26; 



NATURE 



[January io, 1895 I 



count of the meeting, in September last, at Vienna, of the 

 German Malh<;iiialical Association. The Bullelin well main- 

 tains its position, and closes with its useful lists of new 

 publications. 



SOCrETIES AND ACADEMIES. 



London. 



Mathematical Society, Dec. 13.1894. — Afajor Macmahon, 

 F. R. S. , rrcsuient, aid suhsequcnily NIr. X. E. H. Love, 

 F. U.S., Vice-rresidcnl, in ihe chair. — The following cummuiii 

 caliiins were made: — On .MaxAell's law of partition of energy, 

 by Mr. G. H. Bryan. In liis recent report to the British Asso- 

 ciation, ihc author had shown ihai if a large number of dynamical 

 systems of any kind be taken, all similar in every respect, it is 

 always posMble to distribute their co 'rdinates and momenta so 

 that the distribution shall remain permanent, and shall satisfy 

 Maxwell's law of partition of energy. By this is meant that if 

 the kinetic energy of each system be reduced to a sum of 

 squaret:, the mean values o( these squares are equal. But the 

 a ilhor had duuhted whether it was possible in any case to inlcr 

 thai the lime averages ol the squares forming the kinetic energy 

 of a single S)stein were equal. In the preent p^per the con- 

 nection lie' ween lime azera^es, and av rages taken over a lar^^e 

 nuj'tber of diffireni systems, is examined more tully by means of 

 an artifice sugi;ested by Prof. Uolizmann's paper "O.i the 

 applicalim of the determinantal relation to the theory of poly- 

 atomic gases" (publi-hed as an appendix to Mr. Br)an's report). 

 Instead ol a vessel containing gas (as taken by Prof. Bolizmann), 

 any single dynamical system is taken whose coordinates and mo- 

 menta return 10 their oiiginal values after a long lime T. If the 

 time be divided into an infinitely laige number («) of infinitely 

 short intervals (;), we can derive a stationary dislribulion by 

 taking » systems and staiting them, the first at time o, the 

 second at time ;, the third at time 2i, and so on, giving e%'ery 

 system the same coordinates and momenta at the time of 

 starting it. At ihe end of the time T, we shall have the sy.slems 

 dislnbuiC'l according to a permanent or stationary law, and at 

 any sub-cquent instant the mean value of any function of the 

 coordinates and momenta lor the different systems will be equal 

 to the time average of the corresponrling func ion for the original 

 single s)Slem. If, however, we ilarl with a number of systems 

 distribu'ed according to a permanent law, we eaiinH pass back 

 to ihe original single system unless we can show that Ihe law of 

 fermancnt distribution is unique. Now in any simple lest case, 

 such as that afforded by rigid bodies movatile about fixed 

 points or particles moving in planes after the manner of a 

 Lissajou'., curve tracer, a stationary distrihution exists satisfying 

 Maxikell's Law of Partition, but other stationary distiil.uiicms 

 are possible which do not satisfy the law. lUnce the author 

 concludes that the time averages of the squares into which the 

 kinetic energy of a single system can be divided, are not in 

 general equal, at any rale inctependenily of initial circum- 

 stances. — Ihe Spherical Catenary, by Prof. A. G. Gieeiihill, 

 F.k.S. The pseudo. elliptic integrals developed in the I'roc. 

 L.M..S. XXV. are applied in this p.iper to the construction of 

 solvable cases of the spherical catenary, given by the relation 



where 



7. - (t - 2'')U - hf - A- 



connecting i|i the azimuih, and z = cos 9, where 9 denotes the 

 angular distance from the lowest point of the sphere, the tension 

 being w{i - //) (Clebsch, Cretle, 57). Putting 



and 



X = * - / 



' = /■ 



dz 



/(' - 

 (I - ■■ 



•■) s/Z 



then X can be idenllficd with the standard form of the pseudo- 

 elliptic of the third kind, of or ler fi, 



_ ,fp(s - a) - MS/C - 2) 



-'/ 



(J - ff) V S 



NO. 13 I 5, VOL. 51] 



'</', 



where 



S = 4t(j + .v)'- - [{y -f i)r + .(^'l- 



by putting 



X = I. / = if^ + aY A = M()' -1- I), h"- = .\^- 2j' - I, 

 ^ V M ' 



where 



W = - -'L±J. 



2X 



and.j-, ^ are Ilalphen's functions, defined in his " Fonctions 

 lilliptiques," I. p. 102. If « = a when : = i, and « = b 

 when c = - 1, then u = ^ [a - b) when : = h ; a and /' are 

 eich of the form/oij, a traction of the imaginary period ojj. 

 Also 



M'p{a + i) =. - i(, - h^) - JA^ 

 or I2p(a + i) = S.v— (j/ + 1)-, so that <r = o, and 



a + b = f"-». 



Thus, for instance, when 



a + /' = 1B3, A = A- - I , / = i A ; 

 and the corresponding spherical catenary is given by 



\' 2 



With 

 and 



With 



\' 



VI- :^" + (/'- 0^ + 



■ i,a + b - Sttij, K- = h- ' I, / = ''A ; 

 z")'^'^' = A(!' - //c= - 2;) + i(hz + O^'Z. 



then y ~ x = - c, suppose ; 



and 



(I - z-)-e'^ 



where 



With 



M= = 



Hr'' -I- 



-,/>= !'3 - '■)\I 



+ H. + i(L:' + . . .+Lj)^/Z, 



II 



_ 2 - 5c -f f 



M, L = 



^h, &c. 



o -f ^ = ~ai^ 

 2 



ihe calculation is rather more complicated, as this case must be 

 derived from ^ = 8 ; but the result is of the form 



Jx> 



.{Hz- H,)n'(S3-*- = - 



with 



+ «XLs - L,)x/(s 



s,> 3>r,>Ji>«> «>3|ii 



»0' *1> *!!■ *J 1 



denoting the roots of the quartic Z = o. — The Trnnsformaiioo 

 of FJIiptic Functions, by Prof. .\. G. Greenhill, F.K.S. The 

 function z., introduced by Prof. Klein in his paper on the i 

 transform.ition of elliptic functions (/V(V. I. M.S. xi. p. JSOi ' 

 and ileveloped in Klein and Fricke's " M.Mhillunctionen," il. 

 part v., is shown here to be connected with lialphcn's7 fuiiclion 

 liy the relation 



p[ - lYz. = \»y., 

 for a lr.in-furination of the Kth order ; and for an odd value of «, 



p. 



ti 



U; 

 tflll 



It 



kt 



y„ •f . t 



AV+I = ■-■ , / = I, 2, 3. 



■yn+j/ - 1 



;; - I 

 2 



in Ibis manner the relation 7,1 = o is satisfied. 



The biquadratic relations satisfied by the function : are 

 now derived from Ilalphen's formula 



Ym + nYm-.i = 7m ) l7>n-l'>'"" ~ •)"+ Tn-lT" 



