378 
MATHEMATICS: H. BLUMBERG 
The product A - B = (ao, ai, - • • , a,) ■ (bo, h,---, b,) of two parentheses 
is the ordinary symbolic product of the linear differential expressions 
A and B and is equal to (co, Ci, • • • , cj, where n = r-]-s and 
9. and rank are defined as in 4; parentheses and products of 
parentheses, as in 8. 
Ill holds in the following situation: 
10. © consists of the collectivity of fractions e = e' /e" , where e' and e'^ 
are power series in x—h { = Xi, of. 5) with arbitrary complex coefficients. 
The rank of e is defined as in 5. The parentheses of @ and products 
of parentheses are defined as in 8. 
We shall now proceed to the statement of certain consequences of 
the assumptions made that relate to factorization properties of a given 
parenthesis C = (co, Ci, • • , €„) of <S. We base all further considera- 
tions on the tentative assumption that C may be expressed as a prod- 
uct A • B = (ao, ai, ■ • • , a^) (bo, bi, ■ • • , bs) of two parentheses, where r^l 
and s^l. When other assumptions to be made in the theorems con- 
tradict this assumption of ' reducibihty/ it must be that C is under 
those later assumptions incapable of being expressed as a product of 
two parentheses whose orders are at least 1. C is then said to be 
'irreducible.' (In general, the terminology emiployed for our abstract 
situation is parallel to that for the ordinary concrete situations.) We 
assume furthermore throughout in what follows that yo, the rank of Cq, is 
finite (i.e., 4^ — 00 ). We introduce the following notations (partly for 
the purpose of indicating our method of investigation and partly for 
the purpose of simphfying certain future statements) : 
jAo = ao — ao = 0, Ai = q;i — o;o, • ' *, IS.r = ar — ao . 
\ V / t V 
