414 



JACKSON. 



from which it was obtained; but the argument is so similar to the 

 corresponding ones already given that its details may be omitted. 



For the proof that the development of a particular analytic function 

 is divergent, the form of the adjoint system is to be taken into account. 

 The adjoint differential equation is of course the same as Ijefore. The 

 boundary conditions may be discussed by an appropriate modification 

 of the method used in the case oi v — l specialized conditions. It is 

 found that v—2 of the adjoint conditions contain the point tt only, 

 while the remaining two not only may but surely do involve the 

 point 0, in such a way that the sets of terms relating to this point 

 in the two conditions are linearly independent.^^ The conditions 

 may be simplified and arranged in a manner corresponding to our 

 previous practice. The characteristic functions may be written in 

 the form 



(44) 



Vn (.r) = [d'l] gP-^^C'r- a;) + [d'2] fP-J^jU- x)^ 



where the numbers pn are those already determined, d'l and d'o are 

 conjugate imaginary quantities different from zero, and the approach 

 of the bracket functions to their limits is uniform throughout any 

 closed interval interior to (0, tt) . This expression, however, is not 

 sufficient for the present purpose; for in determining the order of 

 magnitude of the coefficients in the series it is precisely at the point 

 a; = 0, where the formula breaks down, that the value of I'i(.t) is 

 needed. At this point, terms involving eP"(^'^-"'>^) and eP"(^'^-"'>^) 

 must be explicitly taken into account. 



The characteristic function (44) was obtained from the determinant 

 corresponding to*° (33) by dividing out a factor e''"^'^^, independ- 

 ent of X, and a constant factor depending on neither p nor x. By divid- 

 ing out the same exponential factor, but a different constant factor, 

 a characteristic function is obtained which may be represented ade- 

 quately, in a sense presently to be explained, by the expression 



Vn {x) = 52' 



g — pnWlX gPnlVliir — X) 



-53' 



g — PnWlX ppnlOliir — X) 



39 The latter part of this statement may be proved either by making use of 

 the fact that the matrix corresponding to that which was called /3 must be 

 non-singular, or by observing that if the statement is assumed to be untrue 

 a direct determination of the characteristic values for the adjoint system leads 

 to results inconsistent with those found for the given system. 



40 This determinant in the present case has elements of the form 

 ^kz^PnWiTT ^jj _|_ ^/^^ 7 2j^,^K 2 |^-|^j jj^ j|.g gecond row, where k'2, y'2, and K'2 are 

 integers, of which 7'2 may be negative, and 0'^ is a constant; cf. the deter- 

 minant from which (43) was obtained. 



