404 JACKSON. 



involve only the point tt and not the point 0. The latter point 

 surely will be involved in the remaining condition of the adjoint set. 

 For ^^ the j/th row of the product matrix, like the last v — 1 rows, 

 involves only zeros in the first v columns, and it follows that the 

 remaining v rows, the (v + l)th in particular, must contain elements 

 different from zero in these columns, since the determinant of the 

 matrix can not vanish. The adjoint conditions may be simplified 

 by linear combination to a form resembling that of the original condi- 

 tions in (16). No confusion will be caused if we denote the simplified 

 conditions by Fg (i") =0, 5=1, 2,..., v; the conflicting notation 

 hitherto used will not be referred to again. We may write 



fa'— 1 ;-— 1 



Fi (v) = i;(^-'') (0) + Z a/ v^''^ (0) + Z ft' ^'^^■' (tt) = 0, 



(30) ^- 



Vs (v) = ij(*^') (tt) + Z ^y' ^^'^ (tt) = 0, s = 2,3,...,v, 



3=0 



where 



v-i^kx ^Q, v-i^ w > ^-3 '> • • • > A-; ^ 0. 



For each value of p, a fundamental system of solutions of the equa- 

 tion (29), 2;j(.T, p), s = 1, 2,. . ., V, can be assigned in such a way that 



(31) ^^zsix,p)^{-pws)''e-'>^sxiii k^O,h...,p-l. 



It is important for us now, however, to notice that the bracket symbol 

 may be given a much more precise interpretation than was done before. 

 Both the relations (31) and the relations (17), (18), remain true ^° 

 if [a], where a is any constant, is understood to mean an expression of 

 the form 



I 'AiC^O , hi'i-) , , 'Am-i( .r) E(x, p) 



p p^ p^-^ p^ 



where m is an integer that may be assigned arbitrarily in advance, the 

 functions \pj{x),j — 1, 2, . . ., m — 1, are continuous with their de- 

 rivatives of all orders for ^ .r ^ tt, and E{.v, p) is a function of x 



29 The statement may also be proved by observing that on the contrary 

 assumption the adjoint conditions could be reduced to the form i'(7r) = v'iir) = 

 . . = I' (""!) (tt) =0, and the adjoint system could have no characteristic 

 values. 



30 See Birkhoff, I, II. 



