MATHEMATICS: H. BLUMBERG 379 
The following results (lemmas^^ I- VI incL, theorems I-X incl.) are valid 
for every 
Lemma I. If G^O and G' ^0, 7f — 7o ^ 0 for every v. 
Lemma II. If G ^ 0 and G' ^ 0, 7r+r' — 7o = + ^ 7f — To for 
every v, 
Lemmalll. If M > M' and M>0(M'>M and M'>0),M == '^^^^-^^ ^ 
/ 
JlU^^Im' = ^IflUI^ ^ iLlllP ) ez;e;'3; v 4= 0, the > sign alone holding 
V \ t' V I 
when v> t {y >t'). 
Lemma IV. If M > M' ^0 {M' > If ^ 0) , there is no fixed k such that 
yk > 7. and JlHl^ ^ for every v:^0,k, 
k V 
Lemma V. // M> M'>Q{M'> M>()) there is no fixed k such that 
Ik ^ yv and ^ T^JL« for every v> 0. 
k V 
Lemma VI, IfM = M'>0, ^^+^'~^ .^ = M = M' ( = ^ = ^) ^ 
t-\-t' \ t t' I 
— — — for every v> the > sign alone holding when v > t + t\ 
V 
Theorem I. Ifjor a fixed i^, t^-to> 0 and '^^^^^^^ ^ iLZffl { f = 1, 
k V 
2, ' • ' , n} , at least one of the following n — k-\-\ congruences holds: 
T,.-T,o-O(xnod^), 
where a takes the successive values k — r, k — r + 1, n — r (or k — s, 
k — s-\-l, • • ■ , n — s) and {k, a) represents, as usual, the G. C. D. of k and <r. 
Theorem II. //, for a fixed k, t^-to> 0, l^^l^' ^ T^^^o = 2, 
k V 
' • ' , n] and (7;^ — 70, k) = \, the parenthesis C contains an irreducible factor 
of order "^k. 
Theorem III (a special case of I) . // 7» - To > 0 and IilH^ > I^lZJ^ 
[v=\,2, ■ ■ , n-\], 7„-7o = o(mod--^). 
n 
{n, r). 
Theorem IV (a special case of III). //7n-To>0, ^-2° ^ JjiZJl 
n V 
\v = \, 2, • • • , n — 1} and (t«— To, n) = l, C is irreducible. 
Theorems V-X inclusive materially extend and generalize Perron's 
theorem, /. Math., Berlin, 132, 304 (1907), which Perron designates 
as 'ein sehr allgemeines Kriterium.' 
