SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 23 



The corresponding term in Q (T/t 2 ) is 



C, -l + A, -l 



L Vw L \to ' 





r r 



It follows that in -g Q -g the r" 1 column is a sum of terms belonging to the indices 



If V 



Q\6i, where ,s- = 1, 2...T, and so for each operation Q. 

 Thus finally we have the result 



/r\ 



The term in the i* 1 ' 1 row and/ 1 ' column of H(-a) is a sum of terms belonging to the 



(ft ^ 

 -j + ) the / h column is a sum 

 t t / 



of terms belonging to the indices l l ...0 l j . 



13. Supposing still that p = 1, let the indices l l ...0 l e * cease to differ by other than 

 integers, and likewise the other groups. Let them be arranged in groups differing 

 by integers, so that their real parts are in descending order of magnitude in each 

 group. 



Then no root of ^(^i 1 ) = will exceed Q* by an integer, and therefore ^(0^ + k)^ k 

 a positive integer, so that the equations X f]+ i, ... do not fail to determine x^ 



If, however, 6* = 0i l k, $ (0^+k) ; so that the coefficient of y^ in Y ti+A vanishes, 

 leaving y^ unknown. We take ?/// = as the simplest assumption, and the following 

 equations then give y l k+\, &c., all without ambiguity. We are, however, left with 

 Y tl and Y ei+Jl in general unsatisfied. Of these one can in general be satisfied without 

 affecting the rest of the argument by an adjustment of the element x/ 1 - 



It has been seen that a constant a 21 in the matrix xi occurs first with non- vanishing 

 coefficient in Y I . 



Clearly, then, if we introduce a 21 t k , it will leave all the equations to Y e]+Jl _i unaffected, 

 and add to Y, i+A the left-hand member of Y 6i _j with 0^ + k+l for 6*. 



But Y ! is an equation of degree l 1 of which d* is the greatest root, so that the 

 multiple of a 21 added to Y ei+A is not zero. Thus a proper choice of a 21 satisfies Y, 1+4 . 



Again suppose 0* = 0*^ = l l k l k 2 , ^>0, 2 >0. 



Then the equations Z. i+t , Z ti+ , i+ , a fail to give z\, z\ +ki ; but a 31 , 21 can again be 

 determined so that, if a 31 * 1+ *% a 3 V' occupy the places above 0^ in XL the equations 

 Z. I+AI , Z |+AiH . tj are satisfied, 0^ being unaffected and z l ki , z\ i+tl being taken zero. 



Suppose then l \..0 1 k form the first group of 6^... 6^ differing by integers. Then 

 treating the first h columns all in the same way, the ^h(h l) equations Y I , Z.,, 

 Z,,-!... must be satisfied identically when the 0^ have been all determined, and must 

 be added to the equations of condition already found. 



Suppose now l h * l ...0 l h+k form the next group of roots differing by integers and 

 consider the (h + r) lh column r<k. The equation giving 6>, A+I is that indicated by the 

 suffix ^-(h+r-l), and those following this up to that with the suffix e l are 

 independent of the undetermined elements of r). 



