640 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in terms of the J's. Whence it follows that we may take the J's as 

 new units, and the number system thus transformed is regularized 

 with respect to the idempotent numbers 7i, I2, ... If. ^ 

 Since, f or 1 ^ w £ r, /„ belongs to Tuu, we may put 



(29) Iu= Jumuuu (u= 1,2, ... r). 



If now B' is any number of the system (ei, e2, ... Cm) satisfying the 

 equation h B' = B', then, by (26) and (27), 



B' = Y. H b'vhJuhv; 

 similarly, if B"Iu = B" , we have 



B" = Y. 11 b"vhJvhu' 

 Therefore, by iheorem I, 



r 



(30) 



T 



mS'zIu — y mvu. 



r=o (w = 1, 2, . . . r)- 



Let (m, v), for u, v any two integers from to r denote a number of 

 the aggregate r„„. From (26) and (27), it then follows that the non- 

 vanishing products of numbers in the several aggregates are given by 

 the following equations: 



(31) {u, v) (v, w) = (u, w) 



{u,v, w = 0, 1,2, . . . r); 

 and we further have 



(32) (m, v) {v', ir) = 



{u, V, v', w = 0, 1,2, . . . r; v' 9^ v).^ 



7 When the number system is thus transformed each of the new units is 

 in one or other of Peirce's four "groups" or aggregates with respect to each 

 of the r idempotent numbers 7i, I2, • • • It- Thus, if u is any integer from 

 1 to r and v, iv any two integers from to r other than 71, then the units 

 Juh]u (1 ^ /ii ^ muu), Juhin (1 ^ hi < m.uv), Jvhsu (1 ^^3 ^ "«tu), and 

 Jvh w (1 S fu = '")vu' are respectively in the first, second, third, and fourth 

 groups with respect to lu- See B. Peirce, loc. cit., p. 109. 



We have now 



r r 



"'uv- 



m = S 2 m 



u=0»=0 



8 Cf. B. Peirce, loc. cit., p. 111. 



