SECT. 15.] On certain kinds of Groins or Series. 19 



15. If the reader will study the following example, 

 one well known to mathematicians under the name of the 

 Petersburg 1 problem, he will find that it serves to illustrate 

 several of the considerations mentioned in this chapter. It 

 serves especially to bring out the facts that the series with 

 which we are concerned must be regarded as indefinitely 

 extensive in point of number or duration ; and that when so 

 regarded certain series, but certain series only (the one in 

 question being a case in point), take advantage of the in 

 definite range to keep on producing individuals in it whose 

 deviation from the previous average has no finite limit 

 whatever. When rightly viewed ii&amp;gt; is a very simple problem, 

 but it has given rise, at one time or another, to a good 

 deal of confusion and perplexity. 



The problem may be stated thus : a penny is tossed up ; 

 if it gives head I receive one pound ; if heads twice running 

 two pounds ; if heads three times running four pounds, and 

 so on ; the amount to be received doubling every time that 

 a fresh head succeeds. That is, I am to go on as long as it 

 continues to give a succession of heads, to regard this suc 

 cession as a turn or set, and then take another turn, and so 

 on ; and for each such turn I am to receive a payment ; the 

 occurrence of tail being understood to yield nothing, in fact 

 being omitted from our consideration. However many times 

 head may be given in succession, the number of pounds I 

 may claim is found by raising two to a power one less 



ellipse in astronomy. In the cases on that supposition, 



in which these conceptions are made * So called from its first mathe- 



use of we have a phenomenon which matical treatment appearing in the 



is continuously varying and also Commentarii. of the Petersburg Aca- 



changing its rate of variation. We demy;- a^ variety of notices upon it 



take it at some given moment, sup- will be found in Mr Todhunter s 



pose its rate at that moment to be History of the Theory of Probability, 

 fixed, and then complete its career 



22 



