EXPANSION PROBLEMS. 403 



in (16), be supplemented by v linear forms Wp+i(;u),. . ., Woy^u), in 

 such a way that all the 2v forms are linearly independent. If 21^ 

 linear forms Vi{v),. . ., V^viv) are determined in such a way that 



2" < ) X = 7r 



Z W.{u)V,{v) = P{u, v) 



s=l ' ) x = 



the equations ^^ 



Vsiv)=0, s=v+1,p+2,...,2p, 



constitute the boundary conditions adjoint to (16). We will denote 

 the matrix of the forms W by a. and that of the forms V by /?. What- 

 ever these matrices may be, the matrix of the bilinear form -lWs{u) Vs{v) 

 is a'/3, where a' is the conjugate of a, and so the determination 

 of the adjoint conditions amounts to the solution of the matrix equa- 

 tion a'i3 = T], which gives 



i5 = (aO-^r?. 



Of course the reciprocal of a is the same as the conjugate of a~^. 

 Let us examine the form of the matrix /?. 



The choice of the v forms Wy+i,. . .,W-2v, is arbitrary, provided that 

 the condition of linear independence is fulfilled. We will choose them 

 in a particularly simple way. We will set Wv+i equal to that one of the 

 variables ui, . . . , ih, which does not appear as leading term in any of 

 the conditions Wi,. . ., Wv-\, in (16), and will let Wv+2,- ■ ■, Wiv, be 

 equal respectively to the variables Mj/+i, . . . , Woi/, with the omission of 

 the one which forms the leading term of Wv It is clear that if the 

 variables u are expressed in terms of the quantities W by solving the 

 linear equations which give the latter in terms of the former, the first 

 V of the It's will be completely determined by the fu-st v — 1 equations 

 and the {v -{• l)th, and so will not involve Wv nor Wv+i,- ■ ■, Wov 

 That is, since the matrix of the new equations is a~^, the last v — 1 

 columns of a~^ (and the vth column) contain no elements different 

 from zero in the first v rows. The last v — 1 rows of the conjugate 

 of a~^ contain no elements different from zero in the first v columns. 

 Consequently, since the first v columns of rj contain non-vanishing 

 elements only in the first v rows, the last v — 1 rows of the product 

 (a)~^7] will have all their elements in the first v columns equal to zero. 

 This means that v — 1 oi the adjoint boundary conditions F = 



28 The use of subscripts here is different from Birkhoff's; the change is 

 made for the sake of convenience in using the language of matrices. 



