907 



(tu — .^)'«+i — ( — ^)'«+i 



ttm = — . 



m-[- 1 



1 

 The quantity a^ ^= Urn |(2„i|'« is therefore equal to 



|(o — A'l or 1^1 , 

 according as the former or the latter expression has the greatest 

 value. The maximum value a{a) of a^^ in the domain («) is therefore 

 equal to 



I w (xm) — «,ft I or «, 



according as the maximum modulus of w — x on the circumference 

 of (tt) is greater or less than a. We have therefore in these two 

 cases resp. 



/?! = « -|- \a) {x,n) — a)m\ or /?j =z 2« . . . (65) 



For the series P, belonging to St^ the number that corresponds 

 to a is now equal to the first amount, and for the series F^ belonging 

 to D—^ the number that corresponds to a is equal to the second 

 amount. We maj therefore say that for the series F, which belongs 

 to the combination T=iSo>D~^, the number corresponding to a is 

 equal to the greater of the two numbers that correspond to a for 

 the separate series. 



In any case, the verification required has been accomplished, for 

 each of the two amounts (65) is less than (64). 



28. Finally we consider the combination D—^So,- Here too it 

 may suffice to verify point 2 of the theorem. For the three numbers 

 «, 7, ^, we may take here «, 2«, j3, where /? is determined by the 

 formula 



^=2«+ I a>(.r',„)— ^'m I , (66) 



if x\n is the point on the circumference of(2fi:), whereto — a- assumes 

 its maximum modulus. The resulting series P we determine again 

 directly. Bourlet's formula, which in the preceding case would still 

 have been efficient, though less easy to handle, produces such 

 an intricate form here that it is difficult to be surveyed. It is on 

 the other hand very easy to work with the quantities §. We have 



X 



h = F>-^ So> {.v^) =: D-^ [a>* {x)-\ = CoyJ^ (t) dt, 







and from this by means of formula (24) 



X 



am {x) = (i — x)"* =i[(o (0 - '^l'" dt 



