Miscellaneous Papers. 



239 



quarter of the square. A columnar pair are those in the same 

 vertical line; a linear pair are in the same horizontal line. A 

 couplet should never be a pair. 



A comjjlement of a number in any square is the difference 

 between that number and half the sum of a line in that square. 



SCHEDULE OF SCHEMES. 



Scheme I (fig. 17). — Cowpling -arrows joining alteimate col- 

 umns and opposite lines. Numbers of trial arrangements gov- 

 ern two upper lines. This places 15 adjacent to 1 in same 

 column. 



In order to produce a perfect square from scheme 1, the 

 eight coupling-arrows of the scheme are numbered at their 

 upper ends in such a way that each of the two rows of arrow 

 numbers adds 18 horizontally and 9 vertically. There are six 

 possible combinations, each of which will produce a perfect 

 square differing from every other. From these trial arrange- 

 ments the following six model perfect squares are produced, 

 strictly according to the scheme : 



FIRST ARRANGEMENT. 



SECOND ARRANGEMENT. 



THIRD ARRANGEMENT. 



No. 4. 



No. 5. 



No. 6. 



