SATISFYING LINEAll DIFFERENTIAL EQUATIONS. 21 



of condition " (p. 15) for each root are satisfied, a formal solution of the linear system 

 has been found in the form 



nti(^+Xz+ +Xi\ 

 ^ \t p+l V tr 



where the elements of rj are series of positive integral ascending powers of t, and 

 XI---XP-I have all elements zero save those in the diagonals, which are made up of 

 determinate constants ; and X P consists merely of square matrices about its diagonal 

 of CD e 2 ... rows and columns respectively, each of which has zero everywhere to the 

 left of the diagonal and determinate constants everywhere else. The elements of 77 

 are in general divergent. 



The matrix II above will be known as the " determining matrizant." As occasion 

 will be found later to discuss a more general matrizant, nothing further will be said 

 of it here except for the case in which p = I , which will be worked out fully in order 

 to make clear the march of ideas in the more general case. 



12. For p = 1 the equation X ti is an equation of degree e l for 0^, Y fi _! is of 

 degree e l I for 0^, and so on. 



-.6(8*)- theiY = 

 , <P \Vi ) - u, t i ,,-i 



_ , i 



so that the remaining roots are those of Y^ diminished by unity. 



Similarly the roots of Y I _J are those of Z ei _ 2 diminished by unity, and so on, so that 

 the roots of iffi) = are <V, 0, 2 -l, ..., ^"-e.-l. 



The equation 



, i+1 s .r 1 



X, i+a is z^0 1 1 + 2) + .r 1 V(0 1 1 ) + X 1 (0i 1 ) = 0, 



and so on. The equations for y^... are of the same form, with 0^ for 0, 1 . We 

 suppose therefore in the first place that #/, 9-?... do not differ by positive integers or 

 by zero, so that the coefficients of the first terms in these equations are all other than 

 zero, and all the x's and y's are determined uniquely. The quantities a being then 

 determined, as above, the solution is altogether determinate. 



If p = l and l l ...0 l ' 1 do not differ among themselves by integers, then the solution 

 is of the form 



21 



in which a. i] = 0, unless # 2 * = 



+ + sav 



+3+ y< 



