402 JACKSON. 



the reservation that only characteristic functions corresponding to 

 simple roots of the determinant equation are to be taken into account, 

 we are entitled to make the assertion : 



The statement of Theorem I is applicable to the series considered in the 

 present section. 



There is a corresponding extension of Theorem 11. Let the left- 

 hand member of the differential equation (15), without the term p^u, 

 be denoted by ^(m). The differential expression adjoint to this is 



and the equation adjoint to (15) is 



(29) M{v) + p^v = 0. 



The determination of the adjoint boundary conditions ^^ must be 

 carried through in some detail. 

 In Lagrange's identity, 



vL{u) — uM{v) = -^ P{u, v), 



the expression P(u, v) is a bilinear form in u(x), u'{x),. . ., u^"'^^ (.r), 

 v{x), v'{x), . . ., t)^""^^ (.t). If this identity is integrated with regard to x 

 from to TT, the resulting expression on the right, which may be de- 



a;=5r 



noted by < P{u, v)\ , is a bilinear form in the 2v pairs of variables 

 ' ) x=Q 



Ml = m(0), M2 = w'(0), • • • u^ = w'""^^ (0), 



U„+l = u(t), U^+2 = u'{t), • • • U2:, = W^""^^ (tt), 



vi = v(0), V2=- v'iO), '■■ v, = I'^^-i) (0), 



lV+i = r(7r), v,+2 = v'{,Tr), ••• r,. = ^''^"'Utt) ■ 



Let 7] denote the matrix of its coefficients, the coefficient of ngV/ 

 standing in the 5-th row and the ^-th column. It is clear from the 

 way in which rj was obtained that the last v elements in each of 

 the first V rows vanish, and likewise the first v elements in each of the 

 last V rows.^^ Let the linear forms Wi{u),. . ., WvCu), which occur 



26 Cf. Birkhoff II, p. 375; Bocher, Transactions of the American Mathe- 

 matical Society, 14, 403-420 (1913); pp. 404-405. Also, LiouviUe II, p. 604. 



27 As a matter of fact, nearly half of the remaining elements also vanish; 

 but this point is not of importance for the present discussion. 



