March 14, 1901] 



NA TURE 



465 



been born in Cork, and who published his work on the natural 

 history of Spain in Paris in 1776. It would be very interesting 

 to trace any conclusions as to extinct volcanoes that were current 

 in Bowles's mind as early as 1750. Ozy is not likely to have 

 made a show of his previous knowledge to men of established 

 position like Guettard and Malesherbes, who had honoured the 

 local apothecary with a call. He listened to their exposition of 

 the craters, and "avoua ingenument " that he was much 

 surprised at what he heard. After all, it was Guettard who took 

 the matter up absolutely fresh from the beginning, and whose 

 memoir made it for the first time public knowledge. 



Bowles may have come from Catalonia, and may have formed 

 his opinions there. I make no mention of Olzendorff, of whom 

 I have no further trace ; but the fact that Bowles instructed 

 Ozy at Clermont, if the above contention is correct, a year before 

 Guettard had formed his conclusions, and in an age rife with 

 unfounded speculation, marks him as a geological observer 

 deserving of more credit than he has yet received. I called 

 attention to Ozy's letter in Knowledge for 1898, p. 266, and 

 have since made inquiries through friends in Cork and else- 

 where ; but the present family of Bowles in Cork, formerly 

 Boles, can furnish no data as to the life of the mineralogist. 



An interesting inquiry also arises as to when the Giant's 

 Causeway in Antrim was first regarded as a lava-flow. What 

 did Bowles know of this phenomenon ? Its detailed apprecia- 

 tion, from a geological point of view, is usually ascribed to 

 Whitehurst, in the second edition of his work in 1786. But 

 Faujas de St. Fond in 1778 calls many of the French lava-flows 

 " chaussees," and clearly shows his own conclusions when he 

 styles certain examples with good columnar jointing " paves des 

 geans." Grenville A. J. Cole. 



Royal College of Science for Ireland, Dublin, March i. 



Probability — James Bernoulli's Theorerti. 



It may possibly be of some little interest to notice that the 

 thedrem in probability, which goes by the name of Jaines 

 Bernoulli's theorem, alluded to in my letter to Nature of 

 December 13, 1900 (p. 154), admits of a treatment somewhat 

 more elementary than the usual one. 



The theorem may be stated thus : — W p is the probability of a 

 given event, and n the number of times considered ; as n in- 

 creases without limit, the probability that the ratio of the number 

 of times in which the event happens to the whole number of 

 times («) will only deviate from / within limits of excess and 

 defect, which decrease indefinitely as n increases without limit, 

 is a probability which approaches indefinitely to unity as its 

 limit. 



In Laplace's demonstration (see Todhunter's " History of 

 Probability," art. 993) Stirling's theorem, for the evaluation 

 of factorials, is used in the first step ; in the second step the 

 theorem of Euler, 



rky=^ydr-\y^h^^- ... 



which is also implied in the usual proof of Stirling's theorem ; 

 and, finally, the result depends on the evaluation of the well- 

 known definite integral! e~^^dt. 



\r 



Further, it is essential to this demonstration to make the limit 

 of deviation in excess from the ratio / equal to the limit of 

 deviation in defect, for then as members of the series, which 

 represent the probability sought, equidistant from the middle 

 of the series contain certain terms equal in magnitude and of 

 contrary sign, these terms cancel in the addition of such pairs, 

 and are thus got rid of. 



It may be shown that the general result of Bernoulli's theorem 

 may be got without the above described use of Euler's theorem 

 (i.e. the second use of it), without using the evaluation of 



I e~*\it, and without making the limits of excess and defect 



Jo 



equal. These limits may have any ratio whatever. 



Let q be the probability that the event does not happen, so 

 that p + q—i. 



Let the whole number of times considered hty+x. Since 

 this is to increase without limit, we may suppose /{jr -J- j/) and 

 q[x-\-y) always integers. 



Let P be the probability that the number of the times in 

 which the event happens be between f[x +y) -H ax and 



NO. 1637, VOL. 63] 



p{x+y)-bx where {a-irb)=i, so that x represents (so to 

 speak) the range of the variation, a and h may have any ratio 

 to one another. Assume that_y=/«jr-<^~'', where k maybe as 

 small as we please, but finite. Thus P is the probability that 

 the ratio of the times when the event happens to the whole 



number of times shall not exceed p by more than J?:^, or fall 



x+y 



bx 



short of / by more than ; limits which vanish when x and 



x-vy 

 y are infinite. 



Let P] = probability that the number of times in which the 

 event happens is less ^k\?iW p{x -^ y) - bx , and Po the probability 

 that it exceeds p{x ^y) + ax. Then I - P = Pj -h Pj. 



Pj=/'+2' + (;c-t-;/);)^+s'-V+ ... 



|/(j:-fj/)-|-ajf+ 1} ! \q(x->ry)—ax- 1} !* 



Now P2 evidently = the probability that the number of cases 

 in which the event does not happen is less than q(x -f-^) - ax, 

 and therefore the series for Pj is derivable from that for Pj , by 

 interchanging / and q, and by interchanging a and b. These 

 values of Pi and Pg may, of course, be also got from the equation 



Pl + P2 = (25 + ^)»-P. 



Pj is evidently less than the geometrical progression of which 

 the sum is 



i^X-^y) \ pp{x+y)+ax^q(x+y)-ax 



{fi(x+y) + ax'( ! {q(x+y)-ax\ ! 



I- 



q{x +y) -ax \ 9{^+»)-<w+i 



q p(x+y) + ax 



X V 



i) q{x +y) — <fx 

 q p{x +y) -f- ajT + I 

 By Stirling's theorem, x and y increasing ad. inf. 



{x + y)\ =(x-f->')*+»+i^-(^+*);^2ir/'l-f 



(■ 



...) 



I2[x+y) 

 \ yJ \ i2{x-\ry) } 



and similarly for the other factorials. 

 Thus the above expression becomes 



slzicpq 



\\2{x+y) '") 



1 + 



I2(p(x+y)+ax) 



■X 



1 + 



I2\q{x+y)-ax) 



(-;) 



v+y+i 



/ J (a4-/))jr \P"+y)+'^+Y 



i-f 



(5^ - a).r\ »(*+*''-'•*+* • y^ 



ly / 



q(x-iry) -ax 

 p{x+y) + ax 



+ 1 J 



q{x+y^-ax+l 



/ q(x+y) -a x 

 q p{x ^-y) + ax->r\ 



The limit of the second factor is unity, 

 be shown to become in the limit 



The third factor may 



^ianpq 



The limit of 



/ j(x +y) -ax \ st^+yJ-'-^+i 



q p{x+y)+ax+ 1 / 



is the limit of e~'''', and the limit of 



q{ x+y)-ax _^\ .^^,^ 



q p(x +y) + ax+i J 



where c and /i are positive constants. Hence the limit of the 

 product of the factors is zero— that is, the limit of Pi is zero, 

 and evidently also the limit of Pg. 



Hence the limit of P is unity. 



The range of the deviation (x) is greatjer in this proof than in the 



usual one, for in the latter x would vary as yi as against ^2-2* 

 where k may be as small as we please. 



-ex 



UP 



