Prof. Sylvester on Tactic. 



49 



it is clear that, as regards distinction of type, r)=/3, and conse- 

 quently there are only at utmost five types remaining, which may 

 be respectively described by the symbols 



1 u« 





u« 





u« 



1 u« 





U« 



1 ii« 





U/3 





Uy 



us 





Ue 



It must be noticed that a comprehends or typifies the squares 

 numbered 1 ; /3 those numbered 7, 8, 9 ; y those numbered 

 4, 5,6; S those numbered 10, 11, 12; e those numbered 2, 3. 



I say designedly that the number of types is at utmost limited 

 to these five. But it by no means follows that the number is so 

 great as five ; for it will not fail to be borne in mind that these 

 differences have reference to the peculiar mode in which we have 

 chosen to decompose in idea each syntheme, by viewing it as a 

 symbolical product of an arrangement containing only the ele- 

 ments of the second and third nomes by an arrangement con- 

 taining only those of the first nomes. But the nomes are inter- 

 changeable, and therefore it may very well be the case that two 

 types which appear to be distinct are in reality identical, their 

 elements in the groupings appertaining to such types having 

 absolutely analogous relations to different orderings of the nomes, 

 so that the groupings will be convertible into each other by per- 

 mutations among the given elements. We must therefore ascer- 

 tain how the above types, or any specific forms of them, come to 

 be represented when we interchange the first nome with either 

 of the other two, or, to fix the ideas, let us say with the second. 



To effect this, let U«, U«, U/3, Uy, US, Ue be actually ex- 

 panded; by the performance of the symbolical multiplications 

 we obtain — 



U*= 



U/3= 



4.5.7 8.9.1 6.2.3; 4.5.8 7.9.2 6.1.3: 

 5. 6. 7 8. 9. 2 4. 1.3; 5. 6. 8 7. 9. 3 4. 1.2: 

 6.4.7 8.9.3 5.1.2; 6.4.8 7.9.1 5.2.3 



4.5.9 7.8.3 6.2.1 

 5.6.9 7.8.1 4.2.3 

 6.4.9 7.8.2 5.1.3 



Uy= 8 



Uo = 



Ue = 



8.9.6 4.5.1 7.2.3 



8.9.4 5.6.2 7.1.3 



8.9.5 6.4.3 7.2.1 



I 8.9.6 4.5.1 7.2.3 



8.9.4 5.6.2 7.1.3 



| 8.9.5 6.4.3 7.2.1 



18.9.6 4.5.2 7.1.3 

 ,9.4 5.6.1 7.2.3 

 ,9.5 6.4.3 7.1.2 



.9.6 4.5.2 7.1.3 

 .9.4 5.6.1 7.2.3 

 .9.5 6.4.3 7.1.2 



,9.6 4.5.2 7.1.3 

 ,9.4 5.6.3 7.1.2 

 .9.5 6.4.1 7.2.3 



7.9.6 4.5.2 8.1.317.8.6 4.5.3 9.2.1 



7. 9. 4 5. 6. 3 8.1. 27. 8. 4 5. 6.1 9.2.3 



7.9.5 6.4.1 8.2.317.8.5 6.4.2 9.1.3 



7.9.6 4.5.3 8.1.217.8.6 4.5.2 9.1.3 



7.9.4 5.6.1 8. 2. 317. 8. 4 5. 6. 3 9. 2.1 



7.9.5 6.4.2 8.1.3 7.8.5 6.4.1 9.2.3 



7.9.6 4.5.1 8.2.317.8 



7.9.4 5.6.3 8.1.2 7.8 



7.9.5 6.4.2 8.1.3|7.8 



7.9.6 4.5.3 8.1.2|7 



7.9.4 5.6.2 8.1.3 7 



7.9.5 6.4.1 8.2.3|7 



7.9.6 4.5.3 8.1.2)7. 



7.9.4 5.6.1 8.2.3 7. 



7.9.5 6.4.2 8.1.3 7. 



6 4.5.3 9.1.2 

 ,4 5.6.2 9.1.3 

 ,5 6.4.1 9.2.3 



8.6 4.5.1 9.2.31 



8.4 5.6.3 9.1.2 



8.5 6.4.2 9.1.3) 



8.6 4.5.1 9.2.31 



8.4 5.6.2 9.1.3 



8.5 6.4.3 9.1.2 



Phil. Mag. S. 4. Vol. 22. No. 144. July 1861. 



