444 



NA TURE 



[April 9, 1874 



Similarly, Jor the probability that all the particles of nitrogen 

 are in the space B, we find 



( I' yx 



Hence the probability that all the oxygen is in A and all the 

 nitrogen in B is 



/ fl \2 X lo'- ^ / li \S X lo' 



\a + b) 

 Now by hypothesis 





and therefore 



a + , 



herce the required probability is 



-26 X io'= 



Call this _, and let log denote common logarithm. We have 



log N - ic" - 26 X iQi- X log. 2 = (10 - 26 log. 2) X 10^= = 

 2173220 X lo°- This is equivalent to the result stated in the 

 text above. The logarithm of so great a number, unless given to 

 more than thirteen sigrificaiit places, cannot indicate more than 

 the number of places of whole numbers in the answer to the pro- 

 posed question, expressed according to the Arabic notation. 



The calculation of T^ when / and « - i are very large numbers 

 is practicable by Stirling's Theorem, ^according to which we 

 have approximately 



1.2 . . . . »' = !' + i f - ' \/2ir 



and therefore 



n{n--\). . . ■ (H- /+ 1) ^ "" + i 



1 



Hence for the case 



V2T «■(?■+ 4 )(?;-;■)« 



+ b 



which, according to the preceding formula, gives 7*,- its greatest 

 value, we have 



V2ir»^ 



where 



e = and / = ■ , 



a -i- 6 a -V b 



Thus, for example, let « = 2 X 10'^ ; 



c = -2, / = -8 

 we have 



T, = ■— = ^L_ 



8ooooo^y^ 1418000 



This expresses the chance of there being 4 X 10" molecules of 

 oxygen in W, and 16 X to" in B. Just half this fraction ex- 

 presses the probability that the molecules of nitrogen are distri- 

 buted in exactly the same proportion between A and B, because 

 the number of molecules of nitrogen is four times greater than of 

 oxygen. 



Il II denote the molecules of one gas, and h' that of the mole- 

 cules of another, the probability that each shall be distributed 

 between A and B in the exact proportion of the volume is 



The value for the supposed case of oxygen and nitrogen is 



23r X '16 X 4 X iC 4021 X lo" 

 which is the result stated at the conclusion of the text above. 

 WiLLWM Thomson 



LIVIAGSTONE'S WORK IN AFRICA 

 'T^HE daily papers have published some extracts from a 

 -•• letter ot the late Dr. Livingstone to Mr. H. M. 

 Stanley, which have been kindly furnished by the enter- 



prising proprietor of the New York Herald ; we reproduce 

 here so inuch of the letter as bears on the geographical 

 work done by Livingstone. 



" The Chainbezi was crossed long ago by the Portu- 

 guese, who have thus the merit of its discoveiy in modern 

 times. The similarity of names led to its being put 

 down in maps as ' Zambesi ' (eastern branch) and I 

 rather stupidly took the error as having some sort of 

 authority. Hence my first crossing it was as fruitless as 

 that of the Portuguese. It took me twenty-two months 

 to eliminate this error. 



" The Cazembe who was lately killed was the first who 

 gave me a hmt that Chambezi was one of a chain of 

 rivers and lakes which probably forms the Nile ; but he 

 did it in rather a bantering style that led me to go back 

 to the head waters again and see that it was not the mere 

 ' chaff' of a mighty potentate. There is Omar Island in 

 the middle of Bangueldo, with 183° of sea horizon around. 

 The natives, slowly drawing the hand around, said — ' That 

 is Chambezi flowing round all this space and forming 

 Bangweolo before it winds round that headland and 

 changes its name to Luapula.' That was the moment of 

 discovery and not the mere crossing of a small river. 



" The late Cazembe I found sensible and friendly. His 

 empire has succumbed before a very small force of Arab 

 slaves and Banyamwezi. Pereira, the first Portuguese 

 who visited the Cazembe eighty years ago, said that he 

 had 30,000 trained soldiers, sacrificed twenty human 

 victims ever)' day, and that the streets of his capital were 

 watered daily. 1 thought that my late friend had 30,0(00), 

 diminished by two co's, and sacrificed five or six pots of 

 pombe daily ; but this may have been only a court scandal 

 — the streets of his village were not made. So I was 

 reminded of the famous couplet about the Scotch 

 roads : — 



" ' If you had seen these roads before they were made, 

 You would lift up your hands and bless General Wade.' 



"I have been the unfortunate means of detnolishing two 

 empires in Portuguese geography — the Cazembes and that 

 of the Emperor Monomotapa. I found the last about 

 ten days above Tette. He had too {it'f! men to make the 

 show Cazembe did, but I learnt from some decent 

 motherly-looking women attached to his Court Zembere (.') 

 that he had 100 wives. I have wondered ever since and 

 have been nearly dumfoundered with the idea of what a 

 nuisance a man with 100 wives in England would be. It 

 is awful to contemplate, and might be chosen as a theme 

 for a Young Men's Debating Society. I wish someone 

 would visit Mtesa, or Uganda, without Bombay as an 

 interpreter. He (Bombay) is by no means a sound 

 author. The King of Dahomey suffered eclipse after a 

 common-sense visit, and we seldom hear any more of his 

 atrocities. The inightiest African potentate and the most 

 dreadful cruelties told of Africans owe a vast deal to the 

 teller. 



" You and I passed the islet Kasenge, where African 

 mothers were said to sell their infants for a loin-cloth 

 each. This story was made to fit into another nice little 

 story of ' a mother bear ' that refused to leave her young. 

 A child that cuts its upper front teeth before the under is 

 dreaded as unlucky and likely to bring death into the 

 family. It is called an Arab child, and the first Arab who 

 passes is asked to take it. I never saw a case, nor have 

 the Arabs I have asked seen one cither, but they have 

 heard of its occurrence. The Kasenge story is, therefore, 

 exactly like that of the Frenchman who asserted that tlie 

 English were so fond of hanging themselves in November 

 5 ou might see them swinging on trees along the road. 

 He may have seen one ; 1 never did. English and 

 American mothers have been guilty of deserting infants ; 

 but who would turn up the whites of his eyes and say, as 

 our mothers at Kasenge did, these people are no better 

 than, or not so good as, she-ljears .' 



" This lake, so far as I have seen it, is surrounded by 



