J 127 



We must now try lo express ll»e cocffieieiils h^^ and hji- f'oi- 

 / = 1, 2, . . . (>, I' = 1, 2, . . . <), in terms of the coefficients of the 

 given equations of substitution : 



x\z= 2 axivi, . . . , Xp = JS" a^ivi. 



The conditions hjh-=zO for h = o -{-1, ... o are equivalent with 

 the conditions 



but 



:E AjiAhi=^^ for 



() -f 1, . . . tf; 



/=1 



- ^^'«^7= A 



/=i 



^ Aji an = for / =1^.;. 



1=1 



are also alw^ays satisfied. 



So we have the following set of equations, 



2Ajiau= 0, 2Ajia.>iz= 0, . . . ^ Ajiaj-x^iz=i 0, :£ .1^7 a,/ = A, 

 2 .4^7«j+i.t-= 0, . . . ^ Ajia^i^O, 2 AjiAM= A^ hjk, 

 2 Aji Apj^i^i = 0, . . . , ^ Aji A,i — 0. 

 Hence 



A^i , A^i ,..' A^^ 



= 



or 



A6jA: 



«11 

 «21 



«12 



«;1 ? «i2 » 



.4.1 



^4,2 



«l7 

 «27 



«;7 



.1.+ .. 

 ^77 



{-\)r+J 



«11 

 «21 



«/— 1. 1 



«;Cl 

 ^tl 



Aci 



«12 » 



«22 > 



«/—I, 2 J 

 «^ + 1,2, 



«/:2 , 



At> , 



^52 > 



. Oi, 

 • «2t 



