EXPANSION PKOBLEMS. 417 



verges uniformly, its sum must necessarily have continuous deriva- 

 tives of all orders. 



Of the adjoint boundary conditions, v — fx, depend on the point 

 TT only, while the remaining fjt, contain linearly independent combina- 

 tions of the values of v and its derivatives at the point 0. The char- 

 acteristic functions of the adjoint system have an expression 



Vn (x) = [d'l] e'"'«'M('^^) + [d'o] eP''«'M+i('^^), 



the interpretation of which is similar to that of (48). Finally, the 

 general coefficient in the formal development of the function (14) 

 may be treated as in the previous cases, and it may be shown that for 

 infinitely many positive integral values *^ of q the series diverges at 

 every interior point of the interval (0, tt). 



It has been pointed out that the numbers pnWf^ and pmOf^+i, which 

 figure in the formula (48), are situated along rays which become 

 less and less inclined to the axis of pure imaginaries as [x increases. 

 If an analogous formula were to hold in the case that v is even and 

 IJ. = ^v, which has been excluded by our hypotheses, the corresponding 

 numbers would be distributed along the axis of imaginaries itself, 

 and the formula would resemble those which Birkhoff finds in the case 

 of regular boundary conditions.*^ As a matter of fact, the boundary 

 conditions do become regular when (x = ^v, ii the order of the deriva- 

 tives at the point involved in the last /x conditions is not too high. 

 This is precisely what happens in the case of Liouville's conditions, 

 with /i = 1, when v is equal to 2. 



41 In the formula corresponding to (46), there will be a determinant contain- 

 ing wi,..., Wfi-i, w^, and one containing Wi,. . ., w^-i, w^i. When /x is odd, a 

 common factor is disclosed by dividing the first determinant by a power of wi, 

 and the second by the like power of w^; when jj. is even, the second determinant 

 is to be divided by a power of Wi, and the first by this power of W2. Among 

 V successive values of q, there will be just m — 1 values for which the determi- 

 nants vanish. Not more than one other value is excluded by the requirement 

 as to the order of magnitude of the quantity corresponding to (47), and there 

 remain at least p — /j. oi every v consecutive values of q, which are available 

 for the purposes of the demonstration. 



42 In the paper II, p. 389. The analogy is not perfect; the determination 

 of the characteristic values becomes less simple in the regular case. 



