( 808 ) 



vvitli the exception however of the first term of the series which 

 now assumes the form (13). In calculating An we have therefore to 

 apply a correction to the expression for An which is easily found 

 by remarking that : 



(n 4- 1) j .^'"' R'n dx = j .r"'— ±^ dx 



= ( ,7:'« Qn^i J —ml /l""-! Q,,_l_i d.X 



en) 



For m < ?i + 2 the last integral vanishes and, R, being of the 

 second degree, we have to consider this case only. 

 We have, therefore : 



(71 + 1) ^^"« R'n dx = (x"' Qn+i j , m < 3 



By (6) we find : 



/ \+i 2 2"{n'\-2)!n! 



\^ ^ y_ kn-^i (2n + 1) / 



whilst for n odd the expression vanishes. 

 Hence also : 



( .^•'« Qn-^i ] — 7 — ( "^ + ^^ even j 



and equal to zero for m-{-n odd; in calculating the constant A» we 

 have, therefore, only to apply a correction such that, instead of 

 (12), now is used, for n odd : 



J2"+i Z> n! n! C 2^'(u,—u„)?i!n! 



and for n even : 



This example of adaptation, of which many variants might be 

 given, will suffice to demonstrate the applicability of the method to 

 special cases. 



§ 3. Development between definite limits on the one side 



AND indefinite LIMITS ON THE OTHER. 



a. JSfo given value for the limit. 

 As has been noticed above, frequencies of duration and quantities 



