SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 13 



Similarly the equation Z l gives 



= a/' "-y,'' = /" + < "-'-a^'- 1 = a/^' + a/'''-'*^'.'.- 3 , 



and so on for the equations for each of the first e t columns of r). 



For the (ej + l)" 1 column, however, the non-diagonal term of x*+i = and the 

 equations for this column do not contain the elements of the preceding column. 



In fact, the e 2 columns beginning with the (cj + l)" 1 form a group related to one 

 another in just the same manner as the first e x are. We obtain from them, as from 

 the latter, the conditions 



, + !,<! + t, A 



a. p \J, 



and so for the columns associated with each group of equal roots. These and other 

 conditions which arise in the course of the investigation will be called " equations of 

 condition." Supposing those already found to be satisfied, we may return to the 

 solution of the equations X, and of these the following statements are to be 

 proved : 



I. The first e[ 1 equations of the (r+l) th set determine x r 2 ...x r f < in terms of 

 ^...aj 1 ,-! and l p+l ...0 l p - r+1 ; the equations from the (ej+l)" 1 onwards give .r/ 1+I ....r r " 

 in terms of the same quantities. 



II. When the values thus found are substituted in the equation X r+1 , the resulting 

 left-hand member is independent of the undetermined quantities .r, 1 ...^ 1 , B p l ...0\- r for 

 all values of r up to (e 2 2), but for r = Ci 1 it is independent of all save B p l ; in fact, 

 the equation X ei is an algebraic equation of degree e t for p l and contains no other 

 undetermined quantity. 



III. Supposing one root of this to be chosen for the value of B p , and the equations Y 

 to be treated in the same way, Y ei _j will be an equation of degree (e l 1) in B p 2 , 

 whose roots are exactly the remaining roots of X v 



IV. Similarly Z E| _ 2 furnishes an equation of degree EI 2 for B p \ whose roots are the 

 remaining roots of Y,^, and therefore of X i , and so on. 



Thus B p l ...B p ' are the roots of the equation X ei . 



V. The values of #,,_!... subsequently obtained in association with each of these 

 roots will be the same in whatever order they are taken. 



Of these I. does not require proof. 



With regard to II., the proof that the equations do not contain the undetermined 

 a;'s follows exactly the same lines as the corresponding proof when the characteristic 

 equation has its roots all diiferent (vide pp. 7-8). 



The proof that they do not contain 6 P . . . until r = ej 1 requires considerations of a 

 different kind involving the equations X, Y, Z... simultaneously. 



