644 PROCEEDINGS OF THE AMERICAN ACADE]\IY. 



1 to r. For, in this case there are two pairs of numbers, namely, 

 (w, v)', {v, u)' and (i', w)', {w, v)' such that 



(w, v)' {v, u)' = lu, {v, ^ly {u, v)' = /„, 



(y, w)' {w, v)' = I„ {tv, v)' {v, to)' = ly,. 



Therefore, if 



(w, w)' = {I'i, v)' {v, wY, {w, u)' = {w, vY {v, u)', 



we have, by (26), 



(w, wY {iv, uY = {u, vY. {v, «')' (^''j ^')'- ('i'» w)' 



= (w, vY iv (i', uY = (it, vY {v, 'uY = h, 



(w, uY {u, wY = {w, v)'. («, uY (u, vY. (v, ivY 



= {w, v)' I, (i', wY = (w, v)' (v, ivY = Iw 



For u, V any two distinct integers from 1 to r, let 7„ and 7„ be con- 

 nected. Thus let 



(u, vY (v, ^lY = h, (''. ")' (u, vY = Iv 



Let k = m^u - 1; an<^l let the nilpotent units of Tuu be denoted by 

 Nu^'\ iVV'\ • • • Nu^'^Y Then {u, vY and the products Nj^^-{u, vY, 

 ior h = 1,2, . . . k, are numbers of the aggregate r„„ linearly inde- 

 pendent. For, if 



k 



90 ■ {u, vY + L ghNu^'^ ■ {u, vY = 0, 

 then 



jr. Z- 



9oIu + I QhNu^'^ = [go • (u, vY + I gkNu^"^ ■ {u, vY] (v, uY = 



h = l h=l 



which is impossible, unless the gf's are all zero. Therefore, 



muv ^k -\- I = niuu- 



Moreover, there is no number in the aggregate T^v linearly indepen- 

 dent of these k + 1 numbers of this aggregate. For, if {u, v) is any 

 number of this aggregate, since (w v) {v, uY belongs to the aggregate 

 Tuu, we have 



k 



{u,v)iv,uY = coIu+ L ChNu^''^; 



