19. The following stars have also been observed: numerous lines are seen in the spectrum of each; and in some, several of the lines were measured; but we have not yet instituted parisons with the metallic spectra. any com Castor; e, 5, and n Ursa majoris; a and e Pegasi; a, B, and y Andromeda, the last an interesting spectrum; Rigel, a spectrum full of fine lines; n Orionis; a Trianguli; y and e Cygni; a, B, y, e, and n Cassiopeia; y Geminorum; B Canis majoris; B Canis minoris; Spica; y, d, and e Virginis; a Aquila; Cor Caroli; B Auriga; Regulus; B, y, d, e, , and n Leonis. η η [To be continued.] LXII. On a Problem in the Calculus of Variations. may be HE problem to which the following remarks relate THE thus enunciated :-To determine the greatest solid of revolution, the surface of which is given, and which cuts the axis at two fixed points. The problem has been discussed in the Philosophical Magazine for July 1861, August 1861, September 1862, and March 1866; but something may, I think, be added to what has been already published. π a Adopting the usual notation, we have to make Syd maximum, while 2 Sy√(1+p2)dx is given. Let a be a constant at present undetermined; then we have by the received theory to make u a maximum, where u denotes We obtain where A stands for S {y2+2ay √(1+p2)} dx. du= SA(dy—pdx)dx +B, d аур 2y+2a√(1+p2) −2· and B stands for certain terms which are free from the integral sign. The expressions for u and Su are both supposed to be taken between limits which correspond to the fixed points on the axis. Now by the known principles of the subject we put A=0; this leads in the usual way to The whole spectrum of Sirius is crossed by a very large number of faint and fine lines. It is worthy of notice that in the case of Sirius, and a large number of the white stars, at the same time that the hydrogen lines are abnormally strong as compared with the solar spectrum, all the metallic lines are remarkably faint. On the 27th of February, 1863, and on the 3rd of March of the same year, when the spectrum of this star was caused to fall upon a sensitive collodion surface, an intense spectrum of the more refrangible part was obtained. From want of accurate adjustment of the focus, or from the motion of the star not being exactly compensated by the clock movement, or from atmospheric tremors, the spectrum, though tolerably defined at the edges, presented no indications of lines. Our other investigations have hitherto prevented us from continuing these experiments further; but we have not abandoned our intention of pursuing them. 15. a LYRE (Vega).-This is a white star having a spectrum of the same class as Sirius, and as full of fine lines as the solar spectrum. Many of these we have measured, but our investigation of this star is incomplete. We have ascertained the existence, in the stellar spectrum, of a double line at D corresponding to the lines of sodium, of a triple line at b coinciding with the group of magnesium, and of two strong lines coincident with the lines of hydrogen C and F. 16. CAPELLA.-This is a white star with a spectrum closely resembling that of our sun. The lines are very numerous; we have measured more than twenty of them, and ascertained the existence of the double sodium line at D, but we defer giving details until we have completed our comparison with the spectra of other metals. From this star we obtained (on February 27, 1863) a photograph of the more refrangible end of the spectrum; but the apparatus was not sufficiently perfect to exhibit any stellar lines. 17. ARCTURUS (a Boötis).—This is a red star the spectrum of which somewhat resembles that of the sun. In this also we have measured upwards of thirty lines, and have ascertained the existence of a double sodium line at D; but our comparisons with other metallic spectra are not yet complete. 18. POLLUX.—In the spectrum of this star, which is rich in lines, we have measured twelve or fourteen, and have observed coincidences with the lines of sodium, magnesium, and probably of iron. At any rate there is a line which we believe occupies the position of E in the solar spectrum. a CYGNI and PROCYON are both full of fine lines. In each of these spectra we observed a double line coincident with the 19. The following stars have also been observed: numerous lines are seen in the spectrum of each; and in some, several of the lines were measured; but we have not yet instituted any comparisons with the metallic spectra. Castor; e,, and n Ursa majoris; a and e Pegasi; a, B, and y Andromeda, the last an interesting spectrum; Rigel, a spectrum full of fine lines; n Orionis; a Trianguli; y and e Cygni; a, B, y, e, and ʼn Cassiopeia; y Geminorum; B Canis majoris; B Canis minoris; Spica; y, d, and e Virginis; a Aquila; Cor Caroli; B Auriga; Regulus; B, y, d, e, 5, and n Leonis. THE [To be continued.] LXII. On a Problem in the Calculus of Variations. HE problem to which the following remarks relate may be thus enunciated :-To determine the greatest solid of revolution, the surface of which is given, and which cuts the axis at two fixed points. The problem has been discussed in the Philosophical Magazine for July 1861, August 1861, September 1862, and March 1866; but something may, I think, be added to what has been already published. a Adopting the usual notation, we have to make Syd maximum, while 2 Sy√(1+p2)dx is given. Let a be a constant at present undetermined; then we have by the received theory to make u a maximum, where u denotes We obtain where A stands for S{y2+2ay √(1+p2)} dx. Su= √ A(dy—pdx)dx+B, d аур 2y+2a√/(1+p2)—2 'dx √(1+p2)' and B stands for certain terms which are free from the integral sign. The expressions for u and Su are both supposed to be taken between limits which correspond to the fixed points on the axis. Now by the known principles of the subject we put A=0; this leads in the usual way to where b is another constant, which is introduced by the integration. Since the curve is to meet the axis, we have y=0 at certain points; hence b=0, and the equation becomes 2a Thus we have either √(1+p3) +y=0, or y=0. The first corresponds to a sphere, and the second to a cylinder, of indefinitely small radius. This solution is due to the Astronomer Royal; it is given in the Philosophical Magazine for July 1861, and illustrated by figures. There is, however, one difficulty which still exists. On proceeding to verify the solution, we find that the terms denoted by B vanish at the limits; also A vanishes for all that part of the solution which corresponds to the sphere; but A does not vanish for the remaining part of the solution. In fact, if we put y=0, we find that A=2a. Thus instead of having du=0, we have since p=0 when y=0. The integration is to extend throughout the range which corresponds to the cylindrical portion of the solution. It appears to me that the following is the explanation of the difficulty. It is seen on examination that a is a negative quantity. Also by the nature of the problem, corresponding to y=0, the value of dy must be positive. Thus du, instead of being zero, is essentially a small negative quantity; this, however, ensures that u is a maximum, and therefore we obtain all that is required. The following consideration will, I think, show that it is in vain to seek for any other solution instead of that proposed by the Astronomer Royal. It is an admitted result that among all figures of given surface the sphere is that which has the greatest volume. Now, in the solution of the problem which we are considering, we obtain a figure the surface and volume of which differ infinitesimally from those of a sphere; and hence we conclude that the volume we obtain is greater than that of any proposed figure which differs to a finite extent from a sphere with the given surface. It should be observed that there are two forms in which the problem may be proposed: we may suppose that the solid is the axis of revolution and which pass through the fixed points; or we may suppose that this restriction is not imposed. We have hitherto considered the problem without the restriction; it is easy to see that the investigation will require only a slight modification if the solid should stretch beyond the space bounded by the two fixed planes just mentioned. If, however, the restriction be imposed, the solution already obtained holds so long as the given surface is not greater than that of a sphere described on the given length of axis as a diameter; when the given surface is greater than this, the solution is, I believe, that which was proposed in the History of the Calculus of Variations,' page 410, and adopted in the Philosophical Magazine for August 1861 and September 1862. In conclusion I will state briefly the grounds on which I consider that the results published in the Philosophical Magazine for March 1866 are unsatisfactory. In the first place, the solutions there obtained do not satisfy the fundamental condition A=0; and in the second place, they are liable to the objection which I have drawn from the admitted result respecting a sphere. Cambridge, May 9, 1866. T LXIII. On the Heat of the Electric Spark. test an electrical exploding-apparatus, the fuse of which is to be ignited by one spark, we must possess some knowledge of the heat of the electric spark itself. Investigations have already been made on this subject; the earlier results will be found collected in Riess's work†, whilst later researches on the question have been communicated by Poggendorff and Reitlinger§. Their experiments have shown that when a series of sparks from the electrical machine or the induction-coil is discharged between wood and wood, or wood and metal, a considerable rise of temperature is the result. The production of heat, however, by the single sparks of a battery of Leyden jars could not be proved. Neither a mercurial thermometer nor a thermopile with galvanometer, when in the immediate neighbourhood of the discharge, was in the slightest degree affected. When, however, the platinum wire of Riess's air-thermometer * Translated from the Monatsbericht of the Academy of Sciences of Berlin, November 1865, p. 563. † Lehre von der Reibungselektricität, §§ 550 and 700. Berliner Monatsberichte (1861) pp. 349-377; Pogg. Ann. vol. xciv. pp. 310 and 632-637; vol. cxxi. p. 307. § Zeitschrift für Math. und Phys. vol. viii. pp. 146 -149. |