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MATHEMATICS: H. BLUMBERG 
I a) T„S ^™i,(ax + /3M), {>'=0, 1,2,...,«!, 
if + /3ju attains its maximum, X and varying so as to satisfy the rela- 
tion X + M = for a single pair of values of X and ix. Here a^, = 
rank^ of a^, h^, Cp. 
n a) 7.^x+;|.(«x + /3m). {^ = 0,1,2,...,^}, 
h) yv= x + r=''^''^ + ^^)' {^^ = 0, 1, 2,' ' w}, 
if aj^ + jg^ attains its maximum Mv (X and /z varying so as to satisfy 
\-\- ix = v) for a single pair of values of X and /x, and if My ^ a;^ + 
whenever \-{- ix < v. 
HI «) y,^^^;- {. = 0,1,2, •■•,»}, 
0- being an integer such that O-^a-^v. 
b) x.-x:r-(«x+^.), {.=0,1,2, •••,«}, 
if q;^ + attains its maximum Mp (X and fi varying so as to satisfy 
\-\- fi= v) for a single pair of values of X and /x, and if Mp> ax + iSju+o- 
for X + ju + 0- = 1/ and a-> 0. 
It is to be noted that I implies II and that II impHes III. Hence 
every theorem proved for all ©^^^ {i.e., for all © having property III) is 
valid for every and and every theorem proved for all holds for 
every 
We shall now describe various important situations where I or II or 
III holds. For this purpose, we must define in each case the aggregate 
®, the parentheses of (S, the product of two parentheses and the rank of 
every element of @. 
I holds in the following situations (1-7 incl.): 
1 . © consists of the set of rational numbers. We understand by the 
parenthesis {cq, Ci, e^,' ' Cra) of the rational polynomial Coy"^ + eiy"""^ + 
■ • • + e,n in the letter y. The product of two parentheses (ao, ai, • • 
ar) • (bo, h, • • •, bs) is defined, as usual, to be equal to the parenthesis 
(co, Ci, • • • , Cn), where n = r -\- s and Co = aobo, Ci = aobi + aibo, • ■ • , 
On = drbs. The rank of an element e=e' /e" (where e' and e" are integers) 
is defined with reference to a fixed prime p. First let ^ 0; let e' be 
divisible by /' but not by + e", by but not by + ^ We 
define the rank of e by the equation 77 = d" — d'. Moreover, we (natu- 
rally) define the rank of 0 to be — 00 . 
2. (5 consists of the class of the Hensel ^-adic numbers. The pa- 
rentheses of @ and the product of two parentheses are defined as in 1. 
The rank of the ^-adic number e is the negative of Hensel' s^ 'order' of e 
with respect to p. 
3. @ consists of the class of rational fractions e = e' {xi, • • •, XKj/e" 
{xi, ' ' Xk) in k letters Xi, x^, ■ • Xk, e' and e" being polynomials with 
