SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 27 



But in the same way 



so that !<*_! (0 2 2 ) must also vanish when 0./ is found and thus iw A _ 1 (^ 2 1 ) == 0, so that 2 a> k 

 is also independent of 0^. 

 Ultimately we have 



and o> k _ h (0 2 h ) = and is independent of 0f. 



Thus h G> k (Oi] is an equation of degree h for (V, since 0% = 2 h , 3 l = 3 h .... 



Suppose its roots are v lf v 2 , ... v h - 



Take 6 to be v^ Then A-IW*-! is an equation of degree h l for 0*, and its roots 

 are (p+v a )(p+v a )..., as shown by effecting the operation Aj. 



Choosing one of these again to be 0*, say p + v 2 , A-aw*-a is an equation for 0, 3 , whose 

 roots are (2p + v s ), (2p + v 4 ).... Thus the quantities l l ...0 l ' 1 are r 1( (_p + r a ), (2^+r,)..., 

 (h-l)p+v h . 



In order to particularise the order of the roots v 1} v 2 , ..., they are arranged as soon 

 as found as follows : 



Let all those roots which differ from one another by integers be grouped con- 

 secutively and let the arrangement in each group be such that 6 r ~ l 9 r = or a 

 positive integer. Suppose, now, the equation k (a k (0i l ) = is the equation X ? . 



Then the coefficient of x in X ?+1 is A w /l (# 1 1 +l) which, since no root of A w*(0) = 

 exceeds 0, 1 by a positive integer, does not vanish. X 7+1 is moreover independent of 

 x a l , ..., owing to the equations k -i<a k = 0, ..., being satisfied independently of 0,\ 



Thus X ?+1 determines x^. 



Similarly, X ?+2 gives x a l , the coefficient being A w A (0 1 1 + 2), and so forth. 



Of the Y equations, that determining 0^ is obtained from X 7 by the operator A t . 

 It is, in fact, Y ? _ p . 



The coefficient of y^ in Y^.^+j is equal to the coefficient of o^ 1 in X ? _ /)+1 with 0^, 

 0./, ..., substituted for 0^, 2 l , ____ But X ? _ p+1 vanishes identically as far as 0^ is 

 concerned and 2 , ..., are the same as 0^, .... Thus Y g _ p+1 is independent of?/! 1 . 

 Similarly, Y g - p+2 ...Y q - l are all independent of the undetermined elements of y. 



Suppose now 0\0i X (a positive integer). 



The equation Y ?+ft contains y^, ..., y k , the coefficient of the last of these being 

 A eo A (0! 2 +&) which vanishes for k X. 



If, now, in the matrix *%r( r = P> P~ 1--.2) the second element of the first row be 

 c r 21 , and in ^ be c 21 ^, the constants c will, as before (p = 1), affect first the equations 

 Yj-p+j..^.! and Y ?+A respectively, entering into these with non- vanishing coefficients. 

 Let c r 21 be determined then to satisfy the first pl of these equations; Y ?+1 ...Y ?+x _j 



E 2 



