Jan. 2 1, 1886] 



NATURE 



271 



responding degrees of freedom of the same or of other 

 particles. For instance, Maxwell's Theorem itself is not 

 proved if the spheres of one system have not as perfect 

 freedom for mutual collisions as for collisions with those 

 of the other system. We are not entitled to assume that 

 they can then acquire, much less that they can maintain, the 

 "special ■' state on which the further argument is based. 

 This is one of the traps into which Clerk-Maxwell fell ; 

 for he assumed that the result could be deduced from the 

 consequences of a sort of typical impact between two 

 particles, one from each system, moving in directions at 

 right angles to one another, and each having the mean- 

 square speed of its own system. 



Let the masses of two impinging spheres, whose co- 

 efficient of restitution is unity, be P and Q ; and let u 

 and V, measured towards the same parts, be the com- 

 ponents of their velocities along the line of centres at 

 impact. Let these become, after impact, ic and v' . Then 

 we have, as in the text-books, 



P{u' - ti) 





Qiy' 



P{u"^ 



-, {Pu- - Q-"- - (^ - Q)uv) 



which gives, at once, 



IP+Q) 

 = - Q(v''^ - V-). 

 Each of these equal quantities is double the amount of 

 energy transferred from one sphere to the other. 



Now, when kinetic equilibrium has been (at least 

 approximately) arrived at, such transference must (on 

 the average) cease : — so that the equilibrium condition 

 will be 



Pi? - QV' - [P - Q)Tv = o, 



where the bars indicate average values. 



Everything turns on the proper estimation of these 

 averages For, if the average of uv be taken as zero, we 

 have Clerk-Maxwell's result ; provided that Ptr and Q'n^ 

 be proportional to the average energy of a /" and a Q 

 respectively. This is a comparatively obvious trap. 



But if we consider that collisions are more likely to 

 occur between two particles, h3.\'ing ^'iz'ea speeds, if they 

 be moving towards opposite parts than if towards the 

 same parts, we see that, on the average, te and v are 

 more likely to have unhke, than like, signs ; and 

 therefore that the value of uz' is negative. It is not 

 so easy to see, beforehand, what sort of changes 

 this consideration may produce in the values of u- 

 and v^. 



This leads to an inquiry as to the relation between the 

 relative speed of two particles and the probability of their 

 collision, and the formula become complicated. 



I found, by an approximate investigation in which the 

 above consideration was given effect to, that, if the aver- 

 age energies of a /" and a Q be called, as usual, 3/^11-/2 

 and 2>Qfi-l2, we have, nearly, 



H- = d-;2, v- = li-JT. ; itv = - Cii!i ; 



where e depends only on the relative magnitudes of a 

 and /3. If this were true, it would follow at once that the 

 average energy per sphere would be less for those of 

 greater mass.' 



But I soon found that at least part of this must be 

 erroneous, because though many of its consequences 

 would require a mere modification of the mode 0/ stating 

 certain well-known theorems, others were incompatible 

 with physical principles. 



Yet it seemed (and this is a specially good instance of 



' This conclusion, after I had seen it to be erroneous, and had taken 

 timely precautions, sufficient (as I thought) to prevent its appearing in 

 Nature, was unfortunately published as a definitely-ascertained fact : 



the pitfalls I ha-,c r.' ,iJed to) hardly possible that, as uv 

 is certainly negative, we could get Pa- - 23' = o for 

 the form of the a' ove expression, except when P = Q. 



When I revised my calculations, dispensing with 

 incthods of approximation, I found that, strange as it 

 appears, the average value of u{u — v), the P part of 

 the above expression, depends on a only, and does not 

 involve ! Its value is 2a-/}, of which 



li^ = ff-+^a^ and - P, = "'■^' . 

 6(«^ + /3^') 6(a^,+ /3-') 



If the above result, which has been obtained by the 

 evaluation of a number of troublesome definite integrals, 

 be correct, there must be some very direct and simple 

 proof that u{ii — v) depends on a only. 



REPORT TO THE TRINITY HOUSE ON 

 J HE INQUIRY INTO THE RELATIVE 



MERITS OF ELECTRICITY, GAS, AND OIL 



AS LIGHTHOUSE ILLUMINANTS 

 T^ HE Committee appointed by the Trinity House to 



^ report on the merits of electricity, gas, and mineral 

 oil as lighthouse illuminants have recently issued a 

 valuable report giving an account of the investigations 

 carried out under their directions, and the conclusions 

 they have arrived at. The Committee consisted of 

 Elder Brethren of the Trinity House. They were assisted 

 by Air. A. Vernon Harcourt, who v/as appointed by the 

 Board of Trade to cooperate with the Committee, and 

 by Prof. W. Grylls Adams and Mr. Harold Dixon, in the 

 more purely scientific part of their investigation. 



Three temporary lighthouses were erected on the 

 South Foreland, and fitted up for electricity, gas, and 

 mineral oil ; the optical arrangements were "multiform" 

 in all three — that is, consisted of several similar sources 

 of light, each with its own condensing lenses, super- 

 posed ; in the case of the electrical tower there were 

 three superposed lamps, as was also the case with the oil 

 tower : but in the gas tower there were four lamps ; the 

 two former were therefore " triform," whereas the latter 

 was a " quadriform " light." Any one lamp in either 

 tower could be lighted independently of the others, so as, 

 for instance, to permit biform electricity to be compared 

 with triform oil and quadriform gas. 



The lamps for the electric light, and the magneto- 

 electric machines for working them, were suppHed by M. 

 de Meritens ; the gas apparatus was that of iVIr. Wigham, 

 each burner consisting of loS jets in concentric rings, of 

 which a part only might be employed ; the oil lamps in 

 the third tower during the greater part of the trials were 

 six-wicked Douglass pattern, but burners of this descrip- 

 tion with seven and eight concentric wdcks were also 

 tried at various times during the progress of the experi- 

 ments. 



In addition to the temporary lighthouses, three observ- 

 ing huts and a photometric gallery 380 feet long were 

 erected. 



The actual observations that were made may be 

 divided into two classes — eye estimations, and photo- 

 inetric measurements. The former were made by the 

 Elder Brethren, by officers on board the light-vessels in 

 the neighbourhood, by merchant officers in passing ships, 

 and by the coastguard officers at those stations from 

 which the lights were visible. These eye- observations 

 were of two kinds : — (i) Estimations of the comparative 

 brilliancy of the lights ; (2) definite statements as to the 

 various distances at which the lights were visible in hazy 

 or foggy weather. 



With reference to observations of the first kind, they 

 were conducted in accordance with regulations issued by 

 the Trinity House Committee : the observers were in- 

 structed in filling in the books of forms which were issued 

 to them, to put down in one column the light from the 



