Theorem in Probabilities. 247 



The process is given in detail after Laplace in Todhunter's 

 •History of the Theory of Probability/ p. 549, from which 

 the above paragraph is quoted. The expression for the rth 

 term after the greatest is 



.,r2 



v^ j~ fxrz r(n — ni) r z ?- 3 \ ,-. 



mf\ + »w i ~2mJi bW + 6V J *' ' *' ' 



</{27rm 



and that for the rth term before the greatest may be deduced 

 by changing the sign of r in (4). 



It is assumed that r 2 does not surpass fi in order of mag- 

 nitude, and fractions of the order l//x are neglected. 



There is an important case in which the circumstances are 

 simpler than in general. It arises when p = q^ J, and /jl is 

 an even number, so that m = /z — J/*. Here z disappears 

 ab initio, and (4) reduces to 



P 



representing (2), which now becomes 



(6) 



An important application of (5) is to the theory of random 

 vibrations. If /x- vibrations are combined, each of the same 

 phase but of amplitudes which are at random either -f- 1 or — 1, 

 (5) represents the probability of ifi> + r of them being positive 

 vibrations, and accordingly \y^— r being negative. In this 

 case, and in this case only, is the resultant + 2r. Hence if x 

 represent the resultant, the chance of #, which is necessarily 

 an even integer, is 



2e-* 2 /V 



n/(27T/*)' 



The next greater resultant is (^' + 2); so that when a is 

 great the above expression may be supposed to correspond to 

 a range for x equal to 2. If we represent the range by dx, 

 the chance of a resultant lying between x and x + dx is given by 



e- x2 ^dx 

 s/(?tth) {) 



Another view of this matter, leading to (5) or (7) without 

 the aid of Stirling's theorem, or even of formula (1), is given 



* Phil. Mag-, vol. x. p. 75 (1S80). 



