58 



KANSAS ACADEMY OF SCIENCE. 



umne of scheme I are converted into lines and the lines into columns. Fig. 34 is 

 simply fig. 33 reversed, as fig. 26 was fig. 25 inverted, and as in that scheme seven 

 of the derived squares will be like 33; eight of them like fig. 34. The coupling 

 arrows in this scheme will be numbered at their left-hand end. 



The first six arrangements, corresponding to the six harmonic arrangements 

 of scheme I, are here shown with their corresponding squares: 



1st Arrangement. 

 No. 38. 



2d Arrang-ement. 

 No. 39. 



3d Arrangement. 

 No. 40. 



1 8 



7 2 



6 3 



4 5 



1 8 



7 2 



4 5 



6 3 



1 8 



6 3 



7 2 

 4 5 



4th Arrangement. 

 No. 41. 



5th Arrangement. 

 No. 42. 



6th Arrangement. 

 No. 43. 



1 8 



6 3 

 4 5 



7 2 



1 8 



4 5 



7 2 



6 3 



1 8 



4 5 



6 3 



7 2 



All the other arrangements of scheme I, up to the 96th, are equally applicable 

 to this scheme; and from each arrangement 16 squares can be constructed with 

 equal facility with those of scheme I. 



L>. 33. 



"isJi^. 



^^. 3&: 





^^. 3 b. 



Scheme III (No. 45 of Schedule). 



This scheme (figs. 35 and 36) is like scheme I, except that the medial lines 

 are transposed. In the trial arrangements the lines apply to the first and third 

 (or upper alternate) lines of the square, instead of the first and second, as in 

 scheme I. The upper line of the arrangements answers to the upper half of the 

 square; the lower line to the lower half. All the harmonies and characteristics 

 spoken of scheme I are equally true of this. 



Samples of the harmonic squares yielded by this scheme are here shown: 



1st Arr. No. 44. 6th Arr. No. 45. 



(17 6 4 

 i 8 2 3 5 



6th Arr. 

 (14 7 6? 

 < 8 5 2 3 J 



Comparison of these squares may be made with similar arrangements in 

 scheme I. 



