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MATHEMATICS: H. F. BLICHFELDT Proc. N. A. S. 
€2€3- -e n + €163. .€ n + ... + 6i€2. .€ w _i < /( j W I ). (2) 
The conditions (D) referred to are: for every positive integer w, f(w) 
is a positive number which decreases when w increases, and approaches 0 
as a limit when w tends to 00 . 
4. We proceed to give an outline of the proof, limiting ourselves to the 
case n = 3, which is sufficiently illustrative. The function }{w) being given 
subject to (D), we shall take as our equations the following set (under an 
obvious change of notation) : 
x — aw — ra = 0, y — fiw — s/3 = 0, z — yw — ty = 0. (3) 
Here r, s, t represent three arbitrary, but different integers, while a, 7 
are defined as the limits approached when / tends to 00 by three series of 
generalized (Jacobian) continued fractions Xj/wj, yj/wj, Zj/wj,'j = 1, 2, .... 
We take at the outset any four sets of non-negative integers (#1, . ., Wi), 
. . , (#4, . ., W4), such that the determinant (^2^4) = 1; succeeding sets 
(xj, . ., Wj' f j >4) are constructed from previous sets by the rule: 
Xj = Sj-iXj-x + Xj-4, yj = Sj-iyj-i + yj- 4, etc., involving a series of positive 
integers 54, 5 6 , 56, . . . . , which, in turn, are expressed in terms of another 
series designated by [3], [4], [5], . . . . ; namely, we put sj = 2 ^ ~ 
The members of the new series are any set of positive integers satisfying 
the following conditions: 
[3] = 1; when / > 3 take [/] > 3[; - 1], f(2 [j] - [j - 1] ) < 2' 6U ' 1] . 
5. For convenience in the further development we make use of the 
letters M, m, p to represent certain non-negative coefficients (constants 
or variables), the actual values of which are not required: M for a number 
having a lower bound > 0, m for one having an upper bound, and p for 
one having both a lower bound > 0 and an upper bound. 
We note the following preliminary relations: 
xj = M r 2 y " 1) , y s = /x.2 y -^etc.; 
- awj = =±m/s j} 77/^-/^ = *=m/s jy ^ = Zj -yw 5 =±m/s j ; 
XiWj — XjWi = (xiWj) = =tm.2 y ~ 1] " " 1] when / > i, y(U0j — yjWi = 
(yiWj) = etc., etc. Moreover, (xfJUj), etc., do not vanish when / is taken 
sufficiently high. 
(The condition (C), §1, is satisfied by a, /3, y. For, an equation a 0 
+ aia + a 2 |8 + a s y = 0 would imply a Q Wj + aiXj + a 2 yj + a z zj = 0 for every 
subscript / above a certain number. But this requires a 0 = ai = a 2 = a 3 = 0, 
since the determinant (Xjyj + \Zj + 2Wj+z) — =*= 1.) 
6. If x, y, z, w represent any four integers, then at least two of the 
numbers 
A = xwj — xj (w + r), B = ywj — yj(w + s), C = zwj — Zj(w + t) 
are of magnitude =*= M . 2 b '~ 11 " 2[j ~ 2 \ when / exceeds a certain number. 
For selecting any pair, say A, B, we have 
(Aq — Bp) Wj-i + (r — s)pq=0 (mod. Wj), 
