Comparing Le Gentil's computed results, the method of obtaining which he does not explain very clearly, we get

True Zenith Distance of the Sun's upper limb when first visible.

The discrepancy between the two columns, in the case of refraction, is I own very serious; but I cannot change my own. figures, or see exactly how Le Gentil got his own. The normal refraction according to Bessel for Z. D. 89° 30' is about 29′ 3′′, with a probable error, from experiment, of ± 20": Ivory making it 28′ 31′′. The former gives nothing below this, but at 90° Z. D. according to Ivory, with the Bar. at 30°, as I have assumed throughout my calculations, and the Therm. at 60 F. in winter and 70 F. in summer, which seems reasonable, we get 33′ 50′′ and 33′ 10′′ respectively, Bessel's system probably giving in each case about 40′′ more. My own mean results being 33′ 19′′ and 32′ 7′′: it would seem that the allowance for temperature made by Ivory is hardly sufficient, as Le Gentil says (p. 342) that the difference between the summer and winter temperature at Pondicherry was not more than 5° to 6° (R.) in winter in the morning and 13° to 14° in the afternoon. Any how I do not think the refraction can have been as small as Le Gentil makes it in his summary (p. 248): it is to be regretted that he kept so much in his mind the phenomenon said to have been seen by the Dutch in Nova Zembla: which cannot in strictness be thought more than interesting.

I shall conclude this paper with a brief résumé of the observations on horizontal refraction made by M. Bouguer in Peru; which are given and discussed by him in two articles in the Mémoires of 1739 and 1749. It should be noticed that he only records his actual results, and not his complete observations, or his methods of procedure.

Experiments in Peru.

M. Bouguer (9-23 April, 1736) found, as he asserts, the refraction for the setting Sun, on the sea-shore, in 1° 1' S. Lat. to be 25′ to 29'... at 1° alt. 201... The modern tables giving, as the normal refraction, 33′ and 24' about, in these cases, respectively. He thus agrees more or less nearly with Le Gentil in his estimate of refraction generally.

On the sea-coast, on another occasion, at an alt. of 1o, but at an elevation of 40 toises (= 256 ft.), he found it 22′ 15′′. At Quito, at an altitude of 1400 toises (=8960 Eng. ft.), he says it was for 2o 20′′ alt. 12′ 1′′ (the normal being 17'), at 3° alt. 9′ 33′′ (norm. 14' 36"), at 4° alt. 8' (norm. 11′ 51′′).

On the height of Pichincha, near Quito (elevation 527 toises, = 3373 Eng. ft.), he was unsuccessful in obtaining any observations of value, for various reasons: but on Chimborazo, at an elevation of 2388 toises (15,283 Eng. ft.) he obtained the following results from the Sun:

It is to be regretted that the observations on stars rising or setting near the prime vertical, made by Argelander at Konigsberg in 1820, 1; and referred to in the conclusion of the preface to the Tabula Regiomontana are not in print: though there is no reason to think that they would furnish anything exceptional. I know of no others that would be of value in determining whether under any circumstances horizontal refraction usually sinks to so low an amount as it seems to have done in the opinion of the French observers; or as there seems reason to think, from my own observations of last year, it will sometimes do under very peculiar circumstances. On that occasion it must be remembered that the rays of the Sun were passing through strata of atmosphere lying over a probably much colder portion of the sea than that near my own position. A horizontal line drawn at any point on the earth's surface will be at an elevation of 1000 yards at a distance of about 70 miles, and for 2000 yards at about 100 miles distance: but by the laws of refraction the track of any ray at either of these distances must have been much lower than this: and though I feel myself unable to suggest what its elevation at various distances would have been, I am sure that the distance would have been so great for no very high elevation as to bring it sensibly under the direct influence of the Polar Ice. And although the natural effect of cold is to increase refraction, the circumstances may have been sufficiently abnormal to produce the effects observed.

With these remarks, I feel justified in leaving the question with such of my readers as may feel interested in it.

(2) On the problem of two pulsating spheres in a fluid (Part II.). By W. M. HICKS, M.A., Fellow of St John's College.

In a paper read before the Society in the Michaelmas term of last year, I shewed how the force between two pulsating spheres might be determined, by means of mass-images, and I calculated

* Proceedings, Vol. 1. pp. 276-285.

the term depending on the time variation of the potential. By an oversight I treated the term depending on the square of the velocity as if it had no effect on the resultant force. This is only true if powers of the inverse distance higher than the second be neglected (as was the case in the numerical examples). It is of course not true in general, except in the case of the resultantforce on the system regarded as rigidly connected. In the present communication I propose to shew how the other terms may be found.

7. Using the same notation as in the former paper, the force on B from A is

To find the general term in the series for X directly by means of the infinite series for V would be extremely laborious, but this may be avoided by means of the following proposition.

Let be the velocity-potential of any motion symmetrical about a diameter of a sphere in the fluid, and such that the normal motion at the surface of the sphere is constant over the sphere, and equal to u. Then the force on the sphere is parallel to the diameter and is equal to

provided the velocity, along the surface, is always finite.

For consider the term "b" ["V sin cos 000. At every point

=

of the surface of the sphere Vu2 + 1/b2 (dp/00). Since u is constant the part of the force depending on it is zero, and the integral becomes

I:

Now

and

дф

др

+ cot 0

r is put = b, and therefore

,

дф

= n;

18

1="" {(2bu + b2 3 – 24) sin 20 + 2 cos10 30} de — — [p′].

дф

до

[2 cose. +36 de = ['sin 20 d0+ [ø*]=Q + [ø'],

whence

8. We must now shew how P, Q may be determined when all the images are known. Still considering the general problem of symmetrical motion, it is known that can be expanded in a series of Zonal Harmonics

The values of A, B, being supposed known we have at once P=2.3.C1 = {C,.