SYSTEMS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS. 173 



And, further, the differential equations for determining the variables separately are 

 (since V„ = KjUj + . . . + K„U„) 



(30). 



g; 1 ' G, ! ' G x 2T ^ G 1 



\ 



7=r «8»+ p — Si + 'p — S. 2 + + -p — S K — 



/vi o^Aer ivords, these are the last equations in diagonal systems when x l , x n 



are last in diagonal order respectively. 



It must be noticed, however, that (30), considered as a system, is not equivalent to 

 the given system, although it gives correctly each of the variables separately — that is to 

 bay, it gives correctly a value for each variable which, along with a corresponding set of 

 properly- determined values, will constitute a solution of the system. 



Conditions for the "Simplicity" of a Diagonal System. — Prime Systems. 



When a diagonal system contains only one differential equation, this equation must, 

 of course, be the last (unless it be possible to select from among the dependent variables 

 a set of r which can be determined wholly by non-differential equations, each of which 

 will therefore contain no arbitrary constant whatever in its expression, a case which we 

 may suppose excluded ; see Example 1, p. 175). In this case the order of the system is 

 the order of this last differential equation. All the diagonal coefficients [11], [22], 



, [n— 1, n— 1] reduce to constants, for no one can vanish in a determinate 



system ; and the first n — 1 equations are non-differential equations, by means of which 

 we can calculate successively the variables in terms of those previously found, and of 

 their differential coefficients. Such a system may be called a Simple Diagonal System. 



There are two important criteria for the possibility of reducing a given system to a 

 simple diagonal system. 



Since in all cases we have 



K = [nri] [u-1, n-l] [22] [11], 



and since the operations by which [11], , [mi] are derived from the coefficients 



of the original system are all rational, it follows that the coefficients of all these integral 

 functions of D are rational functions of the coefficients of the original system. It follows 

 that, if K be irreducible in the sense that it has no integral factors whose coefficients are 

 integral functions of the coefficients in the operator-coefficients in the original system, 



then all the functions [11], , \_nn~] , except one (under ordinary circumstances 



the last), must reduce to constants, and this one will then differ by a constant factor 

 merely from K. Hence the following important theorem : — 



If K be irreducible, then every equivalent diagonal system to ivhich a given system 



