2-18 Lord Rayleigh on James Bernoulli? 's 



(somewhat imperfectly) in ' Theory of Sound,' 2nd ed. § 42 a. 

 It depends upon a transition from an equation in finite differ- 

 ences to the well-known equation for the conduction of heat 

 and the use of one of Fourier's solutions of the hitter. Let 

 J(fjL,r) denote the chance that the number of events occurring 

 (in the special application positive vibrations) is \p. ■% r, so 

 that the excess is r. Suppose that each random combination 

 of /jl receives two more random contributions — two in order that 

 the whole number may remain even, — and inquire into the 

 chance of a subsequent excess r, denoted by/(/x + 2, r). The 

 excess after the addition can only be r if previously it were 

 r — 1, r, or r-\-l. In the first case the excess becomes r by 

 the occurrence of both of the two new events, of which the 

 chance is J. In the second case the excess remains r in 

 consequence of one event happening and the other failing, 

 of which the chance is J ; and in the third case the excess 

 becomes r in consequence of the failure of both the new 

 events, of which the chance is 5. Thus 



fQ* + i,r)=if{p,r-l) + Vfar) + lfOhr + I). . (8) 



According to the present method the limiting form of /is to be 

 derived from (8). We know, however, that/ has actually the 

 value given in (6), by means of which (8) may be verified. 

 Writing (8) in the form 



we see that when /x and r are infinite the left-hand member 

 becomes 2df/d^i, and the right-hand member becomes ^d 2 f/dr 2 , 

 so that (9) passes into the differential equation 



dfi 8 dv l ••••••♦ \m 



In (9), (10) r is the excess of the actual occurrences over \yu. 

 If we take x to represent the difference between the number 

 of occurrences and the number of failures, x=2r and (10) 

 becomes 



iL-l c ll . . . (in 



In the application to vibrations f(fi, x) then denotes the 

 chance of a resultant + o? from a combination of ja unit 

 vibrations which are positive or negative at random. 



In the formation of (10) we have supposed for simplicity 

 that the addition to /x. is '2, the lowest possible consistently 

 with the total number remaining even. But if we please we 

 may suppose the addition to be any even number yJ , The 



