SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 17, 



substituted for P ', the equation A p _ 1 X, i+2 , so that when 6* is substituted for 6 a 'n\ 

 Y t| it becomes symmetrical in 6 l p ^ and a p -i. Y I must therefore be of the form 



A(<r lf ir,)tf l r . 1 +B(a I ,<r 1 )V.i+0(<r to ir i ) - 0, 

 where 



A(o-i, o-j) = B(o-i, o-j) 

 and 



A(<TI, o- 2 ) + B(o-,, o- 2 ) = (o-j-o-g)... +(0-3-0-.,).... 

 Hence 



A{o-!, o-j) = (o-i 0-3) (o-i 0-4)... = B(cri, o-,). 



Now before the values of #/, #/ are substituted in Y I the coefficient of 6 2 P ^ must 

 be a function of p a only. 



Therefore B must be a function of o- 2 independent of cr,. 



Hence 



B = (cr,-o- :i )...(o-,-o- ti ) 

 and 



A = (cTi 0-3)... (o-j 0- t| ). 



The equation for $ 2 p _i is therefore 



By virtue of the equation giving #^-1, therefore, we have 



(o- 2 -o- 3 ). ..(o 2 -o %i )6V 1 + ^-- = 



tr a o"! 



as the equation for ^_!. 



But this is identically the equation that would have been obtained for l p -i if 

 0-,, o- 2 had been interchanged. Thus the value of 6 P -^ associated with the root cr., is 

 unaltered by this interchange. 



We have further to see that the same permutation does not alter the 6 p ^ l for the 

 subsequent columns. 



In just the same way as above the equation for 6 3 p -i is shown to be 



( W } O 2 + ' } 6 S + ' 2 ~ g ' 1 (r 3~ Cr l _ Q 



which gives 



independently of the order of o-j and o- 2 . 



The same holds for 6 k p -i, k < c^ and a similar proof for the interchange of any other 

 pair of the roots o-. 



Thus supposing the roots of <f> (6 P ) = to be all different, there is associated with 

 each a unique determinate value of f -i. 



VOL. CCV, A, D 



