SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 33 



(&\p) n -^fo (A'."-^ _ 1+1 + A 2 '". _ ) = 

 K" Ic 



and since the last written coefficient is not zero, the roots of this equation are all 

 different, and therefore the transformed system possesses normal integrals free from 

 logarithms. 



Again, suppose a. p <n = and that a. p l -*~ 1 + a. p * 1 also vanishes; then the roots of this 

 equation all become zero, and we find that the condition a. p ' n ~* + a. p ' n ~ l + a*' n = is 

 also necessary. If this is not satisfied and the original system be transformed by 

 x = 2*, k = n 2, the equation for 6 l Kp becomes 



ai \ //}i \2 



V kp\ I u kp \ ( \l,n-2 , A 2 '" -1 J. A :! . " x 



i I ~~ r~ I \- a - Dtt-sm-r-tt- (/t-2>+i + - rt - : 



p(k-2) + \ ~ V> 



which has two zero roots and the rest all different. If the one condition necessitated 

 by the equality of the two zero roots is satisfied, the solution is again found. If this 

 condition is not satisfied, the transformation z = effects what is required. 



Suppose now all the conditions of p. 1 3 are satisfied. Then whatever value of k be 

 taken, we have 6\ p = 0, so that, as fur as we have seen, the transformation does not 

 render the solution any nearer. 



We must, in fact, proceed to consider the further conditions for the case when the 

 roots Op are equal (p. 13). 



Suppose, for instance, the first of these conditions is not satisfied, then putting 

 k = n, we shall have 9 l kp = ... d" Kp , the conditions then necessary before the deter- 

 mination of Oty-i will be satisfied, and we shall eventually obtain a binomial equation 

 for 8 kp -i of degree n in which the constant term does not vanish ; the roots of this 

 equation being all different, the subnormal integral exists. 



Thus we may go through all the equations of condition in turn. 



In the more general case, where the roots of the equation for 6 P fall into more than 

 one group of equal roots, the procedure is exactly similar. 



For example, suppose that a. p '*\ a/ 1 " 1 " 1 '' 1 ' 1 '' 4 , ... are all different from zero. Then the 

 preceding work suggests that a solution may be ftnind in which the first e l rows- 

 proceed according to powers of ' % the next e 2 according to powers of i; ' a , and so on. 



The whole would thus be of normal form with the variable t lik , k being the least 

 common multiple of e^ e 2 



In fact, if we change the independent variable to t llk , k having this value, the matrix 

 A ip -, 1+ i is identically zero, and the indices G^p.-.O'^ are the roots of (^)'' + = 0. 



These roots are all equal, and the corresponding equations of condition are all 

 satisfied owing to the vanishing of the matrices other than A A . r+I . 



If, now, we form the equation for #%,_! we have 



T. v 2 ffL _ Z 3_f)l 2 _ A 



KX 2 V k p -\ U, /CX 4 (7 Ap-l.t-2 - - U, 



~" /cp-l X "2< l -2 + A ''kp-2^ + 1 = 0, 



VOL. CCV.^ A. P 



