SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 9 



If the series t) happen to be convergent, the solution which reduces to unity at 

 t = t can at once be written in the form i)<ato ~ 1 r} ~ 1 . 



Applied to the system formed from a particular linear ordinary equation we have 

 at once the result referred to on p. 3 (v. SCHLESINGER, ' Lin. Diff.-Gleichungen,' 

 vol. I, pp. 341 ff.*). 



As a simple example of the application of the method we may take the well-known equation 

 Putting Wi = w and W-T, = s 2 dw/dt 



dz* 



. w. 



which in matrix notation is 



The equation p 



a. -p 



to canonical form is 



du/dt = 



= VM, say. 

 The subsidiary equations are 



t = - - + + y w l + w,, 



\Z Z j Z 



w. 



= gives p = \/a, and the matrix //, needed to transform the first matrix 



, so that the equation is transformed to 



i 



+ JL I___N 



which give as the general relations connecting the coefficients of the first column of ?;, putting i> = 1 + -J A, 



(i.)- 



- Tia^, 1 + ^ 2 + j-Zj- (a; 1 ,,.! - x 2 ,,,!) = 



- 2 



-> - m 



^-^ (^,.-1 - 2 -i) = I 



- 2 



Hence 



and therefore 



- 



which with the first equation gives 



2 V . n (x 



Thus a recurrence formula is established for the quantities x n l -x n 2 in terms of which a; 1 ,^! and a; 2 ,,+i 

 can be at once expressed. 



* With reference to SCHLESINGER'S demonstration of this result, see a note by the author in the 

 'Messenger of Mathematics," January, 1905. 



VOL. CCV. A. C 



