512 



NATURE 



[March 31, 1892 



old numbers came on the coasts they would probably die in our 

 waters for want of food. And this sustenance was evidently one 

 of the greatest delicacy. Full-grown pilchards have been known 

 to feed up to yielding from three to seven gallons of oil to the 

 hogshead of 3000 fish when having their fill of it. Their food 

 was young Crustacea, and evidently was the larval forms of some 

 crab or crabs which live on our coasts. 



I think a few words will make this plain ; in considering the 

 great crab — Cancer pagtirus — in the sea, the sexes stand in rela- 

 tion to each other of about one male to eight or ten females, the 

 latter spawning from one to two million eggs. These, when 

 hatched out, pass through several distinct larval changes in 

 the surface of the sea before dropping down on the sea bottom. 



Creatures having such vast procreaiive powers, when all the 

 conditions of life are favourable, must produce more young than 

 are wanted to make up for the wear and tear of the race ; hence 

 in our first outlook we seem in danger of having a plethora of 

 crabs. But Nature, true to herself, has a police force at hand 

 to prevent overcrowding. This is found in the pilchard, who 

 attacks the crabs in the surface of the sea when in their zoe 

 forms ; while at the sea bottom, if they are yet too plentiful, 

 those powerful skates {Raia batis and Raia lintea), with their 

 long, sharp, hard noses, make their appearance among them, 

 routing them out of their hiding-places among the rocks, and 

 with their powerful jaws and teeth making short work with these 

 crabs. 



Hence, in the olden times, when there was no demand for the 

 female crab, even at sixpence per dozen, and when they lay off 

 our coasts in millions, and again throwing off their countless 

 millions of eggs, there was seldom any lack of either large or 

 small pilchards in our bays in the summer months of the year. 



But since the extension of the railway systems throughout our 

 land, and the demand came for all the crabs, not only have the 

 large pilchards been scarcer, but they have so fallen off in con- 

 dition as not to yield above one gallon and a half of oil to the 

 hogshead, and the French sardine-sized fish has disappeared 

 altogether. 



It is certainly very satisfactory at this date to know that Mr. 

 Cunningham has found them in their new haunts further out at 

 sea ; and that he has also verified the facts of the size and the 

 ages of the pilchards given in my exhibits at the Falmouth 

 Polytechnic so long ago. Matthias Dunn. 



Mevagissey, Cornwall, March 22. 



On the Boltznmann-Maxwell Law of Partition of Kinetic 

 Energy. 



In the very valuable Report on Thermodynamics drawn up for 

 Section A of the British Association by Messrs. Bryan and 

 Larm^r, and now recently published, there is a remark upon the 

 Boltzmann-Maxwell law of partition of Kinetic Energy, upon 

 which I should like to be allowed to make a few comments. 

 The Report says, in fact, after noticing the attempts to extend 

 the theorem from the case, originally contemplated by Boltz- 

 mann, of molecules composed of discrete atoms under mutual 

 forces, to the general case of dynamical systems determined by 

 generalized co-ordinates: It has now been proved beyond doubt 

 that the theorem is not valid in this general form ; and quotes as 

 a test case a paper by Prof. Burnside to the Royal Society of 

 Edinburgh, on the collisions of elastic spheres, in which the 

 centre of mass is at a small distance, c, from the centre of 

 figure. In this paper, doubtless, results are arrived at, after a 

 vigorous and able treatment, inconsistent with the law now 

 under consideration ; but there is, I think, an oversight, pointed 

 out by Mr. Burbury in a paper recently read to the Royal 

 Society of London, which vitiates these conclusions and leaves 

 the matter where it was before. 



Prof. Burnside, in fact, has omitted to introduce the fre- 

 quency factor of collisions in proceeding to take his average, so 

 that, whether his result be correct or not, for the average of all 

 possible collisions, it is not correct for the average of all collisions 

 per unit of time, and it is this last which is important for the test 

 of permanence of distribution. 



When this frequency- factor is introduced and the approxima- 



tion carried, as in Prof. Burnside's paper, to the second power 



of c, it will be found, I believe, that - = _ and not ?, so that 



/C'l h h 



if this statement is correct, we are hereby furnished with a 

 confirmation of the Boltzmann-Maxwell law by an independent 

 treatment. 



The process is somewhat intricate, and too long for insertion 

 here. 



I should like to make a few additional remarks on a view 

 expressed by Prof. Burnside, which is doubtless widely, but I 

 think not quite reasonably, shared by many eminent mathe- 

 maticians, to whom this theorem of partition of Kinetic Energy 

 is a stumbling-block. 



He says, in the paper referred to, "Themethodof proof adopted 

 by Watson, following Boltzmann, is so vague as to defy criticism 

 or attempts at verification," but I really think the vagueness 

 consists in the generality of the conclusion and not in the method 

 of proof. To establish a proposition applicable to all conceivable 

 cases of collision, either in a field of no force, or of forces of any 

 kind, requires a method of proof which, whether true or false, 

 must of necessity be as general, or, if you please, as vague, as the 

 conclusion ; but, in point of fact, Boltzmann's method adapts 

 itself readily to every case which, like this of Prof. Burnside's, 

 admits of practical treatment. For example, in this very case 

 of the colliding spheres with centre of mass distance (c) from 

 that of figure, Boltzmann's method would assume that the 

 number of spheres with lines of centres in any direction, and 

 with component velocities of translation of C.G. and of angular 

 velocities round the principal axes lying between «, u -h du, 

 &c., &c., W3, W3 -f- (/coj, was 



<p{u, V, w, Wy, (11.2, wg) du . . . rtwj. 

 Suppose, then, the circumstances of the two spheres to be dis- 

 tinguished, as in Prof. Burnside's notation, by the great and 

 small letters, U, ti, &c., H, w, &c., and let the corresponding, 

 dashed letters denote these respective quantities after collision. 



Then, as proved in Prof. Burnside's paper, we have — 



„, ^ 2U + C^(K+k)\J-2CW ^^, ^2V + C^ + k)u + 2C-V . 



2 -f r^(K -H /&) ' " 2 -t- <:-^(K. -F k) 



- , 2irP U-«<-f-cxir , , 2cp \5-u-cw 



' ' A 2 + c\\^ + k)' ' ' A 2 + C-'([!i + ^) ' 



- , _ _ 2cQ \] -u + cw , _ , 2cq V -u-cw 



n' -o +2^1^ V-u + cm 



u 



p. Q. R2 



A B C' 



' C 2 + c\K + i) ' 

 where A, B, C are principal moments of inertia through C.G. % 

 P, Q, R, /, q, r are quantities depending on the relative situa- 

 tions of the principal axes, the line joining the centres of figure 

 and mass, and the line of centres at collision, and not affected 

 by that collision. 



-or = PXlj 4- Qn^ + RHj -f /wj 4- qca.-, ■\- rw^, 



ABC' 

 and the velocity of approach therefore equals \J - u ■{■ cvr. 



The Boltzmann method, therefore, would require, for the 

 permanent or special state, the condition that 



^{u . . W3)(^(U . . . n^) du . . . dn^ {U - tt + c-ar) 

 should be equal to 



(p{u' . . . a>'3)<|)(U' . . . n'-^) du' . . . dn'2 {\J'-t{' + c-ar'), 

 because, when this condition is satisfied, and only then, can the 

 average number of spheres with velocity components in the un- 

 dashed state and line of centres parallel to the x axis (which 

 may be any direction), be equal before and after collision, 

 inasmuch as those in ihe dashed state with velocities reversed 

 enter into the undashed state. 



In determining the multiple differential du' . . . dn,'^, we 

 may neglect the consideration of the resolved velocities in the 

 tangent plane, v, w, V, W, inasmuch as they are unaltered at 

 impact, and we have to evaluate the quantity — 



NO. TI70. VOL. 45] 



and so on for eight lines. 



