90 



±S, = An-X.Sn (5) 



+ .So = An.Sn (6) 



the last of which shows that U, the particular integral of equa- 

 tion (1), is determined if Ua, Mi, u^. .. iin-\i the particular inte- 

 grals of (2), be known. The remaining equations indicate 

 the relations which exist between A^, A2, . . . A^ and Mq, Wi, 

 . . . M„.i. We are not able to derive the integrals Uq, Mi» • • • 

 Un-i from the equations just given, any more than we can 

 determine the roots of an algebraic equation from the well 

 known relations between them and its coefficients. In fact, 

 if we were to multiply the equations (3), (4), (5), (6) by 

 2)""^Mo-D"'^Mo • • • Duo, Uo, and add to their sum the identical 

 equation, 



we should eliminate the other roots Mj Mg • • . Mn-i, but at the 

 same time reproduce the original differential equation. 



All the preceding reasoning applies whenever D denotes, 

 not merely the operation of taking the differential coefficient, 

 but any distributive operation such that 



Z)" {cq + CiX + .. . + c„.ia;"-') = 0. 



The results obtained hold good, therefore, in the case of equa- 

 tions in finite differences. 



As regards the case of diflferential equations, it is worthy 

 of notice that the equation (3) admits of integration inde- 

 pendently of any relation between the functions Mq, Ui, .. . Un.\. 



Since 



'J 71-1 = D Sn, 



we have 



S (+ UoDuiD^u^ . . . D"-^Un.2D'^Wi) = e~^^''^^' 

 And it follows from this that the left hand member of 



