30 



NATURE 



[May 12, 1892 



parallel straight lines, A and B, and we introduce such a relation 

 between .vo,^^) -*. and^ as will express that when the former is 

 a point on the line A the latter will be a point on B, each of the 

 four quantities x, y, u, v may then be expressed as a function of 

 Xfi, yo, ^o» z'o> and it may be proved that 



V 



dx dy dti dv= _j-dxQ dy^ duo dvo ; 



where Vo and V are the resolved parts of the projectile's 

 velocity perpendicular to the two lines as it crosses A and B 

 respectively. 



For instance, let the lines be vertical x = a and x = b, 

 where b — a = c. Our equations are — 



X — Xq = c = 



2 



U — Uq 



.-./=_, v = v^-^ ,u = u^, x = x^^c,y =yQ + -2- -*-- 

 ' ■ ■ «o «o - «o 2Mo' 



and 



Also here / = — is no( constant, as it depends upon z^q. 



, Next let the lines be horizontal, y = a, y =■ b, b - a - c. We 

 then have 



(2) x -x^= u,,t 



(3) u = «/„ 



zf' 



(4) z' = Z'o 



From (i) 



^^gg--^^"'-^-, x = x 



i 



y = y^ + c, u = ti^ 



and our determinant A is 





VV - 2C^ 



•'H 



2Cg' 



O, I, 



o, o. 



V/V - 2Cg 



If our lines were y = mx and^ = inx + c, our additional 

 condition would be 



y -yo = m (x-Xq) + c ; 



and the result mentioned could be arrived at, although with a 

 little additional work. 



The actual problem proposed by Boltzmann is the same as this 

 in principle, although of much greater complexity, and it is 

 treated by him with the utmost generality. The important 

 thing here is to show that the S function with i constant is of 

 no application, inasmuch as in both of these very simple illustra- 

 tions we have t a dependent variable depending upon Uq or v^. 



I- am only pointing out that the S method, with i inde- 

 pendent, would not help to establish the particular proposition 

 to which I am referring. It may lead to the determination of 

 a law of permanence of distribution independently of this 

 proposition and by a simpler treatment. The Boltzmann treat- 

 ment, however, avoids the difficulty which may arise from the 

 fact that encounters, whether of finite or infinitely short dura- 

 tion, involve the assumption of discontinuous forces, and, 

 therefore, of a corresponding discontinuity in the form of the 

 S function. 



A little consideration shows that the condition E constant 

 cannot lead to any determinate relation between the differential 

 products dp-^ . . . dqn and dV^ . . . dQn. 



For to take again the simple case of the projectile. Here we 

 get four equations between the nine quantities, xq yo «o '^ot 

 X y u V and t, whence it is clear that the elimination of t 



does not enable us to arrive at more than three equations 

 between the remaining eight quantities, and therefore that we 

 cannot express x, y, u, v separately as determinate functions of 

 J'oJoWo^O' To enable us to do this we need one additional 

 condition, and this may be supplied in an infinite number of 

 ways. It may be one of the conditions above considered lead- 

 ing to the equation dx dy du dv = ^ " dxo dy^ du^ dv^, or it may 



be the condition t constant leading to the equality of these 

 differential products, and so forth ; but the condition E constant 

 supplies no additional relation between the eight variables. 

 This conclusion holds equally for n degrees of freedom, follow- 

 ing from the two partial differential equations in q^ . . . qn, 

 Qi , . . Q«, to which the characteristic function A is subject, so 

 that the condition E constant leads to no determinate relation 

 between the differential products. 



This conclusion is not inconsistent with Maxwell's proof. 

 That proof takes the form — 



dpi 



dqn 



A 



^'Q^ 



NO. 1 1 76, VOL. 46] 



where A is equal to A', but it may be proved that in this case 

 A and A' are separately zero, and therefore that, as stated 

 above, no relation can be established between the two differential 

 products, 



H. W, Watsox. 

 Berkeswell Rectory, Coventry. 



Palaeonictis in the American Lower Eocene. 



Tal/EONTOlogists will welcome Dr. T. L. Wortman's 

 discovery of a nearly complete skull of Palmonictis in the 

 Wahsatch Lower Eocene of Wyoming. The only specimens of 

 this form known hitherto are the two fragmentary lower jaws 

 from the Suessonian lignites of France upon which De Blain- 

 ville founded the genus in 1841. This specimen includes the 

 facial region of the skull and the complete lower jaws in fine 

 preservation. We owe it to the expert skill of Dr. Wortman, 

 for the fossil was found completely dissociated ; he carried 

 several sacks of the debris surrounding the fragments fifteen 

 miles to the nearest river, and by careful washing recovered all 

 the teeth. 



The skull is about the size and form of that of the Puma 

 (Felis concolor), without the long muzzle so characteristic of all 

 the early Carnivores or Creodonts. The dental series is remark- 

 ably compressed and reduced, especially in the upper jaw, the 

 formula being : I ^, C i, P |, M f. The third upper molar has 

 entirely disappeared, the second is as small as the little tuber- 

 cular in the modern cats, the first is smaller than the fourth 

 premolar. The latter tooth, in conjunction with the first true 

 lower molar, is in course of transformation into a sectorial. 

 This and many other features point to the conclusion that 

 PalcEonictis is closely related to the Eocene ancestors of the 

 Felidaa — which have hitherto been considered a gap in the 

 fossil series. 



The type, which we may call P. occidentalis, will soon be fully 

 figured and described. Henry F, Osborn, 



American Museum of Natural History, April 19. 



WATERSTON'S THEORY OF GASES. 



ON the nth of December, 1845, a paper by Mr, J, J. 

 Waterston, entitled " On the Physics of Media that 

 are composed of Free and Perfectly Elastic Molecules in a 

 State of Motion," was communicated by Captain Beaufort, 

 R.N., to the Royal Society, 



This paper was not published at the time, but was 

 relegated to the Archives. It now, however, has just been 

 issued as a part of the current volume of Philosophical 

 Transactions. 



It is preceded by an introduction by Lord Rayleigh, 

 one of the Secretaries of the Royal Society, and we can- 

 not do better — in order to call attention to this remarkable 

 paper, which anticipates the present theories in many 

 respects, and to explain how it is that it now appears — 

 than print Lord Rayleigh's introduction as it stands, and 

 also the introduction to the memoir itself. 



