340 Normal Series satisfying Linear Differential Equation*. 



$ _ o are all different, a matrix, x can be found, of the form 



in which all elements of \n save the diagonal elements, are zero, such 

 that the system of equations 



dij/dt = UT} - r)X 



is formally satisfied by a square matrix whose elements are series of 

 positive integral powers of t, generally divergent. 



The elements of the matrix y = ijtt(x) are therefore of the form 



Secondly it is found that, if the roots of the equation | a p+ i - 6 \ =0 

 are not all unequal, provided certain conditions attached to each group 

 of equal roots are satisfied, a matrix 



Y _ . . Xi 



x - ^ * 



can be found, having all elements to the left of the diagonal zero, such 

 that the equations 



drj/dt = ui) - T?X 



are still formally satisfied by a matrix of series of positive integral 

 powers of t. 



The matrix 12(x) can, in this case also, be completely integrated in 

 finite terms ; the result may contain a certain number of powers of 

 log (/). 



Applied to the single linear equation of order n it is seen that the 

 n integrals fall into groups of the form : 



log (x) + . . . . + </> r (log x) r } . 



Lastly, when the conditions above mentioned are not satisfied, the 

 system of equations is transformed by the change of variable t z k to 

 a new system of the same type in which the coefficients have poles of 

 order k p +\ at z = Q. 



It is then shown that in all cases an integer k can be found, such 

 that the conditions that the new system can be solved by expressions 

 of the above form are all satisfied. 



The groups of subnormal integrals thus found agree with those 

 obtained by Fabry* directly from the linear equation of order n. 



* These, 1835, Paris. 



