Prof. Sylvester on lactic. 51 



tains 1.2.8, and to take out the figure in that syntheme asso- 

 ciated with 7 . 9, which is 6 ; and finally to seek the syntheme 

 which contains 2.3.7, and then to take out the figure associated 

 with 8 . 9, viz. 5 ; we thus obtain the three corner figures of the 



square which represents _ as thus : 



4 6 . 



5 ! . . 



of which the six remaining figures are given by the condition 

 that in no line and in no column must the same two figures be 

 found. In order to compare these quotients, or rather the rela- 

 tions of the first of them to the remaining five with those of « 

 to a, /?, <y, 8, s, it will be convenient to subtract the constant 

 number 3 from each figure, and to transpose the first and second 

 columns ; we thus obtain 



Ua 



V 



12 3 



2 3 1 



3 12 



— 7T l — a, 







12 3 



2 3 1 



3 12 



= 77!=*, 





3 2 1 

 13 2 

 2 13 



12 3 

 3 12 

 2 3 1 



= 7T n =S, 



V = 



3 2 1 

 2 13 

 1 3 2 



3 12 

 12 3 

 2 3 1 



= 7T 3 =€. 







= 7T 9 i=/3, 



= 7T 6 = 7, 



Y 



v - 



Ef- 



v = 



Thus, for greater brevity, considering the five types to be re- 

 presented by 



a. a. a. a. a 



a /3 y 8 e, 

 or still more briefly by 



a jS 7 8 e; 



and calling the nomes Nj, N 2 , N 3 , we find that the effect of in- 

 terchanging N, and N 2 with each other is to change 



a /3 7 8 e 

 into 



a j3 8 7 e. 



In like manner it may be ascertained (and the student is ad- 

 E2 



