106 BIRKHOFF AND LANGER. 



(90) (chj) (c„) = 0, 



n 



or 2 d a c tj = 0, i, j, = 1, 2, . . . n. 



Now it was shown that (D i; -(Xjt)) ^= 0, i.e. that the rank of D(\k) is 

 (n — 1). It follows that the solution of the system of linear equations 



(91) 2(foci = i=l,2,...n 



w 



is unique, namely that the Cy of system (90) is the c t of system (91) 

 for all j. Consequently the solution of system (89) is unique and of 

 the type Y(x)C-. One solution of (89) is known, however, namely 

 Y {k) (x) ■ . Inasmuch as G (lc) (.r, t) was also seen to be a solution in x it 

 follows that each column of G'^Or, t) must be the same as the general 

 column of Y (lc) (x)- except possibly for a factor independent of x. 

 Accordingly gf{x,t) = cfHt) y\ k) (x), 



from which G {k \x,t)= - Y {k) (x)- -C {k \t). 



n 



But from (88) it is also seen that G {k) (x, t) is, for fixed x, a solution 

 in t of the system adjoint to (89). Since this solution is again unique 

 and -Z ( (t) is a solution we have 



c?\t) =c<*> #>(*). 



Hence gf (x, t) = c ( *> yj»(*) z?\t) 



or 



r (*) 



(92) G m (x,t)= — Y {k \x)--Z {k) (t), 



n 



the value of c ( being as yet undetermined. 



Writing equation (46) in the form 



Y m '(x)> = \R(x) X + B(x)} Y w (x)-+ {X,- X} R(x) Y {k \x)-, 



and considering this as a non-homogeneous equation, we have 



