In this formula denotes the temperature of the surrounding medium, t the excess of the temperature of the heated body, p the pressure of the surrounding air; the other signs denote constants. From the formula adduced, the value of the coefficient of thermal radiation is obtained by division by the value t, assumed to be very small. We have thus

From Dulong and Petit's determination we have, in Centigrade degrees,

a=1·0077,

b=1.233,

c=0.45.

I have to thank Professor Neumann for the statement that for a blackened surface

m=3.6;

and the coefficient n, which is independent of the nature of the surface, in case p is expressed in atmospheres, is for atmospheric air

n=0.0168.

The first number is deduced from his own observation, the latter from occasional statements of Dulong and Petit. Both numbers refer to Paris lines and minutes as units.

Using these values, and taking from Stewart and Tait's ob

servation

p=0·010 atmosphere,
t=0°.45 C.,

and putting = about 20° C., we get from the above formula the value

h=0.0013,

expressed, again, in millimetres and seconds of time.

The concordance of the value deduced above from Stewart and Tait's observation with this directly found is greater than was to be expected from the multifold uncertainty of the observations. Another beautiful agreement is also met with. According to a communication of Professor Neumann, for a metallic surface

m= about 0.5,

that is, about one-seventh that of a lampblack surface. We get from this for the radiation-constant h of a metallic surface the value

h=0·00023;

and it is therefore found that the radiation-coefficient for metals is about five- or sixfold smaller than that for a blackened surface. Stewart and Tait observed that the aluminium disk with a pure metallic surface radiated about one-fourth the heat of a blackened one.

Breslau, September 11, 1868.

V. On Hansen's Theory of the Physical Constitution of the Moon. By SIMON NEWCOMB*.

THE great reputation of the author has given extensive currency to the hypothesis put forth by Professor Hansen some years since, that the centre of gravity of the moon is considerably further removed from us than the centre of figure. The consequences of this hypothesis are developed in an elaborate mathematical memoir to be found in the twenty-fourth volume of the Memoirs of the Royal Astronomical Society. But the reception of the doctrine seems to have been based rather on faith in its author than on any critical examination of its logical foundation t. Such an examination it is proposed to give it. An indispensable preliminary to this examination is a clear understanding of what the basis of the doctrine is. Let us then consider these three propositions:

(1) The moon revolves on her axis with a uniform motion equal to her mean motion around the earth.

(2) Her motion around the earth is not uniform, but she is sometimes ahead of and sometimes behind her mean place, owing both to the elliptic inequality of her motions and to perturbations.

(3) Suppose her centre of gravity to be further removed from us than her centre of figure, and so placed that, when the moon is in her mean position in her orbit, the line joining these centres passes through the centre of the earth.

Let us also conceive that these two centres are visible to an observer on the earth. Then a consideration of the geometrical arrangements of the problem will make it clear that when the moon is ahead of her mean place the observer will see the two centres separated, the one nearest him being further advanced in the orbit; while, when the moon is behind her mean place, *From Silliman's American Journal for November 1868.

† In this connexion it is curious to notice that on page 83 of his memoir Hansen appears as the first of the independent modern discoverers of Cagnoli's theorem of spherical trigonometry

cos a cos b cos C+ sin a sin b=cos A cos B cos c+ sin A sin B. This was about three years before the above formula was published as new by Mr. Cayley, and geometrically demonstrated by Professor Airy, in the Philosophical Magazine.

the nearest centre will be behind the other. This apparent oscillation of the two centres is indeed an immediate effect of the moon's libration in longitude.

Now the inequalities in the moon's motion, computed from the theory of gravitation, are those of a supposed centre of gravity. But the inequalities given by observation are those of the centre of figure. Hence, in the case supposed, the inequalities of observation will be greater than those of theory. Also their ratio will be inversely as that of the distances of the centres which they represent.

Professor Hansen, in comparing his theory with observations, found that the theoretical inequalities would agree better with observation when multiplied by the constant factor 1.0001544. Supposing that this result could be accounted for on the hypothesis of a separation of the centres of gravity and figure, he thence inferred that the hypothesis was true. But the result cannot be entirely accounted for in this way, because the largest inequality of theory (evection) has a factor (excentricity) which can only be determined from observation; and therefore even the theoretical evection is that of the centre of figure, and not of the centre of gravity. It must not be forgotten that the excentricity, which is not given by theory, is subject to be multiplied by the same factor that multiplies the other inequalities. To be more explicit,

Let e be the true excentricity of the orbit described by the moon's centre of gravity. Then the true evection in the same orbit will be

ex A,

A being a factor depending principally on the mean motions of the sun and moon. And on Hansen's hypothesis, the apparent evection, or that of the centre of figure, will be

ex A x 1·0001544.

On the same hypothesis, the excentricity derived from observation, being half the coefficient of the principal term of the equation of the centre, will be

ex 1.0001544,

and the theoretical evection computed with this excentricity will be

ex 1.0001544 × A,

which is the same with that derived from observation.

Hence

The theoretical evection will agree with that of observation, notwithstanding a separation of the centres of gravity and figure of the

moon.

Phil. Mag. S. 4. Vol. 37. No. 246. Jan. 1869.

D

That Hansen overlooked this point is to be attributed to his method of determining the lunar perturbations by numerical computation from the various elements of the moon's motion, so that the manner in which the inequality depends on the elements does not appear. It is only when we determine the perturbations in algebraic form that this dependence appears.

Passing now from the evection, the next great perturbation of the moon's motion is the variation. But the value of this perturbation has not been accurately determined from observation, because, attaining its maxima and minima in the moon's octants, it is complicated with the moon's semidiameter and parallactic inequality. Even if the semidiameter is known, the two inequalities in question cannot be determined separately with precision, because their coefficients have the same sign in that part of the moon's orbit where nearly all the meridian observations are made. From this cause Airy's value of the parallactical inequality from all the Greenwich observations from 1750 to 1830 was 3" in error. And when, in his last investigation*, Airy rejected the observations previous to 1811, owing to some uncertainty as to what semidiameter should be employed, the result was still a second too small. It is therefore interesting to find what value of the variation will result if we substitute the known value of the parallactic inequality in Airy's equations for the determination of that element. Neglecting those unknowns which have small coefficients, these equations are, from 1806 to 1851,

In these equations W x 0.73 represents the correction to the coefficient of variation, and V x 377 that to the coefficient of parallactic inequality. We now know from recent special investigations that the latter coefficient is very near 125"-50. Airy's provisional one was 122"-10, whence

The sum of the preceding equations gives

W=2.15-2·90V = −0·46.

The resulting correction to the provisional variation (2370"-3)

* Memoirs of the Royal Astronomical Society, vol. xxix.

is therefore

−0·46 × 0′′·73=0"·34,

Making the variation derived from observation.
While Hansen's theoretical value is

And Delaunay's

2369.96

2369.86

2369.74

The differences are too minute to found any theory upon.

Leaving the evection and variation, the other inequalities are so minute that their product by Hansen's coefficient is altogether insensible.

Summing up the results of our inquiry, it appears that in the case of the evection the supposed discordance between theory and observation would not follow from Hansen's hypothesis, and therefore, even if it exists, cannot be attributed to that hypothesis as a cause. In the case of the variation no such discordance has been proved. In the case of the other inequalities the discordance would be insensible.

The hypothesis is therefore devoid of logical foundation.

VI. On Extraordinary Agitations of the Sea not produced by Winds or Tides. By RICHARD EDMONDS, Esq.*

NE of those not infrequent agitations of the sea, which are

always accompanied by earthquakes or thunderstorms, or great maxima of the thermometer, or considerable minima of the barometer and sometimes by all these together-but which are never occasioned by winds or tides, was observed in Mount's Bay on the 6th of May, 1867, and another early on the following morning at Plymouth.

At Penzance Pier, on the first of these days, at 5 A.M., a tidelike "wave 4 to 5 feet high, without a moment's notice, swept into the harbour. A vessel in the act of moving from the new pier to the old was whirled round, and the pilot feared she would have become unmanageable. The large trawlers were swept against each other; and the sand at the entrance to the harbour was washed up, so as to colour the water for a considerable distance." The agitation continued nearly two hours; and a friend to whom I wrote for information replied that he was informed, by an eye-witness who had watched it for an hour after the first influx, that the duration of each efflux as well as of each influx was from three to five minutes. "The sky at the time was very overcast, and at 11 A.M. there was thunder with three or four flashes of lightning away to the S.E." The barometer at 9 A.M. was 29 in., the maximum of the thermometer 64°, which are * Communicated by the Author, having been read before the Royal Geological Society of Cornwall on the 3rd of November, 1868.