcomb thinks that one fourth of the angle POP' may be attributed to the influence of the Ocean. Further, as the product of inertia E standing in the numerator of the right-hand member of VI. depends principally on the deformation of superficial layers, and for that reason the mean density in E must be smaller than the mean density of the Earth-we must multiply the right-hand member of VII. by a factor smaller than unity. We take, with Prof. Newcomb, the mean effective density to be 0.6 of that of steel, i. e. 468, and multiply the right-hand member of VII. It is to be remarked that by the meaning of

with

4.68

5.5

mean densities p contained in VII., they are taken inferior to the mean density of the Earth. formula VII. changes to

all to be Now the

p=4.68 [Prof. Newcomb's effective mean density]. With these data,

n=1615 x 109.

This coefficient of rigidity is nearly twice as great as the coefficient of rigidity of steel after Everett [819 × 109]. By certain combinations of the chosen effective mean densities in the numerator and denominator of VII., also putting again instead of in the left-hand member, we may render the coefficient of rigidity smaller. But it remains much greater than the coefficient for steel.

The proof of the rigidity of the Earth from the tidal phenomena was subject to certain doubts *, but now it is strongly supported by the test of the phenomenon of Variation of Latitudes.

M. Gylden† has presented some objections to the views of Lord Kelvin and Prof. Newcomb. Without discussing his paper, we remark only that we can interpret his analysis as corresponding to the case of an absolutely rigid earth with certain fluid or generally mobile parts. Of course he has found that these mobile parts must be greater than the Oceans. We have taken the Earth as incompressible. It is known that this assumption has a very little influence on the final results. In a quite similar case, that of the tidal problem, Mr. Love ‡ has obtained nearly the same results on the hypothesis of compressibility [m=2n] as on the hypothesis of perfect incompressibility [m=].

*See Prof. Darwin's paper, Proc. Roy. Soc. London, Nov. 1886. + Compt. rendus, vol. cxvi. (1893), pp. 476–479.

Transactions Cambr. Phil. Soc. vol. xv. pp. 107-118.

XXIV. On the Electrification of Air. By Lord KELVIN, P.R.S., and MAGNUS MACLEAN, M.A., F.R.S.E.*

§ 1. HAT air can be electrified either positively or negaTHAT tively is obvious from the fact that an isolated spherule of pure water, electrified either positively or negatively, can be wholly evaporated in air†. Thirty-four years ago it was pointed out by one of us as probable that, in ordinary natural atmospheric conditions, the air for some considerable height above the earth's surface is electrified §

* Communicated by the Authors; having been read before the Royal Society, May 31, 1894.

This demonstrates an affirmative answer to the question, Can a molecule of a gas be charged with electricity? (J. J. Thomson, Recent Researches in Electricity and Magnetism,' § 36, p. 53), and shows that the experiments referred to as pointing to the opposite conclusion are to be explained otherwise.

Since this was written we find, in the 'Electrical Review' of May 18, on page 571, in a lecture by Elihu Thomson, the following:-"It is known that as we leave the surface of the earth and rise in the air there is an increase of positive potential with respect to the ground...... It is not clearly proven that a pure gas, rarefied or not, can receive and convey a charge. If we imagine a charged drop of water suspended in air and evaporating, it follows that, unless the charge be carried off in the vapour, the potential of the drop would rise steadily as its surface diminished, and would become infinite as the drop disappeared, unless the charge were dissipated before the complete drying up of the drop by dispersion of the drop itself, or conveyance of electricity by its vapour. The charge would certainly require to pass somewhere, and might leave the air and vapour charged."

It is quite clear that "must" ought to be substituted for "might" in this last line. Thus the vagueness and doubts expressed in the first part of the quoted statement are annulled by the last three sentences of it.

‡ “Even in fair weather the intensity of the electric force in the air near the earth's surface is perpetually fluctuating. The speaker had often observed it, especially during calms or very light breezes from the east, varying from 40 Daniell's elements per foot to three or four times that amount during a few minutes, and returning again as rapidly to the lower amount. More frequently he had observed variations from about 30 to about 40, and back again, recurring in uncertain periods of perhaps about two minutes. These gradual variations cannot but be produced by electrified masses of air or cloud, floating by the locality of observation."Lord Kelvin's 'Electrostatics and Magnetism,' art. xvi. § 282.

§ "The out-of-doors air potential, as tested by a portable electrometer in an open place, or even by a water-dropping nozzle outside, two or three feet from the walls of the lecture-room, was generally on these occasions positive, and the earth's surface itself therefore, of course, negative-the common fair-weather condition-which I am forced to conclude is due to a paramount influence of positive electricity in higher regions of the air, notwithstanding the negative electricity of the air in the lower Phil. Mag. S. 5. Vol. 38. No. 231. Aug. 1894 Q