( 394 ) 



llial if llic miinbei' is lar<:>e, and Ihe i)laiiots do nol iiiierfbre witli 

 each otlier, in llic long- nin (lie [>lanels will most likelv get about 

 nnifonnly scattered over the ecliptic. 



If' we shonld wish to treat the problem iji exactly the same waj 

 as (xiBiJs, we shoidd have to consider an ensemble of systems each 

 consisting oi' n ])lanels. As, however, the planets do not interfere with 

 each otlier, we may also take an ensemble of systems of only one 

 planet, in which case the ensemble represents all j)ossibi]ilies which 

 may occiu- in the placing of a })lanet. When now such an ensemble, 

 satisfying cerlain sim|>le coiidilions, gradnally spreads uniformly over 

 the ecliptic, there is foi- evei-y planet chosen at random from this 

 enseml)le, linally an e(pial chance to any place of the ecliptic, so 

 that, if we have to choose a planet from such an ensemble w times, 

 they will most probably l>e distributed about uniformly over the 

 ecliptic, if // is lai'ge. 



It is assumed that the orbits are circular, and He in the plane of 

 the ecliptic, so that c\'ery planet is determined by the variables 

 / (length) and m (angular velocity), in which vj is constant, and 

 I z= I^ -\- o)f, if also larger angles than 2.t are admitted. The function 



;S' (Poincark's entropie tine) is, accordingly, here t i P loa rjliho 



integrated over all the phases. 



As A\ convspoudiiig to / is 0(|ual to <f), and A', cori'es|»onding 



^-» ()A'/ óio 

 to (o is e(|ual to 0, Ikm-c > ^ -=-= 0, so the function ,S' I'emains 



.— «^ d.c,- ()/ 



constant. 



Vet the ensemble a|)proachcs uniform distribution ovei' the ecli|itic, 



which, however, is an altogether ditlei-ent thing IVom the density J* 



becoming constant. Then S would, of course, decrease (it must be 



observed that | I /V/.7(f> and I I (//(Uo remain also constant). This 



approach to uniform distribution is perha[)s most readily seen, when we 

 consider oidy that part of the enseml)le that had originally a length 

 between /, and /„ -f r//„, the angular velocities lying between tOj 

 and V).,. This part of the ensemble, being originally found in one 

 point of ihe ecliptic, will get disintegrated by the ditferent velocities, 

 and gradually spread over the ecliptic, till tinally the ecliptic is taken 

 up a very large number of times. At a definite point of the ecliptic 

 theie are now parts, which were originally s|tread oxer a large 

 number of elements of the extension, always at equal distances from 

 each other, and it is easy to see, that if the function representing 

 the original density, and its derivative are tinite and continuous, the 



