In a recent paper (Reed, 1920) I showed that the growth of shoots on Bartlett pear trees followed the course of an autocatalytic reaction very closely, except in the first few weeks of the season. During the early part of the growth period the calculated values were larger than those observed. Using the observed data I have computed values of K and c from the equation

Plotting the values of this equation (fig. 2) the value of log K (the intercept) is -1.75 (K= .0178) and c (the slope) is 1.09.

When the size of the pear shoots is computed from the equation

we obtain values agreeing more closely with the observed, as shown by table 2, where x = observed mean length of the young shoots at successive intervals; x1 = length calculated from

01=deviations of x from x; x=length calculated from

Graph of values for pear shoots. Points represent values of log (t-47.4) when t<t; crosses represent same when t> ti.

x is 2.38 cm., and that of x2 from x is 3. 14.

Finally, a growth process which closely coincided with values computed from the usual formula, may be mentioned. In the paper cited (Reed, 1920) the average height of a population of young walnut trees (J. regia) was presented. The calculated values had a root-mean-square deviation from the observed of only 3.12 cm. The equation for growth as derived by graphic method was

FIG. 3

Graph of values for walnut trees. Points represent values of log (t −96) when t<t; crosses represent same when t> t1.

The value of K here is only .002 less than that first used and the exponent of (t 96) is practically unity. The root-mean-square deviation of values obtained from this equation was 2.65 cm., which differs so slightly from the first that there is no real advantage in favor of one equation over the other.

The use of the method described does, however, give a more accurate means of computing the growth in cases where the initial growth of the organisms falls measurably below that given by the equation as previously used.

* Paper No. 85, University of California, Graduate School of Tropical Agriculture and Citrus Experiment Station, Riverside, California.

Mitscherlich, E. A. (1919), “Ein Beitrag zum Gesetze des Pflanzenwachstums," Fühling's Landw. Zeit., 68 (130–133).

Mitscherlich, E. A. (1919), “Zum Gesetze des Pflanzenwachstums,” Fühling's Landw. Zeit., 68 (419-426).

Reed, H. S. (1920), "The Nature of the Growth Rate," J. Gen. Physiol., 2 (545–561). Rippel, A. (1919), "Die Wachstumkurve der Pflanzen und ihre mathematische Behandlung durch Robertson und Mitscherlich," Fühling's Landw. Zeit., 68 (201–214). Robertson, T. B. (1908)," Further Remarks on the Normal Rate of Growth of an Individual and Its Biochemical Significance," Arch. Entwicklugsmechn. Organ., 24 (108).

ON THE APPROXIMATE SOLUTIONS IN INTEGERS OF A SET OF LINEAR EQUATIONS

BY H. F. BLICHFELDT

DEPARTMENT of MathematiCS, STANFORD University, California

Read before the Academy, April 26, 1921

1. It has been proved that a set of n+1 integers1 x1, x2, exist which satisfy approximately the n equations2

where α1, ..., αn, B1, .

....

ẞn are 2n given real numbers, the first n of which are subject to a condition (C) stated below. The degree of approximation is as follows: having specified a positive number e as small as we please, then the numerical values of the left-hand members of the above equations (i.e., the "errors") can simultaneously be made smaller than e The condition (C) imposed upon a1, α2, an is this: they are to be irrational numbers of such a nature that a linear expression of the form a。 + a1α1 + ... + anan, with integral coefficients ao, a1, vanish exactly only when these coefficients are all = 0. For a given approximate solution of the equations (1) the errors shall be designated, respectively, by 1,..., n.

=

βε

=

... an, can

...

=

Вп

=

0,

2. In the two special cases: (I) n = 1, or (II) ẞ1 we can make additional demands upon the errors, namely (and this is of importance in applications) we can cause them to be smaller than certain corresponding functions of w. Thus, in the case (I), we can demand that < 1/(4 | w |); in case (II) we can demand at the same time < k/(\w])", where k 1)/n; furthermore, the condition (C)

€ < and at the same time that
that €1,
← be each <e and

=

is not necessary in case (II).3 These results may be stated in another way: in case (I), a being irrational, an infinite number of sets of integers (x', w'), (x", w"), . . . . exist which, when substituted in turn in x απ β = 0, will produce errors e', e", .... of such magnitudes that ' < 1/(4 | w' |), e′′ < 1/(4 | w"), etc. Similarly, in case (II) we have an infinite number of solutions satisfying corresponding inequalities.

3. In the general case the errors cannot be made to satisfy such extra demands. Stated more precisely: no matter what function f(w) be assumed, if only it be subjected to the conditions (D) below, we can always construct a set of equations (1) with attendant condition (C), which do not admit an infinite number of approximate solutions in integers (x1, x2, Xn), .. if we demand the following degree of accuracy for each solution: €1 < ƒ( | w | ), . . ., en < ƒ( | w | ). In fact, we may even substitute the

...

€2€3.. En + €1€3..εn + + €162 . . En _1 < ƒ( | w | ). (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 ∞.

4. We proceed to give an outline of the proof, limiting ourselves to the case n = 3, which is sufficiently illustrative. The function f(w) being given subject to (D), we shall take as our equations the following set (under an obvious change of notation):

x

aw - rα = 0, y

βω

=

0, z

γω ty = 0. (3) Here r, s, t represent three arbitrary, but different integers, while a, ß, y are defined as the limits approached when j tends to∞ by three series of generalized (Jacobian) continued fractions x;/w;, yj/Wj, Zj/wj; j = 1, 2, .... We take at the outset any four sets of non-negative integers (x1, .., wi), .., (X4, (x4, .., w1), such that the determinant (x1223w1) = 1; succeeding sets .., w;; j>4) are constructed from previous sets by the rule:

(xj, X; =

.....

· Sj− 1Xj−1 + Xj−4, Yj = Sj−1Yj−1 + y;-4, etc., involving a series of positive integers S4, S5, S6, which, in turn, are expressed in terms of another series designated by [3], [4], [5], .; namely, we put s; = 2 [j] - [j − 1] ̧ The members of the new series are any set of positive integers satisfying the following conditions:

[3]

=

1; when j> 3 take [j]≥ 3[j lj 1], ƒ(2--1)) < 2−6 [-1]. 5. For convenience in the further development we make use of the letters M, m, μ 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 μ for one having both a lower bound > 0 and an upper bound.

We note the following preliminary relations:

&j=x; — aw; = ±m/s;, n;=y;— Bw; = m/s;, $j=Zj — yw; = ±m/s;; = ±m. 2-1] - [i]+[i-1] when j > i, y;w; XjWj X¡W;= (x;W;)

=

YjW; =

(y;w;): etc., etc. Moreover, (x;w;), etc., do not vanish when j is taken sufficiently high.

(The condition (C), §1, is satisfied by a, ß, y. For, an equation a. +a1α+a2ß+αзy = 0 would imply a。w; +a1x;+a1⁄2y; +a3%; = 0 for every subscript j above a certain number. But this requires a。 = a1=a2=A3=0, since the determinant (x,y;+1%; +2Wj+3) = ± 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 y; (w+s), C = ZW; — z;(w+1) are of magnitude M.2-11-26-21, when j exceeds a certain number.