January ii, 1894] 



NA rURE 



247 



and therefore I admit that if Mr. Bryan attaches his wheel and 

 windlass to my pistin of constant mass, we should get 



/' 



T 



for each complete turn of the wheel. Whether that equation 

 can be correctly said to express the second law, where there is 

 only one independent variable, is a question of definition of the 

 second law. 



I said, also, that if we are at liberty to vary the mass of the 

 pistin, we have two independent variables, but no longer a 

 conservative system. Mr. Bryan, with greater generality, 

 points out that the same effect would be produced by altering 

 the gravitation-potential. 



The objections to Clausius's proof generally cannot be more 

 forcibly stated than they are in the Report. What is required 

 is a definition of the time "?." The absence of that definition 

 is to my mind not only an objection, but a quite fatal objection. 

 If, as I proposed, we make 



v~ 



that answers the purpose for the very limited class of cases 

 in which __ 



ax = $^a-^. 



dv 

 A treatment of the subject in generalised coordinates is as 

 follows : — Let ^i . . . yi. be the unconstrainable, ^i . . . q^ 

 the controllable coordinates, concerning which latter I assume, 



as does Boltzmann, that q or J, and also i-f are to be ne- 

 ^ dt dt- 



glected. Let x be the potential, t the kinetic energy, T the 



mean value of t. Then we have generally 



aQ_ 



aio,T.i{a,,.(|-|),, 



dT 



In some cases the term does not appear, and therefore 

 dq 



in this class of cases 



9Q 

 T 



= a log T + 



T-(9X 



^P 



Now let/^/yi . . . dyn be the chance that in the stationary 

 motion with the qi. constant, the coordinates shall lie between 

 the limits ji and_;'i + dy\, &c., so that 



. xfdy^ 



and 



This makes 



dx 



dq 



dx 

 dq- 



fd}\ 



dyn 



dyn. 



¥'-i\ 



and 



aQ 



= a log T + j Y 



x^fdyx 



-^d/dy. 



dyn, 



dyn- 



In order that -;^ may be a complete differential, we must make 



/= <?*(^ ). where <p is an arbitrary function. That is the 



general solution. 



Now it will be found that this general solution agrees exactly 

 with that given by Mr. Nichols. For his condition is 



T \dv dv) 



dT 



dv 



Now the use of these averages necessarily implies the existence 

 of a function /, such that 



■ihe integrations being with proper coordinates, and therefore 



^X-^=/"x^^cr. 

 dv dv J dv 



NO. 1263, VOL. 49] 



Mr. Nichols' condition may therefore, without loss of its 

 generality, be put in the form 



J T dv J dv dT 



The general solution of this is 



y = .(!), 



as before 



That equation, 



seems to me to define the limits of the second law for all cases 



in which the term does not appear in Lagrange's equations. 

 dq 



I hope if Mr. Bryan comes to discuss the virial proof, he will 



give his opinion on this point. 



I dT 

 I think that the term - — , when it exists, can be made to 

 I dq 



lead to a somewhat similar condition. If, namely, F be the 

 chance of a given combination of the velocities, 



T dq JT dq J T dq 



:dFdff + a log T, 



J 1 



whence F must be a function of - -. 



= -/ 



I have to thank Mr. Bryan for reminding me of Mr. Nichols' 

 paper, which I had forgotten, though I had some discussion 

 with its author at the time. S. H. Bureury. 



The Fauna of the Victoria Regia Tank in the Botanical 

 Gardens. 



Prof. Lankester's account of the "Freshwater Medusa," in 

 Nature, December 7, 1893, shows how with very little trouble 

 the interests of zoologists may be served by those who have the 

 charge of botanical gardens like that in the Regent's Park. All 

 that is necessary is to refrain from periodically cleaning out the 

 tanks in which tropical water plants are grown. When these 

 latter are imported from abroad they often carry with them 

 various aquatic animals of novelty or interest like the medusa 

 mentioned. This particular tank has recently produced quite a 

 number of remarkable animals. Mr. Bousfield, some years 

 since, found certain new or little known species of I}e?v therein, 

 and more recently Prof. A. G. Bourne met with a new form of 

 the Naid genus Pristina in the same tank. I have been able, 

 thanks to the courtesy of Mr. Sowerby, to examine water and 

 decaying weed therefrom on more than one occasion, and I dis- 

 covered a series of rare or novel species of Oligochceta. The most 

 remarkable form was one which I described a year or two since as 

 a new genus Bratic/iiura ; this worm, with the general characters 

 of a Ttibifex, possesses a row of dorsal and ventral branchial 

 processes, besides showing other points of interest. In the same 

 sample of water were large quantities of a Naid, called by its 

 original describer, Prof. Bourne, who met with it in the town of 

 Calcutta, Chatobranchtis semperi. This worm has also a series 

 of branchiae, but they are lateral in position, and enclose the long 

 dorsal setae of the Annelid, thus suggesting the parapodia of the 

 marine Chastopods. I have also found the rare species 

 /Eolosoma nivejim in the same locality, and a freshwater 

 Nemertine ( ? Tetraste^-itma aqiiariun dulciwn), besides a num- 

 ber of Oligochaeta which I dia not at the time identify. 



Frank E. Beddard. 



! Rudimentary (Vestigial) Organs. 



Prof. Hartog's letter in your issue of the 28th ult. is inter- 

 esting as an illustration of the extreme danger of regarding an 

 organ as vestigial and functionless merely because our super- 

 ficial investigations have not revealed to us its use. 



If Prof. Hartog had examined the distal knob of the modern 

 eyeglass, even by the old-fashioned macroscopic methods, he 

 would have seen that there passes through a screw, whose 

 function is to hold the two sides of the oval frame together, and 

 thus retain the lens in place. 



The knob is also useful as a convenient point to lay hold of 



