MATHEMATICS: H. BLUMBERG 
377 
arbitrary complex numerical coefficients.^ The parentheses and the 
product of two parentheses are defined as in 1. The rank rj of an 
element e (=1= 0) is defined as d'~d", where d' and d" represent respec- 
tively the degrees (in the usual sense) of and e" in the k letters Xi, 
• • Xk. The rank of 0 is defined to be — oo . 
00 
4. (S consists of the collectivity of elements e = x^~^, where 
X = 0 
the c's are arbitrary complex numbers, that is, e is a Laurent series 
having only a finite number of terms with positive exponents. The 
parentheses and the products of parentheses are defined as in 1. The 
rank oi e (4= 0) is defined to be the exponent of x in the first non-zero 
term of the development of e. The rank of 0 is (naturally) defined to 
be — 00. 
5. @ consists of the collectivity of fractions e = e'/e", where e' and 
e" are power series in k letters Xi, X2, • • x^ with arbitrary complex 
coefficients: 
00 CO 
Xi, X2, •■;\k=^ Xi, X2, • • Xife = 0 
The rank of e (4= 0) is defined as d" —d', where J' and d" represent the 
lowest degrees (in the usual sense) in the k letters Xi, x^, • • of a non- 
zero term of and e" respectively. The rank of 0 is — oo . 
6. @ consists of all rational fractions e (x) = e' {x) je" (x) in x {e! {x) 
and e" (x) being polynomials) with arbitrary complex coefficients. The 
parenthesis (^o, ^i, • • ^w) of © is the linear homogeneous difference 
expression y^+m-^ yx+m -i ' ' " + The product A - B = 
(ao, ai, • • ■, dr) • (bo, hi, ■ • bs) of two parentheses of ® is the ordinary 
symbolic product of the linear difference expressions A and B and is 
equal to {co, Ci, • • Cn) where n = r -\- s and 
V 
2^x W br;_^(x'^r-\), Ip = 0, 1, 2,' ' ',n}. 
x=o 
The rank of e (x) is what is ordinarily called the degree of e (x), i.e., 
d'— d", where d'= degree of {x) and d"= degree of e" (x). 
7. and rank are defined as in 4; parentheses and products of 
parentheses, as in 6. 
II holds in the following situations (8-9 incl.) : 
8. and rank are defined as in 6. The parenthesis (eo, ei, • • • , 6^) 
of © is the linear homogeneous differential expression 
d^y d^~^y 
