648 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



and the number of linearly independent numbers in F,; /^-'^ will be 

 denoted by w„_/P'«'^^. By what is shown above we now have 



(44) mj^'^^ = 1 .{p=\,2, ...V- u, V = 1, 2, . . . ^p), 



(45) mjp-^^ = 



{p,q = 1,2, ...v; q 3^ p; u = \,2, . . . ixf, v = \,2, . . . n^Y^. 



For 1 ^ p ^v and u and v any two distinct integers from 1 to /Xp, 

 we may now, in harmony with the preceding notation, denote the single 

 unit of Fa/^'^^ by J„/p^; and if, further, we denote by J„a^^^ the idem- 

 potent unit /„^^^ of Vuu^'^\ we shall have as the multiplication table 

 of the system 



(p = 1,2, ... v; u, V, v, to = 1,2, . .. np-, v' y^ v), 



(47) Juv^"^ Ju'/"^ = 



(p,q= 1,2, ... v; q 7^ p; u,v = 1, 2, . . . fxp] u', v',= 1, 2, . . . ^g) 



by (31), (32), and (44), w-here Puuv = Pun = 1- For iS^p^v, and for 

 u, V any two integers from 1 to /Xp, it follows from (44) that 



and thus p^^J^^ ^ 0, otherwise J„„^p^ = I^"^^ and Jj"^ = 7,^-^^ are not 

 connected; and, since 



(n (P)V 1 (P) — (n (P) T iP^)- = ( T W T (p)V- 

 we have pum^'^'^ = Pvm^^^. Further, for 1 ^ w ^ fip, 



= p,J^U,J^P^ r^ 0; 

 and, therefore, puvw^'' ?^0. 



13 Thus, whereas, formerly Vm denoted the aggregate of numbers lueilv 

 for i = 1, 2, . . . m, of which viuv were linearly independent, Fun^^-'^ is now the 

 aggregate of numbers ly^'''^ Cih''''^ for i = 1, 2, . . . m, of which //(y/^'^^ are 

 linearly independent. 



14 Therefore, 



m = S S S S m„,.(p.9) =222 TOu/p-^^ = 2 Mp'. 



• p=l 5=1 u=l ])=l p=l«=l»=l p=l 



