If we write car (<1) for the greatest, and a= yr (y < 1) for the least, this equation becomes It will be found convenient in what follows to replace x, y by u, v where 1 - a3 = u, 1 — y' =v; when the form never departs very largely from the spherical form u, v are not large, in this case we have very approximately whence neglecting cubes and higher powers of u, v Equation (5) can be put into a very elegant form, easily adapted for the use of logarithmic tables, which give the logarithms of secants and tangents, by introducing angles e, & where when it will be found that (5) takes the form cos1 cosec = sin 0 sec log, cot = 2·3025851 sin10 sec log cot 40. Corresponding sets of values of e, are obtained with extreme case by means of this equation, by giving a series of values to 0, finding the value of the logarithm of the right-hand member, and taking as a rough approximation φ = −0. A closer approximation can then be obtained by proportional parts. where it must be remembered that has different forms for the cases of a > r and a <r. We consider these separately and determine the time from the spherical state to the greatest elongation or depression. If g denote the acceleration due to gravitation on the surface of the liquid when it has the spherical form it can easily be shewn that 4k=3gr3. Putting in this value of k, it can be shewn, after some numerical calculations, that if t1, t, denote the times from greatest elongation to the spherical form, and from the spherical form to the greatest compression respectively, then to the second order of u, v whence if T denote the time of semi-oscillation, we have, from (7), the first term of this agrees with the result obtained by Sir W. Thomson for the time of oscillation of a liquid sphere, slightly deformed according to a zonal harmonic of order 2. Similarly in the case of the elliptic cylinder, it follows that * "Oscillations of a liquid sphere." Phil. Trans. Roy. Soc. (1868), p. 608. (5) On functions of more than two variables analogous to Tesseral Harmonics. By M. J. M. HILL, M.A. [Abstract.] A transformation analogous to the ordinary transformation from rectangular to polar variables is applied to an equation similar to that of Laplace, but containing i variables. A solution of the transformed equation is found which consists of products of terms, each term being a function of one only of the new variables. The solution thus found, omitting the first term, is called a Normal Function, and is analogous to a Tesseral Harmonic. The Normal Functions possess a property analogous to the conjugate property of Tesseral Harmonics. The value of an integral corresponding to the integral of the square of a Tesseral Harmonic over the surface of the sphere is next evaluated, Then it is shewn that the solution of the transformed equation is a rational integral homogeneous function of the original variables, and that it includes under its form all the independent rational integral homogeneous functions of the original variables of the same degree which satisfy the original equation, but nothing else. The expansion of an arbitrary function of i-1 variables in a series of Normal Functions concludes the paper. (6) Observations of the Transit of Venus across the Sun, taken near Kingston, Jamaica, at Cherry Garden, the residence of Oscar Marescaux, Esq. Dec. 6, 1882: by Dr Pearson. The Cambridge Philosophical Society will probably be glad to receive an early account of the observations of the Transit taken by one of its members. Their value will not be affected, though the interest taken in them may perhaps be increased, by the fact that they were made by one who belongs to the same college in this University as the sole spectator of the first transit ever observed, viz. that of Nov. 24 (0. s.), 1639. The Latitude of my place of observation was determined, though from the nature of my instruments, not with very great precision, by meridian altitudes of the Sun, Fomalhaut and Achernar, the Pole-star not being available: and verified, in a way, by my known distance from the Lighthouse to the east of Port Royal, on the Palisadoes, at a place named Plumb Point: which bore only a degree and a half east of South, at a distance of about seven miles. My observations vary between 18° 24′ and 18° 31' North Lat. I have taken 18° 3′ 20′′ as my basis of calculation, this being about the average given by the stars. Greater accuracy is clearly of no importance. The Longitude is derived from that of Rodney's statue in the town of Kingston, which was determined by Lieut. Green, of the U. S. Coast Survey, two or three years ago, by telegraphic signals made with extreme care, to be 5 h. 7 m. 10-71 s. W. of Greenwich. In my results I assume that I was about 14 s. east of this, or in 5 h. 7 m. 9 s. E. Long. of Greenwich: 5 h. 16 m. 30 s.* E. of Paris. My Local Time I have been able to determine with considerable precision. From about Nov. 23rd to Dec. 2, and from Dec. 8 to 13, I was continually taking morning and afternoon altitudes of the Sun, as well as altitudes of the stars in the evening. From these I inferred the best chronometer of the two which I had with me, to be m. 1. Nov. 25 a.m. 2 45 slow on L. M. T. Dec. 1 3 6 Also, by the kindness of Dr Copeland and Capt. Mackinlay, from the clocks of the Government Transit Expedition at Up Park near Kingston, I obtained, Dec. 1st, Chronometer 3 m. 6 s. slow on their local Time, their meridian being within half a second of my own, and Dec. 13, 9 p.m. by Transits of y Ceti and σ Arietis, their clocks having been dismounted, 3 m. 49 s. which gives the chronometer a losing rate of 344 s. I have therefore assumed my chronometer to have been 3 m. 24 s. slow on L. M. T. at the morning contact; and 3 m. 244 s. in the afternoon; which gives results without fractions of seconds, which situated as I was it would be needless to introduce. The actual times of contact I had intended to take from my second chronometer, which has an extremely clear tick: unfortunately it stopped from the heat when it had been exposed five or ten minutes to the rays of the Sun, and before the time of the first contact; I was therefore compelled to use my watch which I held in my hand, though I do not think that any error * Since my return I have found by the large Chart of Kingston Harbour, to be seen in the University Library, that the Lighthouse which I mention is as nearly as possible in 5 h. 7 m. 6 s. E. longitude reckoned by its position as referred to Rodney's Statue. But 7m. × tan 11o gives 187 m. or nearly 1s. of time (a_sea-mile=4 s.) as the difference between my own meridian and that of the Lighthouse. This would imply that my own W. longitude was 5 h. 7 m. 78. instead of 5 h. 7 m. 9s. The position of the Government observatory at Up Park Camp appears from the chart to be nearly on the same meridian as my own: and this agrees with the result I obtained on the spot, by which I made the two agree within half a second. The L. M. T. therefore of my results must be increased in each case by two seconds. I shall look with interest to see what longitude Dr Copeland gives in his published observations. can have resulted from my doing so, as I habitually observe in this way; the time on my watch I compared with my best chronometer directly after each contact. The first external contact I unfortunately missed in a way which can be easily explained. Though constantly observing the Sun for altitude I have seldom examined its features; and thus failed to notice that a point on its disk, 145° E. of N. the predicted point of contact, would at 9.a.m. be slightly to the right and not to the left of the lowest point of the Sun's limb. Having for the three days previously been rather indisposed and unwilling to expose myself in observing, I had put my preparatory work somewhat aside; and thus failed to look through the question as I might perhaps have done, had I been quite ready for any kind of work. When I saw Venus first she had intruded about one-third of her sphere on the Sun's disk; I watched her carefully until the two limbs were very nearly in contact, from which time I did not remove my eye until the Sun's light appeared to surround the planet; the moment of this phenomenon I fixed at 9 h. 16 m. 26 s. a.m. L. M. T., or perhaps two or three seconds later. I noticed no kind of black drop, or sympathetic attraction, or assimilation between the limb of the planet and that of the Sun or rather the edge of the atmosphere enclosing the Sun. If the slight want of definition from which my vision suffered, be it due to my own eyes or my eye-piece or my object-glass, allows me to give any formulated description of the first internal contact, I should say that when the planet was actually projected on the Sun's disk, say 20 s. before the time I assign for actual contact, the black surface of the planet adjoining the atmosphere seemed to begin to be picked out with little white dots commencing very probably from either side; but as the phenomenon was new to me, I cannot say whether the white spots began at the two ends of the incomplete segment of the planet's disk, or whether they began throughout at once. I cannot say that I actually saw two horns of light gradually advancing until their points touched; but rather, as I have said, the segment of the planet nearest the atmosphere and still obscure began to be speckled with white dots which in not more than 20 seconds, or 25 at the outside, developed into a white line. When the planet was a little way advanced on the Sun's disk, she was well defined in her outline: and no remarkable difference presented itself at 10:52 a.m. when I took a measure of the distance of her limb from the Sun's edge. But at noon, when the distance of the centres of the Sun and Venus was least, the irregularity of her disk, from the boiling of the surrounding solar VOL. IV. PT. VI. 23 |