TABER. — SCALAR TUNCTIONS OF HVPER COMPLEX NUMBERS. 643 



for any two numbers (u, v) of r„, and (c, u) of T^^- I-'Ct (w, v)', 

 (r, ?/)' be any two numbers of T^t, ^vu respectively. Then 



{u, v)' (v, v)' = plu - A'u 



by (31), where N^ is hnear in the nilpotent units of !„„ and is, there- 

 fore, either zero or nilpotent, and thus iV/"^^ = for some positive 

 integer p. Furthermore, 



If now /„ and /^ are connected, then, for a proper choice of {u, v)', 

 {v, u)', we have Si(w, v)' (r, u)' 9^ 0, in which case p 7^ 0: therefore, 

 we may put 



{v, u)" = 4r, iv, u)' (p^Iu + P^-^A^ + . . . + P^\''-' + AV), 

 when we have 

 {u, vY (v, u)" = 4n (P^u - Xu) (p'' lu + P^-'Xu + . . . + pA'u^-^ + Nu") 



and since 



[(v, ur (u, v)J = (v, u)". {u, vY (r, «.)". (u, vY 



= {v,uY'h(u,vY = {t,u)"{u,vY, 

 it follows that 



(r, uY' {u, vY = Iv, 



otherwise, there is more than one iderapotent unit in T„, which is 

 contrary to supposition. Wherefore, if /„ and I„ are connected, there 

 arc two numbers (if, vY and («, ?a)', of Tuv and T^u respectively, such that 



(u, vYiv, uY = lu, («, w)'(u, v)' = I,; 



and conversely, since in this case 



Siiu,vyiv,uY = Srh ?^o. 



// /„ and /p are connected, and /„ and I^ are also connected, then /„ 

 and /„, are connected, where u, v, ic are any three distinct integers from 



11 Further, S2 (u, v)' (v, u)' = pS^h; 



therefore, if Si {u, v)' (v, u)' 9^ 0, then Si {u, v)' {v, u)' 7^ 0, and conversely. 



