TRANSACTIONS OF SECTION A. 579 



where rt„fl, . . . are constants, and A; = AP/PB, A;' = A'P'/P'B, A;" = A"P"/P"B". 

 Starting from this form, various trigrapliic 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 A B C D, A' B' C D', A" B"'C" D" as corresponding tetra- 

 hedrons in the assumed trigraphic figures, and suppose that the variable points 

 P, P', P" of the figures subtend trigraphic pencils at the axes (BU, CA, AB), 

 (B'C, CA', A'B'), (_B"C", C"A", A"B'0 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 MaxivelVs Metliod 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 Max- 

 well 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 hydrodvnamical 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 lew words the results of his calculations, in three short nott^s, 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 manu- 

 script 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, E.A., F.R.S. 



8. Principes fondamentaux de la Geometrie non-euclidienne de Riemann, 

 Par P. Mansion, j^rofesseur ct I'Universife de Gand. 



I. M. Gerard a expose, sous une forme simple et rigourense, les principes 

 fondamentaux de la gt^om^ti'ie non-euclidienne de Lobatchelsky, dans un article 

 ins6r(5 dans la livraison de f(5vrier 1893 des 'Nouvelles Annales de Math^matiques ' 

 (3' s6rie, t. sii , pp. 74-84). 



On peut d^moutrer d'une maniere anahigue les principes fondamentaux de la 

 g6om6trie non-euclidienne de Riemann, en partant dea deux propriet^s fondamen- 



fp2 



