174 PROFESSOR CHRYSTAL ON THE EQUIVALENCE OF 



can be reduced is simple. In such a ease, each of the dependent variables will involve all 

 the m arbitrary constants required by the order of the system. 



Again, if in the reciprocal matrix there be any one column, say the last, which is 

 relatively prime, then, if we take the variable x n corresponding to that column as the last 

 in an equivalent diagonal system, the last equation in that system will, by (30), be 

 Ka"„ + , &c, = — that is to say, [nri] will differ from K merely by a constant factor ; and 

 therefore [11], , [n — 1, n — 1] must all reduce to constants. Hence 



For every 'prime column in the reciprocal matrix of a given system a series of 

 equivalent simple diagonal systems can be formed in ivhich the corresponding variable 

 is the last variable. 



In particular, if every column of the reciprocal matrix be prime, then every equivalent 

 diagonal system will be simple, and the expression for each of the dependent variables 

 will contain all the arbitrary constants of the system. 



This second criterion obviously includes the first ; for, if K be irreducible, all the 

 columns of the reciprocal matrix must be prime. 



By a preliminary transformation ive can always make the solution of any system 

 depend on the solution of another system cdl the columns of whose characteristic 

 determinant are prime. Such a system we may call a prime system. We have merely 

 to introduce new variables X ] , , X„ such that 



g 1 x 1 =X 1 , , . . . .g^'x n =X„ (31). 



Having solved the new system in X l5 , X„ , we pass to the solution of the 



original system by solving system (31), which consists merely of single equations for the' 

 separate variables. 



A prime system is by no means necessarily transformable into a simple diagonal 

 system ; it has, however, the characteristic property that, in every equivalent diagonal 

 system, the first equation is non-differential. 



Practical Methods of Solution. 



The foregoing theory suggests various methods for solving linear systems with 

 constant coefficients. The most natural method, and, if a particular solution of the 

 system cannot readily be guessed, possibly the best, is to transform the system into 

 a prime system, and then reduce the latter to an equivalent diagonal system, simplify 

 this last as much as possible, solve it, and then pass back to the original system. 



We may also proceed step by step, as follows : — Transform to a prime system, 

 separate one of the variables, say x v in this system by reducing it to an equivalent 

 system in which x x occurs in only one equation. The rest of this new system is a 

 determinate system for x. 2 , . . . . , x n . Transform this last to a prime system, then 

 separate one of the variables as before, and so on. 



