358 



NA TURE 



[August 13, 1891 



After a sufficiently great number of crossings and re-crossings 

 across the line X'OX, the particle will cross this line very nearly 

 at right angles, at some point, N'. Vary the position of N very 

 slightly in one direction or other, and re-project in from it per- 

 pendicularly and with proper velocity ; till (by proper " trial 

 and error" method) a path is found, which, after stiil the same 

 number of crossings and re-crossings, crosses exactly at right 

 angles at a point N", very near the point N', Let m continue 

 its journey along this path, and, after just as many more cross- 

 ings and re-crossings, it will return exactly to N, and cross OX 

 there, exactly at right angles. Thus the path from N to N" is 

 exactly half an orbit, and from N" to N the remaining half. 



(14) When <;-E/(o-/3'-) is a small numeric, the part of the kinetic 

 energy expressed by \cx'-y" is very small in comparison with 

 the total energy, E. Hence the path is at every time very 

 nearly the resultant of the two primary fundamental modes 

 formulated in § 13 ; and an interesting problem is presented, to 

 •find (by the method of the " variation of parameters") a, e, b,f, 

 slowly varying functions of/, such that 



X = a sin [at-e), 

 X = aa cos {a( - c), 



y = b sin {&t -/), 

 y = l>&co:.{0t-/), 



shall be the rigorous solution, or a practical approximation to 

 it. Careful consideration of possibilities in respect to this case 

 '[cE/{a^0') very small] seems thoroughly to confirm Maxwell's 

 fundameatal assumption quoted in § 11 ; and that it is correct 

 whether cE/{a-0-) be small or Ir^rge seems exceedingly probable, 

 or quite certain. 



(15) But it seems also probable that Maxwell's conclusion, 

 which for the case of a material point moving in a plane i = 



Time-av. x- = Time-av. j)"^, (i) 



is not true when a^ differs from ,8-. It is certainly not proved. 

 No dynamical principle except the equation of energy, 



i(i-2 + f-) = E - V (2) 



is brought into the mathematical work of pp. 722-25, which is 

 given bySMaxwell as proof for it. Hence any arbitrarily drawn 

 curve might be assumed for the path without violating the 

 dynamics which enters into Maxwell's investigation ; and we 

 may draw curves for the path such as to satisfy (i), and curves 

 not satisfying (i), but all traversing the whole space within the 

 bounding curve 



Ma'x" -1- &y + cxY) = E, 



(3) 



and all satisfying Maxwell's fundamental assumption (§ 11). 



(i6) The meaning of the question is illustrated by reducing it 

 to a purely geometrical question regarding the path, thus: — 

 Calling e the inclination to x of the tangent to the path at any 

 point xy, and t/ the velocity in the path, we have 



i: = ^ cos e, y = t/ sia 9, (4) 



and therefore, by (2), 



^ = V{2(E - V)} (5) 



Hence, if we call s the total length of curve travelled, 



fx\lt = j q cos2 Qqdt =^ f V{2(E - V)} cos- e els ; . (6) 



and the question of § 15 becomes, Is or is not 



i- r<ls ^{2{E - V)} cos-e 



y j 



= i^afs V{2(E - V)} sin2 6? . . . (7) 



where S denotes so great a length of path that it has passed a 

 great number of times very near to every point within the 

 boundary (3), very nearly in every direction. 



(17) Consider now separately the parts of the two members of 

 (7) derived from portions of the path which cross an infinitesimal 

 area da- having its centre at {x, y). They are respectively 



and 



V{2(E 



n/{2(E • 



V)]d(r f Nr/ecos^ e 

 J 



V)} da- r^de sin- 6 

 J 



(8) 



where Nr/(J denotes the number of portions of th2 path, per unit 

 distance in the direction inclined ^-ir + tox, which pass either- 

 wards across the area in directions inclined to x at angles between 



NO. I 137, VOL. 44] 



the values - ^dd and + y0. The most general possible 

 expression for N is, according to Fourier, 



N = A(, + Aj cos 20 + A., cos ^0 + &c. 1 , , 



+ B] sin 20 + B3 sin 4^ + Sic.]' ' " ^^' 



Hence the two members of (8) become respectively 



V!2(E - Y)}d^in{A, -f 4Ai) ^ 

 and '- 



and 



(II) 



■ (10) 



V!2(E - VjjY/crKAo - iAi) J 

 Remarking that A,, and Aj are functions of x, y, and taking 

 d(T — dxdy, we find, from (10), for the two totals of (7) re- 

 spectively 



hnj jdxdy{A„ + 4Ai)V[2(E - V)]] 



hnjjdxdyiA, - 4Ai)V[2(E - V)]j ' 



where / / dxdy denotes integration over the whole space in- 

 closed by (3). These quantities are equal if and only if 

 1 / dxdyA^ vanishes ; it does so, clearly, if a = i8 ; but it 



seems improbable that, except when a = B, it can vanish gener- 

 ally ; and unless it does so, our present test case would disprove 

 the Boltzmann- Maxwell general doctrine. 



THE INTERNATIONAL GEOGRAPHICAL 

 CONGRESS A T BERNE. 

 'T^HIS Congress began its proceedings on Monday. Fourteen 

 -^ countries and forty-six Geographical Societies are officially 

 represented. France has sent 73 delegates, Germany 33, Aus- 

 tria-Hungary 21, Switzerland 87, Italy 21, Russia 13, Great 

 Britain 8, and Spain, America, and the Netherlands two each. 

 Egypt, Portugal, Roumania, Greece, Norway, and Sweden are 

 also represented. There are, in addition, 150 Members and 

 Associates who have not yet given in their names. 



M. Numa Droz, Swiss Minister for Foreign Affairs, bade the 

 delegates heartily welcome to Berne. 



Dr. Gobat, Regierungsrath, Berne, President of the Congress, 

 then delivered his inaugural address. In the name of the 

 Geographical Societies of Switzerland he thanked the savants 

 present for responding so cordially to their invitation. 



Among the good work already done. Prof. Penck, of Vienna, 

 has proposed the following resolution :— " This Congress on the 

 geographical sciences, held at Berne, resolves to take the initia- 

 tive in the preparation of a large map of the earth on a scale 

 of one to a million, of which the various sections shall be de- 

 limited by latitudes and longitudes ; and, with this object, it 

 appoints an international committee to determine the principles 

 upon which the preparation of such map shall proceed. The 

 members of this committee shall arrange that the various States 

 engaged in preparing maps, the societies and periodicals pub- 

 lishing original maps, and all private geographical establishments 

 working in this field shall prepare detached sections of the said 

 map, the sale of which shall also be regulated and arranged for 

 by the committee." 



In the course of his address on the subject Prof. Penck 

 paid a high tribute to the services rendered by Mr. Stanley to 

 the cause of geographical science, directing special attention to 

 the fact that each of the explorer's expeditions across Africa had 

 led to the preparation of from 20 to 30 maps. 



The proposal was referred to a committee of the Congress, 

 which will report upon it. 



The subjects of an initial meridian and universal time, geo- 

 graphical educition, orthography of geographical names, lakes 

 and glaciers, cartography, bibliography, meteorology, com- 

 mercial geography, and voyages and travels are all to be 

 touched upon in the deliberations. 



SCIENTIFIC SERIALS. 



Journal of the Jiussian Chemical and Physical Society, vol. 

 xxiii., No. I. — The chief papers are : — On the molecular weight 

 of albumen, by A. Sabaneeff and N. Alexandroff. Several 

 determinations were made on the method of Raoult, and gave 

 an average of 14,276, the molecular weight thus appearing to be 

 nearly three times as great as that deduced from the formula of 



