Miscellaneous Papers. 237 



HARMONIC FORMS. 



(second paper.) 



By Bernard B. Smyth, Topeka. 

 Read in abstract before the Academy, at Emporia, Kan., November 30, 1907. 



CHAPTER L— PERFECT SQUARES. 

 nPHE requirements of a magic square are that all the col- 

 ^ umns, horizontal lines and two main diagonals of the pro- 

 posed square should add equally. 



In a harmonic square not only do the rectilineal lines and 

 diagonals add equally, but the sum of the vertices of all possible 

 regular quadrilateral figures, as squares, rectangles, rhombs, 

 and rhomboids, add equally. Harmonic squares are possible 

 only when the root of the square, or number of cells on a side, 

 is divisible by 4. 



The term perfect square is applied to a square which adds 

 equally not only in all the rectilineal lines but also in all the 

 diagonal lines, thus making in straight lines a number of sums 

 equal to four times the number of cells on one side of the 

 given square. Thus a square of 4 should give 16 equal sums ; 

 a square of 5 should give 20 equal sums ; a square of 6 should 

 give 24 equal sums, etc. But the perfect squares here shown 

 are of a superior character, and not only add equally in the 

 many ways shown, but also add equally in all possible quadri- 

 lateral figures in any part of the square when as many cells 

 are included in the quadrilateral as the number of cells on a 

 side of the square. 



Perfect squares are now constructed of any number of cells 

 on a side above 3. A perfect square of 3 is impossible, for 

 the reason that the number of sums necessary to entitle a 

 square of 3 to be called "perfect" is twelve, while the greatest 

 number of equal sums that can be obtained from any three 

 numbers of a regular series of nine numbers is eight, as shown 

 in part I of this paper, published in volume XIV of these 

 Transactions (1894), pages 47, 48. 



To form any sort of a magic square the given series must 

 be divided into as many sets as there are cells on one side of 

 the square. Whatever arrangement be given to the first set 

 must be followed absolutely by each of the other sets The 

 members of the first set need not be taken consecutively from 



