near Reading as the only places where shocks were felt in this island; but there is abundant traditional evidence of shocks having been felt in Cornwall; whilst in the Scilly Isles" several persons ran out of their houses for fear they would fall upon them "*.

On the other hand, if the centre of an earthquake be inland and close beneath the surface, the shock would not travel vertically or upwards at all, but horizontally, as the shock in this country of the 6th of October, 1863, appears to have done; for when at Pontypool, in the midst of Welsh mines, soon after its occurrence, I was informed by a gentleman who had ascertained the fact from the miners themselves, that whilst the shock was felt at the mines above ground, it was not felt by any under ground. Nor does it appear to have been felt under ground in any of the mines of Cornwallt.

The two preceding paragraphs may help to account for a striking fact mentioned by Mr. Mallet, viz. that the extraordinary disturbances of the sea now under consideration have "6 never been observed to take place in any earthquake whose centre of impulse was inland, however violent". This fact (which is in perfect agreement with my hypothesis) may perhaps be owing to the centre of impulse, when inland, being generally close under the surface, so that the shocks must travel horizontally, or nearly so; and it is not to horizontal, but to vertical shocks that I have always ascribed these disturbances of the sea.

Another fact not less remarkable is, that, while earthquakes in general take place equally in all states of the atmosphere, those which are known only by the extraordinary agitations of the sea or lakes which they produce occur almost exclusively during storms, or at or near considerable minima of the barometer. Is it because these submarine shocks, as already stated, are almost always vertical, or nearly so, while those on dry land are generally horizontal? In vertical shocks there may be electrical discharges between the earth and the atmosphere which might occasion the attendant minima of the barometer, whereas in horizontal shocks the discharges (if any) may be only between differently charged portions of the earth without much affecting the atmosphere. The first earthquake felt by Humboldt in Cumana was during a severe thunder-storm. "At the moment of the strongest elec trical explosion were two considerable shocks of an earthquake;" but the barometer, which had been previously falling, continued to fall for five hours afterwards, when a third and last shock

* Troutbeck's 'Scilly,' 1794, p. 40.

† Journal of the Royal Institution of Cornwall, No. 1. p. 61.
British Association Reports, vol. xx. p. 46.

reason to conclude that on these occasions, as well as on the day of the great earthquake of 1755, every disturbance of the sea, wherever it happened, was confined to a few furlongs from that part of the bed of the sea over or near which the agitated waters had previously rested; so that all such disturbances in 1755, although observed the same day on most of the European coasts, were perfectly independent of one another, and proceeded not from any distant disturbance in mid-ocean, but from local submarine shocks beneath or close to the disturbed waters. In Falmouth harbour and Plymouth and Mountsbay, for instance, where such extraordinary currents have so often simultaneously occurred, no kind of disturbance has been observed in the offing*. Indeed I have ascertained by personal inquiries from eye-witnesses and other sources, whilst residing in Penzance, that such agitations at the piers of St. Michael's Mount, Penzance, and Newlyn (the first being two miles east, and the last one mile west of Penzance pier) have always been independent of each other, and none of them ever extended beyond a few furlongs from the pier where it was observed.

Assuming that the extraordinary disturbances of the sea in Plymouth and Mountsbay on the days of the two great earthquakes of 1755 and 1761 were occasioned by submarine shocks, and that those in the same places in 1811, 1843, 1847, 1859, and 1862 were in all respects like them (neither of which facts will be questioned), the only reasonable conclusion is, that all these like effects must have resulted from like causes-that is, from submarine shocks, whether any shocks on dry land were then perceived at those places or not. This may appear more clearly by considering some of the phenomena of the earthquake of 1755.

If (as is universally allowed) the centre of this great earthquake was deep in the interior of the earth, most of the shocks therefrom which reached the surface must have proceeded either vertically, or upwards at various angles, and the times of reaching the surface at different places must have been generally according to their distances, or the angles of propagation, and the conducting-power of the intervening ground. The shocks which moved through very bad conductors might have been exhausted long before they could attain the surface. Some reached as far upwards as the bottom of the sea and the basins of lakes and ponds, occasioning extraordinary disturbances of the waters; and yet most of them fell short of the surface of the adjoining dry lands. Whilst five smart shocks were felt in Derbyshire Peak between 11 and 11.20 A.M., sixty fathoms under ground, only one reached the surface there. Mr. Mallet mentions this and two places

near Reading as the only places where shocks were felt in this island; but there is abundant traditional evidence of shocks having been felt in Cornwall; whilst in the Scilly Isles " several persons ran out of their houses for fear they would fall upon them "*.

On the other hand, if the centre of an earthquake be inland and close beneath the surface, the shock would not travel vertically or upwards at all, but horizontally, as the shock in this country of the 6th of October, 1863, appears to have done; for when at Pontypool, in the midst of Welsh mines, soon after its occurrence, I was informed by a gentleman who had ascertained the fact from the miners themselves, that whilst the shock was felt at the mines above ground, it was not felt by any under ground. Nor does it appear to have been felt under ground in any of the mines of Cornwallt.

The two preceding paragraphs may help to account for a striking fact mentioned by Mr. Mallet, viz. that the extraordinary disturbances of the sea now under consideration have "never been observed to take place in any earthquake whose centre of impulse was inland, however violent". This fact (which is in perfect agreement with my hypothesis) may perhaps be owing to the centre of impulse, when inland, being generally close under the surface, so that the shocks must travel horizontally, or nearly so; and it is not to horizontal, but to vertical shocks that have always ascribed these disturbances of the sea.

Another fact not less remarkable is, that, while earthquakes in general take place equally in all states of the atmosphere, those which are known only by the extraordinary agitations of the sea or lakes which they produce occur almost exclusively during storms, or at or near considerable minima of the barometer. Is it because these submarine shocks, as already stated, are almost always vertical, or nearly so, while those on dry land are generally horizontal? In vertical shocks there may be electrical discharges between the earth and the atmosphere which might occasion the attendant minima of the barometer, whereas in horizontal shocks the discharges (if any) may be only between differently charged portions of the earth without much affecting the atmosphere. The first earthquake felt by Humboldt in Cumana was during a severe thunder-storm. "At the moment of the strongest elec trical explosion were two considerable shocks of an earthquake;" but the barometer, which had been previously falling, continued to fall for five hours afterwards, when a third and last shock

* Troutbeck's 'Scilly,' 1794, p. 40.

† Journal of the Royal Institution of Cornwall, No. 1. p. 61.
British Association Reports, vol. xx. p. 46.

occurred, at which "moment the mercury was precisely at its minimum height "*.

Having in 1843, at the request of Sir Charles Lemon, the then President, written an account of the extraordinary oscillations of the sea of the 5th of July in that year, I have considered it due to this Society, as well as to myself, to state thus fully my reasons for still regarding the hypothesis I then advanced as the only one capable of reconciling all the known facts connected with these disturbances.

VII. Astronomical Prolusions: commencing with an instantaneous proof of Lambert's and Euler's Theorems, and modulating through a construction of the orbit of a heavenly body from two heliocentric distances, the subtended chord, and the periodic time, and the focal theory of Cartesian Ovals, into a discussion of motion in a circle and its relation to planetary motion. By J. J. SYLVESTER, F.R.S.†

THE

HE original demonstration by Lambert of the celebrated theorem which bears his name was a geometrical one. See Monthly Notices of the Astronomical Society, vol. xxii. p. 238, where this demonstration is reproduced, or rather recapitulated by Mr. Cayley. See also Lambert's own Insigniores Orbita cometarum proprietates, Augusta Vindelicorum [Augsburg], 1761. It occupies seven or eight pages of this celebrated tract, and, elegant as may be considered the chain of geometrical enunciations from which it is deduced, is, as a specimen of geometrical style, little worthy of the inconsiderate commendations which have been heaped upon it, containing, as it does, a hybrid mixture of algebraical, geometrical, and trigonometrical ratiocination. The late Professor MacCullagh, as I am informed by my ingenious coadjutor Mr. Crofton, one of his hearers at Trinity College, Dublin, greatly improved upon Lambert's method, and succeeded in reducing it to a purely geometrical form. Lagrange has given no less than four distinct demonstrations of the same,* Personal Narrative, vol. iii. pp. 316–318.

† Communicated by the Author. A portion of this paper has appeared in the Monthly Notices of the Astronomical Society of London for December last, viz. so much of it as relates to Lambert's theorem proper. The portion concerning circular motion formed the subject of a communication to the London Mathematical Society at the Meeting of December 18, 1865. The part which presented itself last to the author's mind, and is consequently the least developed, is that which relates to the determination of the forces in any orbit to any two (or more) centres of force. The general expression for such forces will be found stated further on in a footnote, where the equation of radial work is defined and employed to

one a sort of verification by aid of trigonometrical formulæ in which the eccentric anomalies are introduced; a second of a similar nature, but dealing only with the true anomalies; a third founded on a property of integrals*; and a fourth, perhaps the most remarkable of any, derived from the general expressions for the time in an orbit described about two centres of force varying according to the law of nature, but one of them supposed to be situated in the orbit itself, and to become zero. Notwithstanding this plethora of demonstrations I venture to add a seventh, the simplest, briefest, and most natural of all, in which I employ a direct method to prove, from the ordinary formulæ for the time of a planet passing from one point to another, that, when the period is given, the time is a function only of the sum of the distances of these points from the centre of force, and of their distance from one another.

Let p, p' be the distances of the two positions from the sun, c their distance from one another, v, v the true, u, u' the eccentric, m, m' the mean anomalies thereunto corresponding, e the eccentricity, w=m—m', s=p+p', ▲=}(s2—c2); then

p=1-ecos u, p'=1-e cos u', mu-e sin u, m'u'—e sin u', p cos v=cos u―e, p sin v=√1-e2 sin u,

p'cos = cos u'-e, p' sin v1-e2 sin u', c2=p2+p12-2pp' cos (v'—v).

Writing for brevity p, p, q, q' for cos u, cos u', sin u, sin u', we have

s=2-ep,-ep', w=u-u'—eq+eq',

▲=pp'+pp' cos (v' —v)=1+p,p′+q, q' — 2e(g+q′) +
e2 (1—q, q'—pp').

* The property in question, discovered by Lagrange, is that the integral

in applying it to Lambert's theorem a, b, c are made to vanish. This transformation and its consequences appear to us to deserve further study; as far as I know it has not been touched upon by the writers on elliptic functions.