EXPANSION PROBLEMS. 399 



contain roots only in the strip along the axis of reals which r^^z-i 

 has in common with Ta, so that a finite number of roots at most are 

 to be added to those already found. By a change in the value of g, 

 if necessary, the expression given above can be made to account for 

 all the characteristic values. 



It is evident a priori that if the coefficients in the diflFerential equa- 

 tion and in the boundary conditions are real, the conjugate of any 

 characteristic value will also be a characteristic value, and it follows 

 that all the characteristic values with the exception of a finite number 

 will be actually and not merely asymptotically real. We do not 

 restrict ourselves to this case, however. 



To each characteristic value from a certain point on corresponds 

 just one characteristic function. For, ^* as we have already had 

 occasion to observe, the determinant Ai has first minors which are 

 always different from zero when p is sufficiently large, and it follows 

 that the same is true of A. An expression for the characteristic func- 

 tion un{x) may be obtained by replacing the t-th. element in the last 

 row of A(pn) by ?/((.r, pn), t = I, 2,. . ., v; it is seen that the result of 

 subjecting the expression so obtained to any one of the operations 

 Wg, s = 1, 2. . ., V — I, may be written as a determinant in which 

 two rows are alike, while the result of the operation Wv is the vanishing 

 determinant A(p7.) itself. Inasmuch as a factor independent of x 

 is immaterial, a characteristic solution may be obtained equally well 

 by inserting the functions i/f {.v, pn) in place of the elements of the last 

 row of Ai. It is the latter more convenient form of solution that we 



24 As a matter of fact, the mere observation that the characteristic value 

 is a root of the first order for A suffices to show that there can not be more 

 than one characteristic function; cf., e. g., Goursat, Annales de la Faculte 

 des Sciences de I'Universite de Toulouse, deuxieme serie, 10, 5-9S (1908); 

 pp. 37-38; Birkhoff III, p. 118. A proof is as follows: The derivative of A 

 with regard to p may be written as a sum of v determinants, in each of which 

 v — 1 rows are identical with the corresponding rows of A, while the elements 

 of the remaining row are the derivatives of the corresponding elements of A. 

 The cofactors of these derivatives in the several determinants are of course first 

 minors of A, except as to algebraic sign. If p„ is a characteristic value to 

 which two linearly independent characteristic functions correspond, the 

 first minors of A are all zero for p = p„, and the derivative of A consequently 

 vanishes. That is, there can not be two independent characteristic functions 

 for Pn unless p„ is at least a double root of A. This general fact will be assumed 

 in the later discussion of other sets of boundary conditions. It is clear that 

 similar reasoning shows that for a root of anv order of multiplicity the number 

 of linearly independent characteristic functions can never be greater than the 

 order indicates; cf. Goursat, loc. cit., p. 44. I do not know whether the proof 

 indicated here is to be found in the literature or not; it is the one which 

 Professor Bocher gives in his lectures. 



