where a1, a are constants, and k=AP/PB, k' = A'P'/P'B, k'' = A''P''|P”B”. Starting from this form, various trigraphic forms of increasing complexity can be built up, as in the geometry of homographic figures. But it is impossible to finally construct trigraphic figures on the analogy of homographic figures. To prove this, take ABCD, A'B'C' D', A" B" C" D" as corresponding tetrahedrons in the assumed trigraphic figures, and suppose that the variable points P, P', P" of the figures subtend trigraphic pencils at the axes (BC, CA, AB), (B'C', C'A', A'B'), (B"C", C"A", A"B") respectively. Then, if P and P' move on straight lines, it is shown that P" moves on a surface, and not on a straight line, as it should do.

6. On Maxwell's Method of deriving the Equations of Hydrodynamics from the Kinetic Theory of Gases. By Professor LUDWIG BOLTZMANN.

It is well known that the equations of hydrodynamics for a viscous fluid, as Maxwell was the first to show, can be derived from the hypothesis of the kinetic theory of gases. But Maxwell's method is not quite satisfactory. Many terms of the equations must be neglected in order to obtain the hydrodynamical equations in their usual form. Even if this course in most cases is justifiable, it cannot be rigorously proved that such is the case, and the mathematician is not satisfied. The following question arises, Is this a defect of the theory of gases, or is it rather one of hydrodynamics? Are these terms required by the theory of gases not an essential correction of the equations of hydrodynamics? Will it not be possible to find cases where these two theories are not in accord, and to decide by experiments between them? Maxwell himself raised this question, and he found that the ordinary assumption, that in gases which conduct heat the pressure is everywhere equal in all directions, is only approximately true. A short time before his death he published an ingenious method of treating these questions, viz., the application of spherical harmonics to the theory of gases. Maxwell only gave in a few words the results of his calculations, in three short notes, which are included in square brackets in his paper, 'On Stresses produced by Conduction of Heat in Rarefied Gases.' These three notes show evidently that he must have made a long and elaborate investigation on this subject a short time before his death, which, however, has not been published. I have treated the same subject by a different method, and have also found that many corrections of the equations of hydrodynamics can be derived from the theory of gases. It will be not easy, but perhaps not impossible, to test some of these differences by experiment. I have not yet published these results, because they do not agree in all respects with the results briefly announced by Maxwell, and the danger of falling into errors in this subject is great.

With regard to this I beg the British Association to make efforts to ascertain if the manuscript of the investigation made by Maxwell on the application of spherical harmonics to the theory of gases is still in existence, and, if this manuscript should be lost, to encourage physicists to repeat these calculations.

7. On the Invariant Ground-forms of the Binary Quantic of Unlimited Order. By Major P. A. MACMAHON, R.A., F,R.S.

8. Principes fondamentaux de la Géométrie non-euclidienne de Riemann. Par P. MANSION, professeur à l'Université de Gand.

I. M. Gérard a exposé, sous une forme simple et rigoureuse, les principes fondamentaux de la géométrie non-euclidienne de Lobatchetsky, dans un article inséré dans la livraison de février 1893 des Nouvelles Annales de Mathématiques' (3 série, t. xii, pp. 74-84).

On peut démontrer d'une manière anal que les principes fondamentaux de la géométrie non-euclidienne de Riemann, en pa tant des deux propriétés fondamen

tales suivantes: 1° Deux droites riemanniennes situées dans un même plan se coupent en deux points situés à une distance 24 toujours la même quelles que soient les deux droites (voir De Tilly, Essai sur les principes fondamentaux de la Géométrie et de la Mécanique,' dans les Mémoires de la Société des Sciences Physiques et Naturelles de Bordeaux,' 2a série, t. iii. 1er cahier, ch. i.). 2o La somme des trois angles d'un triangle est supérieure à deux droits (voir Mathesis, août 1894, 2o série, t. iv., pp. 181-182).

II. On démontre, comme dans le cas de la géométrie lobatchefskienne, les théorèmes suivants :

1o Dans un triangle rectangle ayant pour hypoténuse z, pour côtés x, y, si tend vers zéro, l'angle (z, 2) restant constant (r) tend vers une limite finie en décroissant; (y: z) tend aussi vers une limite finie, mais en croissant.

2° Si u et u', v et v' sont les côtés opposés d'un quadrilatère, trirectangle en (u, v), (u, v') et (v, u'), et si u tend vers 0, (u':u) tend en décroissant vers une limite (v), dépendant de v seulement; on a d'ailleurs (u':u) < (v).

III. THEOREME FONDAMENTAL.-On a ¢ (x + y) + $ (x − y) = 2¢ (x) $ (y). 1° Considérons le triangle OAa birectangle en A et a; posons AB=x-y, AC=r, AD=x+y, OA = A. Menons Bb, Cc, Dd perpendiculaires à OA et rencontrant Oa en b, c, d; de bet dabaissons bm, dn perpendiculaires sur Cc. Soit encore BB'= =a, B'b' perpendiculaire à OA et rencontrant Oa en b', b'm', une perpendiculaire à Cc.

2o D'après II. 1°, on a cd >cb; par suite, cn>cm, ou 2 Cc>Cm + Cn. Ensuite,

4° On a

cb'>CB'-B'b' - Cc >CB+ a−2Aa = CD + a−2Aa>cd+a-2Aa.

On peut supposer 2Aa inférieur à la quantité fixe a; dans cette bypothèse, on a cb'>cd, cm'>cn, Cm' + Cn>2Cc, puis

6o Des relations (a), (b) on conclut le théorème.

IV. On démontre, comme dans le cas de la géométrie lobatchefskienne, les propositions suivantes:

2o Dans un triangle rectangle ayant pour hypoténuse 2, pour côtés x, y on a

cos C étant une fonction qui ne dépend pas de la grandeur des côtés a = BC, b = AC, mais seulement de l'angle C opposé au côté c=AB.

9. Formula for Linear Substitution. By Professor E. B. ELLIOTT, M.A., F.R.S.

If (Ao, A1, A2, ... An) (x', y)”

respectively, then, P (a) denoting a rational integral homogeneous isobaric function an, whose order and weight are i and w,

of ao, a, 42,

...

DEPARTMENTS II., III.

A joint meeting with Section I. was held to discuss the two following Papers by Professor Oliver Lodge:

10. On Experiments illustrating Clerk Maxwell's Theory of Light.
By Professor Oliver Lodge, F.R.S.

11. On an Electrical Theory of Vision.
By Professor Oliver Lodge, F.R.S.

DEPARTMENT II.

12. On the Velocity of the Cathode Rays.
By Professor J. J. THOMSON, F.R.S.

13. On a Ten-candle Lamp for use in Photometry.
By A. VERNON HARCOURT, M.A., F.R.S.

The author has in former years brought before this Section a burner consuming a mixture of air with pentane vapour, and lamps consuming pentane vapour, which gave a constant amount of light equal to that of an average standard candle.

When so small a light is used in the ordinary photometry of coal-gas, errors from defects in the adjustment of the photometer, or from slight haziness in the atmosphere of the testing-place, are greater than when a light is used more nearly of the same magnitude as that of the gas-flame which is measured. A light of ten candles is well suited for the purpose. It has been shown that such a light can be obtained from an argand gas-burner consuming air saturated with the vapour of pentane; and a gas-burner has been proposed for use which is well suited for this purpose, except perhaps in being of a rather complex structure. But to supply air saturated with pentane vapour at a steady pressure there is needed a gas-holder of a few cubic feet capacity. The addition of such a gas-holder to the apparatus makes it more costly and not easily portable. It has been shown that great variations may occur in the proportions of pentane and air consumed by the gas-burner without materially affecting the light given out by the lower part of the gas-flame. The admixture of air is, however, unnecessary, since at a moderate temperature pentane can be volatilised without there being any necessity for reducing the atmospheric pressure by the admixture of another gas. The lamp shown is on the same principle as the one-candle lamp devised by the author seven years ago. The wick is raised or depressed by the ordinary rack-and-pinion movement, the lower end of it dipping into pentane in the body of the lamp, while the upper end is warmed by the heat conducted down from the flame. In the tencandle burner the wick fills the circular interspace between the central and outer tubes of an argand. The only difference between this and an ordinary lamp is that the wick does not come near the flame, and needs no cutting or renewal; nor does the smoothness or roughness of its upper surface affect the burning of the lamp.

The dimensions of the lamp and chimney upon which the air currents inside and outside the flame depend have been determined by experiment so as to produce a bright and steady flame. The entrance to the inner tube is through a triangular opening as in the usual construction of such lamps. It was found that the free admission of air at this entrance caused an inequality in the flame, two peaks appearing on either side of the place of admission of air. To steady and regulate the admission of air at this point, a cylindrical case is fixed round the lower part of the tube, into which air is admitted through a round hole 15 mm. in diameter. The flame of a rich gas burning from a wide opening is of the colour of candlelight even when an abundant supply of air is provided on both sides of the flame. The light of a good gas-burner is less red, or more blue, than this; and it is

important for the photometry of coal-gas that the standard light employed should be as nearly as possible of the same colour as the light of the gas-flame. The light of the lamp has been brought to this colour by a small admission of air below the point of combustion. Twelve holes, each 3 mm. in diameter, have been drilled through the outer tube 15 mm. below the top. The draught of the chimney is sufficient to determine an entry of air through these holes, which, happening under fixed conditions, is constant in amount. A short cylindrical screen goes round the chimney, the bottom of which is 58 mm. above the surface of the burner; and both screen and chimney are surrounded by a pentagonal shade, four panels of which are filled with blue glass, while the fifth, which is turned towards the disc of the photometer, is filled half-way down with a metal plate, which, overlapping the inner screen, allows only the light from the lower part of the flame to fall on the disc of the photometer. By supporting the lamp at such a height that the bottom of the screen is level with the centre of the disc, errors of parallax are avoided. Mr. Sugg, I believe, observed, and Mr. Dibdin has proved, that with the flame of an argand burner the light thus given by the lower part of the flame is independent, within wide limits, of the total height of the flame. The observation is true, not only for a particular burner, but for many, and probably for most, argand burners. Many alterations have been made in the structure of this lamp, and it has generally been found, though not always with equal exactness, that the height of the flame did not affect the light emitted from its lower part. As the lamp is now arranged the height of the flame can be observed through the blue glass panels, and whether the top is just visible above the circular screen or rises to the top of the panel, the light which falls upon the disc of the photometer does not alter measurably. The adjustment of the light to the value of ten candles-in other words, the adjustment of the height of the circular screen-has been effected by a number of comparisons between the lamp and the one-candle pentane standard.

14. On the Cause of the Spurious Double Lines sometimes seen with Spectroscopes, and of the Slender Appendages which accompany them. By G. JOHNSTONE STONEY, M.A., D.Sc., F.R.S.

Spurious double lines are sometimes seen with the spectroscope. In order to observe the phenomenon it is not necessary to use the complete spectroscope, not is monochromatic light essential. It is sufficient to look with the telescope of the instrument directly down the collimator, so as to see the image of the slit. If the instrument be now pointed towards, or nearly towards, a distant flame, and if the slit be narrowed down to a certain point, a spurious double line will be seen in the observing telescope, instead of a correct image of the slit. The phenomenon will be produced under the circumstances which most readily admit of investigation when the incident light is restricted to a single beam of plane waves, falling on the slit either normally or obliquely; and incident light which sufficiently approximates to being of this kind is easily provided by placing a coarse supplementary slit in front of the lamp flame and allowing only the light which passes through this slit to reach the collimator.

Under these circumstances it will be found that when the light falls normally on the slit of the spectroscope it will form an image in the observing telescope which, with a certain width of slit, becomes a rather coarse double line bordered on either side by an exceedingly fine hair-like appendage, which is visible only when the light is sufficiently intense. By causing the incident light to fall obliquely the two constituents of the double line may be made to thin down, leaving a considerable dark interval between them, and then present very much the appearance of the sodium lines when fine. In intermediate positions of the incident light the interval between the two constituents of the double line is occupied by a bright ruling of hair-like appendages, varying in number with the inclination of the incident light. The conditions of the experiment may be modified in other ways, and other appearances produced-notably a flare consisting of a ruling of bright lines fading out in one direction.