250 On James Bernoulli? s Theorem in Probabilities. 



and, finally, as the chance that the excess lies between r and 

 r + dr, 



\Z&-^ ( is ) 



Another method by which A in (12) might be determined 

 would be by comparison with ((3) in the case of ?' = (). In 

 this way we find 



A 



pi 



1.3.5. 

 I~ 2.4.6, 



by Wallis' t 



...p-1 



s/fi 



2*. tr I & 



= \/fc) 



heorem. 



If, as is natural in the problem of random vibrations, we 

 replace r by x, denoting the difference between the number of 

 occurrences and the number of failures, we have as the chance 

 that x lies between x and x + dx 



(16) 



n/(2tt/*)' 

 identical with (7). 



In the general case when p and q are not limited to the 

 values -J-, it is more difficult to exhibit the argument in a 

 satisfactory form, because the most probable numbers of 

 occurrences and failures are no longer definite, or at any rate 

 simple, fractions of ft. But the general idea is substantially 

 the same. The excess of occurrences over the most probable 

 number is still denoted by r, and its probability by f(fi, r). 

 We regard r as continuous, and we then suppose that /j, 

 increases by unity. If the event occurs, of which the chance 

 is p, the total number of occurrences is increased by unity. 

 But since the most probable number of occurrences is increased 

 by p, r undergoes only an increase measured by 1 — p or q. 

 In like manner if the event fails, r undergoes a decrease 

 measured by p. Accordingly 



f(n+%r)=pf{n r r-q) +qffar+.p). . . (17) 



On the right of (17) we expand f{p,r—q),f(/j,, r+p) in 

 powers of p and q. Thus 



