PHILOSOPHICAL TRANSACTIONS. 



I. On the Normal Series Satisfying Linear Differential Equation*. 



By E. CUNNINGHAM, 13. A., Fellow of St. John's College, Cambridge. 



Communicated by Dr. H. F. BAKER, F.If.S. 



Received December 14, 1904, Read December 15, 1904. 



CONTENTS. 



Section Page 



1. Introductory ........ . .................. 1 



2. The equations to be considered .and the canonical form of a linear system of equations . . 3 



3. The solution in view ........................ 4 



4. The unique determination of the determining matrix when the roots of the characteristic 



equation are unequal ....................... 5 



5. The completion of the solution in the same case ............... 



6. The general case; restriction on systems considered .............. 10 



7. The matrix x ; preliminary assumptions as to its form ............. 11 



8. The difference equations for the coefficients ; equations of condition ......... 11 



9. On certain operators A r and their application ......... ....... 14 



10. The particular case when the roots of a certain equation are unequal ........ 16 



11. The non-diagonal elements of x in this case ................. 19 



12. The complete solution for p 1 in this case ................. 21 



13. The solution forp = 1 in the general case ................ 



14. Resumption of most general form .................... 25 



15. Application of the method to a particular equation .............. 29 



16. On the method to be adopted when certain equations of condition are not satisfied; sub- 



normal forms ......................... 30 



17. Complete solution of a certain equation of the third order ............ 34 



1. THE present paper is suggested by that of Dr. H. F. BAKER in the 'Proceedings 

 of the London Mathematical Society/ vol. xxxv., p. 333, "On the Integration of 

 Linear Differential Equations." In that paper a linear ordinary differential equation 

 of order n is considered as derived from a system of n linear simultaneous differential 

 equations 



or, in abbreviated notation, 



dx/dt ux, 



VOL. CCV. A 387. B 21.6.05 



