and Random Fliyhts in one, two, or three Dimensions. 325 



former alternative there is a unique middle term when 

 a = b = ±?i; but in the latter a and b cannot be equated, and 

 there are two equal middle terms corresponding to a = ^n-f J, 

 b — ^n — ^, and to a — in-~j? } b = ^nA-^. The values of the 

 second traction in (6) are the series of integers in what is 

 known as the " arithmetical triangle." 

 We have now to consider the values of 



to be found in the neighbourhood of the middle of the series. 

 If n be even, the value of the term counted 5 onwards from the 

 unique maximum is 



.... (8) 



(hn-s) ! (irc-M) ! 



If n be odd, we have to choose between the two middle 

 terms. Taking for instance, a = i-n + 5, b = \n — ^, the 5th 

 term onwards is 



... (9) 



The expressions (8) and (9) are brought into the same form 

 when we replace s by the resultant amplitudes. When n is 

 even, x= — 2s ; when s is odd, a? is —2(5 — -J), so that in both 

 cases we have on restoration of the factor (^) n 



n 1 



. . . (10) 



The difference is that when n is even, x has the (?i + l) values 



0, ±2, ±4, ±6, ±n; 



and when n is odd, the (?i+l) values 



+ 1, +3, ±5, . . . . ±n. 



The expression (10) may be regarded as affording the 

 complete solution of the problem proposed ; it expresses 

 the probability of any one of the possible resultants, but for 

 practical purposes it requires transformation when we con- 

 template a very great n. 



The necessary transformation can be obtained after Laplace 

 with the aid of Stirling's theorem. The process is detailed 

 in Todhunter's ' History of the Theory of Probability,' 

 p. 548, but the corrections to the principal term there exhi- 

 bited (of the first order in x) do not appear here where the 



