380 MATHEMATICS: H. BLUMBERG 
Theorem V. // G ^0, ^k>lv for every v^^h and "~ ^ 
k 
^ — ^ for every i' =t= 0, //fe following relations hold: 
k \ t t' I 
k = t + f; t = T, t' = T'; (At=G; Ar = G'). 
Theorem VI. If G^O, G' ^ 0, jk > yv for every v k, llUll > 
k 
^ — —for every ^ =j=0, k, and {jk — ya, k) = 1, C is irreducible. 
V 
Theorem VII. If G>0, G' > 0, jk ^ yv for every v and lAZJ^ ^ 
k 
^ — ^ for every =}= 0, the same conclusions may he drawn as in theorem V 
V 
with the possible exception of t = t and t' = r . 
Theorem VIII. //G>0, G'>0, y^-^yv for every v, JlHl^ > yjLZSf^ 
k V 
for every v^^, k, and (7^ — 70, k) = 1, C is irreducible. 
Theorem IX (generalizes VI). // G^O, G' ^0, 7^> yv for every v^pk, 
• ^ — — ~ for every 4= 0, and if furthermore every parenthesis (eo, 
h V 
' ' ' , ^m) lhat may occur as a factor of C is such that rjv ^ rjo for at least 
one v>0, every decomposition of C into a product of parentheses contains 
at most (yk—yo, k) factors. 
Theorem X (generalizes VIII). // G >0, > 0, 7^ ^ yp for every Vj 
'Yh ' — 'Vn 'Yit — 'Yn 
^ — for every p^O, and if furthermore every parenthesis 
k p 
(eoy ej, • • ■ y e^) that may occur as a factor of C is such that rjv > 770 for at 
least one p, every decomposition of C into a product of parentheses contains 
at most (yk — 70, k) factors. 
The lemmas leading up to theorems XI and XII are omitted. 
Theorem XI. Theorem I holds for every if the inequality 7^^ — 7o >0 
is replaced by yk-~yo< 0. 
Theorem XII. Theorem I holds for every if the inequality 7;^ — 7o>0 
is replaced by > 1. 
k 
The results here outKned will be offered for publication in extenso to 
the Transactions of the American Mathematical Society. 
