568 



NA TURE 



[October io, 1895 



LETTERS TO THE EDITOR. 



[ The Editor dots not hold hinndf responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertaie 

 to return, or to corresportd with the writers of, rejected 

 manuscripts intended for this or any other part of NATURE. 

 No notice is taken of anonymous communications.'^ 



Clausius' Virial Theorem. 



The question raised by Colonel Base\-i, in Nature for August 

 J9, illustrates the importance of keeping in riew a clear state- 

 ment of what a general theorem such as that of Clausius with 

 lres])ect to the viria! asserts, and the essential relativity of the 

 forces which are regarded as acting on the particles, and of the 

 kinetic energy of the system. The theorem asserts, I think, 

 that if the motion of the system of jiarticles be continued over 

 any inter\al of time, /,, the excess of the mean value of the 

 kinetic energy of the s)-stem for that interval of time over the 

 virial for the same interval is equal to the excess of the value of 



_L5/«^[£li at the end of the interval over its value at the be- 

 4/, di 



ginning, p being the distance of a specimen panicle from the 

 origin and m its mass, and the summation being extended over 

 all the particles of the system. 



It may be noticed here that the mean value of the kinetic 

 energy of a system for an interval of time /, is equal to the 

 action of the system for that interval taken per unit of the time 

 in the interval. 



There can be no doubt that the theorem is true, and will be 

 verified by any lest case to which it can \x applied. The i)roof 

 given by Clausius himself is perhaps the simplest, but the follow- 

 ing mode of arriving at the theorem is instructive in some ways. 

 Refer the jjarticles to a system of rectangular axes in the ordinary 

 way, and adopt the fluxional notation for velocities and accelera- 

 tions. Thus taking a specimen [article, which is at the point 

 X, Y, :, at time /, regarding, as we arc at liberty to do, the 

 velocities x, y, I, as functions of the position of the particle in 

 the motion, we have 



V dx -"dy dzj 



and two other equations for V, Z, which can be written down 

 from this by symmetry. Multiplying these equations by x, y, z 

 respectively, adding, and rearranging, we easily find 



'"(r- + r" + i-yt = - i(xx + \y + zAdt 



+ ^d(xS+yy + =:y 



Integrated from / = o to t = 1^, and extended to all the 

 particles, this gives 



i5/H r'(.*5 +f + -'')<'' = - i^('\'^-f + Vj' + Z:)dt 



+ ir2;«(j:i: + yy + ;J)"|''. 



The expression on the left [which may be written 



T.m{[idx +ydy + irfs)] 



is nowhere asserted, so far as I know, to be kinetic energy, but 

 is the lime-integral of the kinetic energy (that is the action of the 

 system) for the time-interval /,. Dividing both sides by /, we 

 get the theorem as stated above, namely 



where T denotes the kinetic energy of the system at the in- 

 Mant /. 



It is clear that if /, Ix- taken very great, and the velocity and 

 the cli.itancc of each particle from the origin be always finite, 



ihi ' • 'he left is neither Infinite nor M;ro, while the last term 



oti comes v;inishin(;ly small. Tnis Is Clausius' case 



lit TV motion," in which it Is justifiable to write 



NO. 1354. VOL. 52] 



The expression on the right is the viria/, and is in the circum- 

 stances stated undoubtedly equal to the time average or mean 

 value of the kinetic energy, as the equation asserts. 



If R l)e the force acting on a |\arlicle In the direction tiKvards 

 the origin along the line joining the origin w ith the particle, and 

 p the distance of the larticlc from the origin, we have 



Xj + Yy +'/.:■= - Rp, 



and the theorem for stationary motion may be stated thus. 



Mean value of T = mean value of ^SRp, 



where the summation takes in each particle once, and onuvonry; 



Let us apply this to the case taken by Lord Kayleigh, and 

 alleged by Colonel Basevi to contradict the theory, of !«■> 

 particles each of mass m, at a distance ajMrt r( = 2p), revolving 

 round their common centre of gravity. Here, taking the origin 

 at the common centre of gravity, we have constant values of the 

 virial and of T, namely i2Rp = Rp and T = ;«\ -. Thus, 

 «;V-/p = R. which, as Lord Rayleigh remarks, agrees with the 

 law of centrifugal force. 



If we lake the motion relatively to one of the two i5articles 

 regarded as at rest, we ijel the same result. The relative velo- 

 city of the other jmrticle becomes 2\', and the corresponding 

 kinetic energy 2m\'-, the distance of the origin from the other 

 particle 2p, and from itself zero. Since the acceleration of the 

 moving |>arllcle relatively to the i>artlcle now supposed reduced 

 It) rest, Is double Its acceleration relatively to the common centre 

 of gravity, the force now^ considered as acting on the moving 

 particle must be taken as 2R. Thus we have2///V- = A2R x 2p, 

 or as before, m\"'lp = R. 



If we do not suppose the origin 10 coincide with one of the 

 |Xirliclcs reduced to rest In this manner, but to coincide for the 

 moment with ihe position of one of the particles, the velocity of 

 each parllcle is V, the force towards the origin on that distant 

 from it r is R, and we have T = ///\'-, i5Rp = fiKr, since now 

 p - r. Hence once more m\'-/p = R. 



Similarly, any other origin and axes of reference would give 

 the same result. Colonel Bascvl has. It seems to me, overlooked 

 the fact that In the theorem it Is the forces acting on e.ach 

 l)article relatively to the assumed axes, and the corresponding 

 motions that must be taken into account, and that In the case of 

 a system of ijarticlcs between which exist forces of mutual 

 attraction, the stress between a given pair can only enter once 

 into the value of i2Rr. -V. CiKAV. 



Bangor, September i. 



I THINK the fort will not surrender at Colonel Basevl's 

 summons. We have 



df dx\ d'x^ ./'«'■'■ V. 



"'dt[-'-dt) = "''dt^'-"\dt) ' 



and if we put .v = u and -^ = v, this may be written 



d, . dv , du 



"'rf/""' = '""7/ + ""',// 

 and 



if you please so to write it. This corresponds to Colonel 1 

 Bascvi's equation, except that I have written v for his .v. 



But now m frdu, or m jv "dl, docs represent kinetic energy.. 



And - /// / 'udv or - /« / a V/ Is the virial. The equation 



J » J i> dt- 



shows that if for a certain lime /, the right-hand member, 

 vanishes, then on the average of that lime /, the two terms on 

 Ihe righ'. are eijual and opposite. 



The form 2K/- Is a rather slippery one. If In the example 

 which Cohmel Basevi cpiotes from Lord Rayleigh, you put 

 X.V + \y for R;-, it comes out easily, l-or we may take for 

 origin the centre of the circle of radius p. Then 



X = •-/ V =-1' /•■""' ^'^ + ^>=/P- 

 P P 



And therefore 



2jwz" = ii»ifp, 

 or 



f- 



S. 11. BURUUKV. 



