solution being in fact represented by r (M+1) which varies from unity at the corners of the triangle up to the very large number at its centre. No simple and easy method of evaluating precise r (M+1) M +] 3 numerical results for the probabilities of the different cases is attainable; but it will readily be seen that conclusions of a general nature, not without interest and importance, may notwithstanding be obtained in any given case. Analogous considerations applied to a four-cornered constituency lead to a representation of the different cases by the portions into which a tetrahedron is divided by planes parallel to its four faces. On the Division of Elliptic Functions. By W. H. L. RUSSELL, F.R.S. This paper was intended to illustrate some of the discoveries of Dr. Weierstrass. It was shown that certain series for sin am. u, assumed to be convergent for cer-- • tain values of (u) when sufficiently small, are true when (u) has any value. The proof depended on an application of the proposition known as Abel's Theorem. On a construction for the Ninth Cubic Point. On Geometrical Constructions involving imaginary data. On a property of the Hessian of a Cubic Surface. On the Successive Involutes to a Circle. By J. J. SYLVESTER. From the first involute of a circle we may derive a family of parallel curves forming the second involutes of the circle; from each of these again families, the totality of which will form the third involutes, and so on continually. The author had been led by circumstances to study the arco-radial or semiintrinsic equation of these curves, and had arrived at certain conclusions concerning its form which subsequent investigations have verified: it turns out that the general equation between the arc s and radius vector r of the general involute of the nth degree will be found by taking F, any rational integer function of x of the ath = 2 degree, and eliminating a between the equations F2+(d)2; 8= dF =fdxF + dx It follows, as the author had surmised, that the general arco-radial equation for the involute of the nth order when n exceeds unity, is of the degree (n+1) in and 2n in s. Of course, in the case of n equal to unity, the degrees sink to 1 in →→ and 1 in s. The second involute formed by unwrapping from the cusp of the first may be termed the natural second involute, but is not the most simple of the family; this, which is at the normal distance of half the radius externally from the one last named, is of the third degree in r and the second in s. It may be derived from the curve which a fixed point in a wall at half the length of the radius of a wheel from the ground marks in the wheel as it rolls along the face of the wall by doubling the vectorial angles and taking the square out of the radii vectores. From the arco-radial it is easy to pass to the general polar equation to the n-ary involute; the equation between p, the perpendicular from the centre and q the polar subtangent, is also very easily obtained, being, in fact, no other than the result of eliminating between Fr=p, Fr=q, Fr being any quantic in r of the nth degree, so that this equation will be of the (n-1)th order in q, and the nth order in p. In the Philosophical Magazine for October and December, and in the Proceedings of the Mathematical Society of London, will be found further developments of the theory of these circular involutes, which it is proposed to term Cyclodes. On the application of Quaternions to the rotation of a Solid. ASTRONOMY. On the extent of evidence which we possess elucidatory of "change" on the Moon's Surface. By W. R. BIRT, F.R.A.S. The questions of change on, or fixity of the moon's surface must be decided, as Webb (Celestial Objects for Common Telescopes, second edition, p. 68) remarks, by observation and not assertion. It is therefore important to gather up the fragments of our knowledge bearing on the evidence which we possess on these questions; and we may remark, in the first place, that our real knowledge of the fixity of the moon's surface does not at present depend upon "evidence," using this term to designate the results of observation and not the deductions of theory; nor can we possess any adequate evidence of this kind, as is manifest from the very circumstance that up to the present moment our records of the physical aspect of the moon's surface are not only exceedingly scanty-in comparison with the countless thousands of objects of every variety of description which are revealed to us by even small instruments, let alone the increasing visibility of smaller objects due to larger apertures-but it is now becoming acknowledged that such records and the delineations accompanying them are not sufficiently precise and exact to enable us to refer to them as reliable witnesses in establishing "fixity;" indeed it is difficult to conceive how the unalterable state of the moon's surface can be determined by "observation." If, as has been asserted, all changes on the moon's surface have ceased myriads of ages ago, we are certainly destitute of the records of "observation" of the real state of that surface at so remote a period. In fact our absolute knowledge of“ fixity can only date from the construction of the first lunar map, since which there are no traces of any grand convulsion. The establishment of "fixity" can only have reference to those objects which have been more particularly observed during the intervening period; and, as shown by Webb (Intellectual Observer, vol. xii. pp. 435, 436), if really established by a long course of observation at any one point, it would be no argument for its universal prevalence, since a state of quiescence might be attained at very different epochs in different regions. Such being the case as regards "fixity," let us now inquire as to what evidence we possess on the subject of "change." The earliest attempts to perpetuate a knowledge of the moon's surface consisted in delineating the disk in the form of maps, accompanied in two instances with "catalogues" of the most striking and prominent objects; but, as might be expected, such maps are greatly destitute of "detail," especially of such a character as is necessary to pronounce on change." Towards the close of the last century Schröter, seeing the importance of perpetuating a knowledge of detail, bequeathed to us the result of his labours in this direction in the form of his 'Selenotopographische Fragmente,' taking up portions of the moon rather than attempting a delineation of the whole in detail. His successors, Lohrmann, Beer and Mädler, and Schmidt, have followed in his steps, and produced works abounding much more in detail than any of the predecessors of Schröter; Lohrmann's sections and map, with the two maps of Mädler, are well known. The larger portion of Lohrmann's Sections, as well as the results of Schmidt's labours, are still unpublished. Webb's index map of the moon in his 'Celestial Objects', with two catalogues of the principal craters, &c., together form an excellent guide to observers who are commencing the study of "detail." It is by means of the study of "detail" that a definitive answer must be given to either of the questions mentioned at the commencement of this paper. The details of the moon's surface are very various; mountain-chains, towering peaks, isolated hills, deep valleys, extensive plains scored with comparatively low ridges, crater-openings, in some localities crowded together by thousands, in others occurring singly, and situated far apart from each other, rings (apparently the highest parts of crater walls) the interiors of which convey the idea of having been partially filled with an injected material, minute craters are not unfrequent on their surfaces; bright spots, often of a dazzling whiteness, marking the tops of mountains and the crests of mountain-chains, as well as others of less brilliancy and greater indistinctness of outline; dark spots surrounded in some cases by bright rims; in others the difference between the dark spots and the comparatively lighter surface is rendered distinctly visible by a sharp outline inclosing the dark surface. these varieties must be carefully studied before a conclusion can be drawn as to the unalterable stability or mutation of such objects. All The means which we possess for the study of lunar objects may be considered as twofold, the examination of delineations and topographical notices on the one hand, and personal observation of the objects themselves on the other, of course including a comparison of one with the other. For example, we may find on the moon a spot darker than any object in the immediate locality surrounded by a rim brighter than the exterior surface, and we record its appearance. It is now desirable to place in juxtaposition all the records we possess of it as under. Beer and Mädler, in Der Mond,' p. 304, thus describe two craters on the Mare Nubium. I am indebted to W. T. Lynn, Esq., of the Royal Observatory, Greenwich, for the translation. "The boundary of the Mare Nubium does not run alongside of Alpetragius, but passes by it under several deep curvatures from three to five miles [German to the eastward. In the Mare itself is situated the bright radiating crater B (9° light) at -14° 55′ lat. and -7° 27' long., and near this the far larger and deeper one a, which, however, is found with difficulty at the full moon. It is about 4, and the interior 3° bright (Insula Lesbos H)." As it is important that nothing should be quoted from memory unless quite unavoidable, the following are extracts from my note-book. 1868. June 29, 8.40. Crossley Equatorial, 7-3 inches aperture, power 122. "B. & M.'s a and B are both quite conspicuous; a is a shallow crater or ring with a smooth floor; interior west shadow very narrow between 0·1 and 0-2, the diameter of a being 10. B is a deep and bright crater, shadow gibbous =0·5." 1868. July 29, 10.30, Royal Astronomical Society's Sheepshanks telescope No. 5, aperture 2.75 inches, power 100. "The interior of a very dark, the darkest surface in the locality, probably 20.” 1868. July 31, 10.15, R. A. S. Sheepshanks No. 5, power 109. "The floor of a is darker than any surrounding part; all three authorities, Lohrmann, Mädler, and Schmidt, make it lighter than the surface of the Mare." The following brightnesses were determined :— Written from memory, Aug. 2 7h. "On the evening of the 1st, the line of demarcation between the surface of the Mare Nubium and the adjacent lighter parts was very distinct." With such facts before us can we decide for "change?" In replying to this question one disadvantage immediately suggests itself. We are uncertain as to the number of observations on which the earlier records rest; but while in doubt on this point, we have it in our power not only to increase our own observations, but also to solicit the aid of others, that in the re-observation of an object the want of confirmatory evidence may not exist to occasion a doubt as to the certainty of what is recorded. Webb says very truly (Celestial Objects, second edition, p. 68), we are scarcely as yet possessed of the means of detecting small changes. The evidence capable of being brought to bear on the question of change is consequently very limited in extent, especially as former records are more or less open to be regarded as inexact drawings or inaccurate statements when they happen to differ from present observed appearances, still as instances such as are given above increase, and they are upon the increase, it will become more and more difficult to put aside the earlier records. It therefore remains rather, as recommended by M. de Beaumont, to increase our observations and compare them with the earlier records than to rest satisfied with the notion that, as no change has been satisfactorily ascertained, it is unlikely upon certain theoretical considerations that we may meet with any. The Meteor Shower of August 1868. By GEORGE Forbes. The author merely stated the results of some observations made on this shower on the nights of the 10th-11th and 11th-12th of August last. The peculiarities of this shower were for the most part the same as last year. The hourly average number on the first night was 21 on a clear night seen by one person, while last year the number was 25 on a hazy night by the same person. The radiant-point was approximately R.A. 2h 16m N.P.D 31°. Last year it was nearly the same. Two meteors traced curves, one of them of a very remarkable form. When a distinct train was left the meteor was generally noticed to pass beyond the end of the train and to become suddenly extinguished without previous diminution of brilliancy. No trace of the radiant discovered last year in Pisces was noticed. The average size of meteors was that of a 4th magnitude star. ACOUSTICS. On a Simple Method of exhibiting the Combination of Rectangular Vibrations*. By W. FLETCHer Barrett. Physicists are well acquainted with the elegant experiments of M. Lissajous, in which the vibrations of two tuning-forks, placed at rectangles, are optically combined by viewing a ray of light successively reflected from a mirror attached to each fork. A regular series of curves is thus obtained which gives a perfect optical expression of each of the musical intervals, the curves augmenting in complexity as the dissonance between the forks increases. Instructive and beautiful as are these experiments, the extreme costliness of the apparatus necessary for their proper exhibition has hiherto debarred many from repeating them. Upwards of two years ago the author found a method of obtaining any desired combination by an extremely simple arrangement. A piece of straightened steel wire, about No. 16 gauge and some 12 or 18 inches long, is first well softened in a flame at a point 6 or 8 inches from the end, which length is then bent downwards. The extremity of the longer portion is fixed in a vice, a silvered bead is cemented by marine glue on to the summit of the bend, and the instrument is complete. The whole system is thrown into vibration by smartly tapping the wire near the point held in the vice, and in a direction oblique to the plane of the two wires. The vibration travels up the wire, rounds the bend, and throws the inclined arm into motion. The latter, being free, vibrates more easily than the portion which is fixed at one extremity; a compound motion is thus the result, and the spot of light, reflected from the bead, describes a curve expressing the resultant action. The ratio between the vibrations of the two parts of the wire can evidently be adjusted, or altered, by raising or lowering the point clamped in the vice. The same end may also be obtained by loading the free portion of the wire by a little * Published in extenso in the Philosophical Magazine for September 1868. sliding weight. But an alteration in the angle of the bent wire yields a more satisfactory result. When the wires are parallel and even in length, a combination of 1 to 1 is obtained, and the bead describes a circle passing into an oblique line; but on opening the free limb to an angle of about 30°, the figure changes into the complex curve given by the ratio of 4 to 5. Opening the angle still further, the curve expressing the ratio of 3 to 4 is obtained; then at 45° 2 to 3; and at an angle of 75° the figure of eight comes out, expressing the ratio of 1 to 2. In fact, by varying the angle an entire series of combinations, more or less perfect, can be produced at will. Figure 1 shows the instrument. The wire is capable of being firmly fixed at any height in a support which is attached to a heavy stand, more convenient in use than a vice. Not only may this arrangment be used for exhibiting the combination of vibrations, but it also shows very prettily the formation of nodes and ventral segments. On the free arm an instructive change is seen to take place in the position of a node which is there formed. When the arms are equal and parallel, and a ratio of 1 to 1 obtained, the node is near the free extremity of the bent wire; as the wire is raised and the angle increases, the node rises nearer to the bend. It is also worth observing that, in any combination, the distance of the node from the free extremity of the wire, compared with its distance from the bend, is approximately the same as the ratio of the interval depicted by the figure. Another arrangement for effecting the combination of rectanglar vibrations (shown in figure 2) has been adapted by Mr. Ladd from an instrument devised by Professor Helmholtz. Two flat pieces of steel are here welded at right angles to each other into a single rod. The upper part (a, fig. 2) is tapering, and on its summit is fixed a polished silver bead. The lower part (b) is capable of being firmly fixed in a suitable support. According to the height at which b is clamped, so a corresponding portion is allowed to enter into vibration. A combination of the vibration of a with that of b can thus be obtained in any given ratio. Complete command of any figure can be had by marking its position on the lower strip of steel; and so nice an adjustment is possible, that an almost absolutely steady figure can be secured with a little care. The author proposes to call the instrument described in this paper a Toncphant. HEAT. On Sources of Error in determinations of the Absorption of Heat by Liquids. By W. FLETCHER BARRETT*. During the autumn of 1865 the author had observed that under certain condiPublished in catenso in the Philosophical Magazine for September 1868. |