78 



NATURE 



[November 22, 1894 



The cogency of the evidence is admitted by every one who 

 takes the trouble to compare a few signatures together, and to 

 try making a few himself. I have taken thousands now in the 

 course of the last twenty years, and (bar smudges and accidents, 

 which are rarely bad enough to be fatal) I am prepared to 

 answer for the identity of every person whose " sign-manual" 

 I can now produce if 1 am confronted with him. 



As an instance of the value of the thing, I might suggest that 

 if Roger Tichborne had given his "sign-manual" on entering 

 the Army on any register, the whole Orton case would have 

 been knocked on the head in ten minutes by requiring Orton to 

 make his sign-manual alongside it for comparison. 



I send this specimen to you because I believe that identifica- 

 tion is by no means the unnecessary thing in jails which one 

 might presume it should be. I don'l think 1 need dilate on 

 that point. Here is the means of verifying the identity of every 

 man in jail with the man sentenced by the court, at any moment, 

 day or night. Call the number up and make him sign. If it is 

 he, it is he ; if not, he is exposed on the spot. Is No. 1302 

 really dead, and is that his corpse or a sham one? The corpse 

 has two fingers that will answer the question at once. Is this 

 man brought into jail the real Simon Pure sentenced by the 

 magistrate ? The sign-manual on the back of the magistrate's 

 warrant is there to testify, &c. 



For uses in other departments and transactions, especially 

 among illiterate people, it is available with such ease that I 

 quite think its general use would be a substantial contribution 

 towards public morality. Now that it is pretty well known 

 here, I do not believe the man lives who would dare to attempt 

 personation before the registrar here. The mitkhttars ' all 

 know the potency of the evidence too well. 



Will you kindly give the matter a little patient attention, and 

 then let me ask whether you would let me try it in other jails? 



The impressions will, I doubt not, explain themselves to you 

 without more words. I will say that perhaps in a small pro- 

 portion of the cases that might come to question the study of 

 the seals by an expert might be advisable, but that in most 

 cases any man of judgment giving his attention to it cannot fail 

 to pronounce right. I have never seen any two signatures 

 about which I remained in doubt after sufficient care. 



Kindly keep the specimens carefully. 



Yours sincerely, 



\V. IlERSCHEL. 



Boltzmann's Minimum Function. 



Mr. CrLVEKWELi., in his letter to Nature of October 25, 

 questions the existence of Boltzmann's minimum function, and 

 asks will somebody tell us what the 1 1 theorem really proves? 



As I have made use of the theorem on several occasions, I 

 may be permitted to say a word in its defence. I will en- 

 deavour to answer Mr. Culverwell's question what the theorem 

 proves for the simple case of equal elastic spheres. If I can do 

 thai, it will probably not be dilTicuU to generalise the result. 



Let then V, or OC in the figure, denote the velocity of the 

 common centre of gravity of two ehastic spheres, each having 

 diameter c Let k be their half relative velocity. If we de- 

 scribe a ipherical surface with radius K about centre C, and if 

 P/ be any diameter of it, the actual velocities of two spheres 

 are OP and O/. 



L;t the number per unit of volume of spheres whose velocities 

 arc represented by lines drawn from O to points wiihin the clement 

 of surface i/S at P be denoted by l'"(/.S. I^ct///S denote Ihc cor- 

 responding number fur the element i/.S at p. Then l-yi/S is the 

 number of pairs whose relative velocity K falls within the cone 

 I Alforneys, 



NO. I30S, VOL. 51] 



described with solid angle (/S about PC/* as axis. Let P'C^' 

 be any other diameter, and let FVS', f'J?> be the cor- 

 responding number of spheres with velocities OP' and 0/>'. 



If a pair of spheres collide the relative velocity assumes, as 

 the result of collision, a new position only, and what that 

 position shall be depends on the coordinates of the collision, 

 i.<. the point in which a line parallel to the relative velocity 

 through the centre of one sphere cuts a circular area of radius 

 c, drawn through the centre of the other sphere at right angles 

 to that line. If the collision coordinates be taken at random, 

 then the following condition holds, viz. : — For any given direc- 

 tion of R before collision, all directions after collision are equally 

 probable. Call that condition A. 



Now assume condition A to be fulfilled, and consider all 

 collisions which take pl.ice between pairs of the V R spheres. 



The number which after the collisions belong to the class 



F/i/S will be on the above assumption '11 Fy</S'. 



A-' J J 

 But before the collisions it is Fy;/S. Therefore, as the result 



of collisions it is increased by (/S 



://(Fy. 



'7f: - A 



4ir / 



That is 



constant. 

 Therefore 



F/)rfS', F/" being treated in the integration as 



and if 



dt 4ir 



_ ir^R 

 4' 



H = /"j>log/-i),/S, 



j j'dsfjiF'J-- F/)Iog(F/)rfS' 



f IjjdSdSWf - F/01og(F/) 



= '"■15 f 1' f fdSJS'(F/ - F'f) log ¥'/ by symmetry 



= 4 ''"^ ( /"{Fy - F/) log ^.'/S./S', 

 4'r J J r / 



which is necessarily negative if not zero. The above is true 

 for all values of V and R, and therefore for the whole system. 



Thus we have proved that if condition A be satisfied, then if 

 all directions of the relative velocity for given V are not now 

 equally likely, the effect of collisions is to make H diminish. 



The objection that I understand to be made is that if you 

 reverse all the velocities after collisions, the system will retrace 

 its course with 11 increasing — which is supposed to be contrary 

 to the thing proved. 



The objection is wrong because in your reverse motion con- 

 dition A is not fulfilled. The proof (is not wrong but) ceases 

 to be applicable by failure of the condition on which it is based. 



Somebody may perhaps say that by this explanation I save the 

 mathematics only by sacrificing the importance of the theorem, 

 because I must (it will be said) .admit that tliere are, after all, 

 as many cases in which H incre.ises as in which it diminishes. 

 I think the answer to this would be that any actual material 

 system receives disturbances from without, the cflect of which, 

 coming at haphazard, is to produce that very distribution of 

 coordinates which is required to make II diminish. So there 

 is a general tendency for II to diminish, although it may con- 

 ceivably increase in particular cases. Just as in matters 

 political, change for the belter is possible, but the tendency is for 

 all change to be from bad to worse. S. H. BURBURV. 



I New .Square, Lincoln's Inn, November 12. 



The Kinetic Theory of Gases. 

 I CANNOT quite agree in Mr. Bryan's remembrance of what 

 took place in the discussion of the Thermodynamics Report at 

 Oxford. As far as I recollect, Prof. Boltzmann's reply wits 

 not in special reference to such a point .as the specific heats of 

 gases, but was in answer to a very vigorous, if somewhat 

 general, onslaught of Prof Fitzgerald, who sim|)ly stated that 

 It appeared evident from the spectra of gases and other con- 

 siderations, that the energy could not be e(|ually divided among 

 all the degrees of freedom of the coirdinates, and said what he 

 w.intcd to know from Prof. Boltzmann was w/un the thtoiy 



