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



By P. A. MACMAHON, Major R.A., Sc.D., F.R.S. 

 Received November 21, Read December 8, 1904. 



SINCE Part II. of the Memoir appeared in November, 1898, the following papers by 

 the author, bearing upon the Partition of Numbers, have been published : 



" Partitions of Numbers whose Graphs possess Symmetry," ' Cambridge Phil. 



Trans.,' vol. XVIL, Part II. ; 

 " Application of the Partition Analysis to the Study of the Properties of any 



System of Consecutive Integers," 'Cambridge Phil. Trans.,' vol. XVIII. ; 

 "The Diophantine Inequality KX^/JHJ," 'Cambridge Phil. Trans.,' vol. XIX. : 

 " Combinatorial Analysis. The Foundations of a New Theory," ' Phil. Trans. 



Roy. Soc. London,' A, vol. 194, 1900. 



In the present Part III. I consider problems of " Arithmetic of Position." In 

 particular. I define a " general magic square " composed of integers and show that for a 

 given order of square it is possible to construct a syzygetic theory. Such a theory is 

 worked out in detail for the order 3 as an illustration. I further discuss the problem 

 of the enumeration of the squares of given order associated with a given sum. I show 

 that there is no difficulty in constructing a generating function for such squares even 

 when the construction is specified in detail, and I obtain an analytical expression for 

 the number when the sum, associated with rows, columns and diagonals, is unity 

 or two. 



9. 



Art. 124. A "general magic square" I take to consist of n 2 integers arranged in a 

 square in such wise that the rows, columns and diagonals contain partitions of the 

 same number, zero and repetitions of the same integer being permissible among the 

 integers. 



An ordinary magic square I define to be a general magic square in which the n 2 

 integers are restricted to be the first n 2 integers of the natural succession. 



We may regard general magic squares as numerical magnitudes. To add two such 

 magnitudes we add together the numbers in corresponding positions to form a 



VOL. CCV. A 388. 30.6.05 



