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2. On Ronayne's Cubes. By Professor H. HENNESSY, F.R.S.

Some years since a box containing a pair of equal cubes was placed in the author's hands, and he found that one of these could have its parts displaced so as to leave a peculiarly shaped shell, through which the second cube passed without any difficulty. Groups of twin crystals of a cubical form have been long known, but their grouping could rarely admit of such a structure as the cubes referred to present.

It is manifest that a cube passed through another in the direction of the diagonal of the square would leave two triangular prisms, but in order to connect these prisms two flanges with interior sloping faces should be attached. The thickness of these flanges in two directions, as well as the angle of inclination of the sloping faces, are all connected by geometrical conditions which permit of the solution of the problem of the construction of the shell of the first cube. The author was unable to find the solution of the problem originally published by the inventor of the cubes, Mr. J. Ronayne, some time about the middle of the last century. Under these circumstances he completed the inquiry, and on comparing the results with the measured dimensions of the prisms and flanges which constitute the shell of the first cube he found the perfect concordance between the calculated and measured dimensions. When represents the distance from a corner of the cube to the edge of the flange, and a the side of the cube, then 6, the inclination of the sloping face to the face of the cube, is found to be represented by 2a (a/2-2.x) 3a2+4.22-4ax/2

sin 0=

The edge of each cube is 1.92 inch, and is found both by measurement and by the above formula to be 9° 45′ nearly.

3. On a Property of the Catenary. By Professor H. HENNESSY, F.R.S.

In the course of inquiry into some hydraulic questions the author found that the catenary of maximum area under a given perimeter may be inscribed in a semicircle. Hence, if the radius of the semicircle is one foot, the chain hung within it when in a vertical plane will be one yard. Thus the two fundamental standards of English measure are connected with the catenary of maximum area.

4. A Complete Solution of the Problem, To find a Conic with respect to which two given Conics shall be Reciprocal Polars.' By J. W. RUSSELL, M.A.

In the author's 'Elementary Treatise on Pure Geometry,' p. 147, a construction is given in the case in which the given conics intersect in distinct points. This construction was extended to the cases of the conics touching or having threepoint contact. The method in the case of double contact was different. Taking U to be the common pole and AB the common chord, through U draw any line meeting the conics in PP', QQ' respectively. Let X Y be the double points of the volution PQ, P'Q' or of PQ', P'Q. Then the conic required is the conic touching UA at A and UB at B, and passing through X or Y. This method holds, when suitably modified, in the case of four-point contact.

5. The Impossibility of Trigraphic Fields of Spaces.
By J. W. RUSSELL, M.A.

The simplest trigraphic form is that generated by three points P, P', P" on three lines AB, A'B', A"B", these points being connected by a relation of the

form

a ̧k' k' k' + a‚k' k" +

+a,k+ •,• +ag=0

1 Proc. Roy. Soc., vol. xliv. p. 106.

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 Lobatchelsky, dans un article inséré dans la livraison de février 1893 des 'Nouvelles Annales de Mathématiques ' (3e série, t. xii, pp. 74-84).

On peut démontrer d'une manière analogue les principes fondamentaux de la géométrie non-euclidienne de Riemann, en partant 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).

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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 z tend vers zéro, l'angle (z, a) 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. THÉORÈME 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=x, AD=x+y, OA = A. Menons Bb, Cc, Dd perpendiculaires à OA et rencontrant Oa en b, c, d; de b et d abaissons bm, dn perpendiculaires sur Cc. Soit encore BB' = B'b' perpendiculaire à OA et rencontrant Oa en b', b'm' une perpendiculaire

à Cc.

= a,

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

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4° On a

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

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

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$ (x−y−a)+(x + y) > 24 (x)
(y+a) φ (ν)

5° Faisons tendre a vers zéro, il viendra

+(x − y) + P(x + y) = 24(x).
(y) φ (y)

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

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

1° On a (a) = cos

().

k étant une constante.

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

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3o Dans un triangle ABC quelconque,

cos (cos(cos()+ (sin()
(4) cos (4) + sin (4) sin (;) cos C,

COS

COS

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 (A, A1, A2,... An) (x, y)"

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da

+ (n−1) α2 da1

+

respectively, then, P (a) denoting a rational integral homogeneous isobaric function

of a。, a1, a2, . . . an, whose order and weight are i and w,

...

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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

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