318 On the Theory of the Partitions of Numbers. 



"Memoir on the Theory of the Partitions of Numbers. Part III." 

 By Major P. A. MACMAHON, Sc.D., F.RS. Eeceived 

 November 21, Read December 8, 1904. 



(Abstract.) 



In this communication a "general magic square" is denned as 

 consisting of n 2 numbers arranged in a square in such wise that each 

 row, column, and diagonal contains a partition of the same number. 



Such squares can be added together, by addition of corresponding 

 numbers, without losing the magic property, and we can thus speak 

 of a linear function of squares of the same order, the coefficients being 

 integers. The squares can, in fact, be regarded as numerical magnitudes, 

 and can be taken as the elements of a linear algebra. Since, moreover, 

 arithmetical addition can be made to depend upon algebraical multipli- 

 cation, the properties of the magnitudes can be investigated by means 

 of a non-linear algebra. 



The properties of a magic square can be exhibited by means of a 

 system of homogeneous linear diophantine equations, so that it 

 immediately follows that there is syzygetic theory of such formations. 



The method of procedure is set forth and worked out in detail for 

 the third order. 



In Section 10 the general question of enumeration associated with 

 a given sum is considered, and some particular results obtained. 



The methods are applicable not only to general magic squares as 

 herein denned, but to all cases of forms in " Arithmetic of Position," 

 which retain their properties after addition of corresponding elements. 



