38 MA JOE P. A. MAcMAHON: MEMOIE ON THE 



magnitude which is obviously also a general magic square. We can, therefore, form 

 a linear function of magnitudes of the same order, n, the coefficients being positive 

 integers, and such linear functions will denote a general magic square. 



The magnitudes, of the same given order, can be taken as the elements of a linear 

 algebra, and since arithmetical addition can be made to depend upon algebraical 

 multiplication, the properties of the magnitudes can be investigated by means of a 

 non-linear algebra. 



Art. 125. The properties of a general magic square can be exhibited by means of 

 homogeneous linear Diophantine equations, and it thence immediately follows that 

 there must be a syzygetic theory of such formations. There exists a finite number of 

 ground forms, corresponding to the ground solutions of the equations, and the method 

 of investigation determines these and the syzygies which connect them. 



Generally speaking, there is a syzygetic theory associated with every system of 

 linear homogeneous Diophantine equalities or inequalities, and it is because invariant 

 theories depend upon such systems that they are connected with syzygetic theories. 



Art. 126. The method of investigation about to be given applies not only to magic 

 squares of different kinds but to all arrangements of integers, which are defined by 

 homogeneous linear Diophantine equalities or inequalities, whose properties persist 

 after addition of corresponding numbers. 



For example, the partitions of all numbers into n, or fewer parts, are defined by the 

 linear homogeneous Diophantine inequalities 



1 >:a 2 >a : ,...5:a,,, 

 and if another solution be 



A=&>&. ..==&, 



we have 



and since the property persists after addition, a syzygetic theory results. 



This is one of the simplest cases that could be adduced and is at the same time the 

 true basis of the Theory of Partitions. 



Many instances of configurations of integers in piano or in solido will occur to the 

 mind as having been subjects of contemplation by mathematicians and others from 

 the earliest times. These when defined by properties which persist after addition of 

 corresponding parts fall under the present theory. 



Art. 127. There is no general magic square of the order 2 except the trivial case 

 a a 



, but we may consider squares of order 2 in which the row and column 



Cf Ct 



properties, but not the diagonal properties, are in evidence. 

 Let such a square be 



2 



a* 



