This gave directly a series of intensity-potential graphs that are represented with fairly good accuracy by the law

where I (V, v)dy is the intensity in the frequency range dv per electron striking the target from potential V, and H is the ratio of Planck's h to the charge of the electron, and k(v), p(v) and q(v) are functions of v. Since p and q are pure numbers they are independent of the arbitrary intensity unit, and can be determined wherever the data are available, though not very accurately because the term containing them is rather small. In the few data available for rhodium, p is of the order of 0.06 to 0.08 and q about 12 to 16, making pq about 1. The work on tungsten and molybdenum gives only a few points on each intensity potential graph, and because of the smallness of the exponential term and its disappearance at potentials large enough for really accurate intensity measurements, it is impossible to get an accurate test of this law except with more points than one can get from these graphs. But the data obtainable show that the relation between I and V is not far from linear, and the only definite curvature seems to be something of the type indicated by the above equation. In platinum, we have data scattered over the range from 1.33 to 0.43 Å, but most of them rather rough. But to an accuracy of 20 or 30%, they seem to indicate constant values for both p and q, with p = 1/5 and q = 13, so that pq = 5/2. Fortunately the smallness of the p and q terms makes their influence on the determination of k also small, although the existence of the terms themselves may be of considerable theoretical importance. For the present, therefore, we shall include these terms in the calculation, but neglect any changes of p and q with v.

An important point to be deduced from the graphs of 'intensity' against is the fact that they are smooth and regular, so that k must have no discontinuities or sharp curvature in its graph against v. As a trial value we shall therefore assume first that k(v) = kv", with k and n both constant. energy is then

The total

If k(v) is not expressible in this way, but as a power series in v, then this equation gives a power series for E in terms of V, and if the coefficients for such a series are found experimentally those of the series for k can be computed from them. If E depends on a single power of V, then k must depend on a single power of v.

Now, Beatty's work indicates that E = constant X V. Hence, we infer that n 0 and k(v) = k constant, and

=

=

In a previous paper by the author' it was shown that, neglecting absorption in the target, the intensity per electron and per atom and per unit interval of frequency from an extremely thin target would be

where b is the coefficient of the Thomson-Whiddington law in the form V2 V2 = bx, and N is the number of atoms struck per unit length of the electrons' path. This equation is of course subject to some error due to absorption in the target, and as one may readily prove, could be corrected by adding to i a term

αμ
I (V, v),
N

where a is the ratio of the distance travelled in the target by the emerging X-rays to the distance travelled by the cathode ray before it emits X-rays, and is the linear absorption coefficient. This correction is important only at low frequencies. Neglecting it, we have

μ

with k, p and q all constant. It must be remembered that the data are very incomplete and this result is unreliable and is presented only for lack of anything better. But the true law must have something of the same. general characteristics as this, and certain conclusions about the emitting mechanism can be drawn from that fact.

First, let us assume that some form of quantum law governs the radiation of frequencies different from V/H as well as at that one. We are then practically though not rigorously led to one of two alternatives. If the quantum law merely regulates the frequency being emitted at any instant in terms of the energy still available at that instant for radiation, so that the radiation by one electron is not monochromatic, then every electron radiating at all must radiate some energy at the highest possible frequency, V/H, and presumably give up all its energy to radiation. But this would make i(V, v) independent of, and is therefore improbable. The more probable alternative is that the radiation by one electron is monochromatic, and the quantum law gives its

frequency in terms of the total energy radiated. In this case only a few of the electrons radiating will do so at frequencies very near the limit, V/H, though more will radiate there than in an equal frequency interval at a frequency not too far below that one. Adopting this conclusion, the chance that an electron radiates a fraction of its energy within a range e to € + de is f(V, e)de, where

Let R be the radius of the atom and r the distance from the particle causing radiation to the path the electron would have taken if not deviated. Then

If we assumed the frequency to be determined by the energy transferred in a collision of two equal repelling particles, one of which was initially at rest, a definite value of could be predicted for any assumed values of r and V, and the forms of f(V, e) and i(V, v) could be found. Several such assumptions have been tried, but none give the forms of ƒ and i required above. A detailed discussion would be out of place with such rough preliminary data, and any such assumption would be improbable, but it seems likely that more exact data of this type ought to throw valuable light on the mechanism of radiation. It is hoped that with apparatus now being constructed it will be possible to get such data.

1 Beatty, R. T., Proc. Roy. Soc., London, A 89, 1913, (314–27).

2 Webster, D. L., these PROCEEDINGS, 2, 1916, (90-4); Physic. Rev., Ithaca, 9, 1916, (599-613).

Webster, D. L., and Clark, H., these PROCEEDINGS, 3, 1917, (181-5).

4 Hull, A. W., Amer. J. Roentgenology 2, 1915, (893-9).

5 Hull, A. W., and Rice, M., these PROCEEDINGS, 2, 1916, (265–70). Ulrey, C. T., Physic. Rev., Ithaca, 11, 1918, (401-10).

7 Webster, D. L., Ibid., 9, 1917, (220-5).

LINEAR ARRANGEMENT OF GENES AND DOUBLE CROSSING OVER

BY HAROLD H. PLOUGH

DEPARTMENT OF BIOLOGY, AMHERST COLLEGE

Communicated by T. H. Morgan, April 11, 1919

In his recent criticism of the theory of linear arrangement of the genes in the chromosome, Prof. W. E. Castle states that that theory calls for a number of unproved secondary hypotheses, among which is the assumption of

double crossing over. There are, however, a number of facts, each independent of the original hypothesis, that require double crossing over in their explanation. Some time ago1 I demonstrated that treatment of female flies with temperatures both above and below the 'normal' (20°-27°C.) resulted in an increase in the percentage of crossing over in the second chromosome. This increase for a short chromosome 'distance' was roughly proportional to the number of degrees above or below the 'normal.' Bridges had already shown that age increased crossing over and more recently it has appeared probable that other environmental agents may cause a similar increase. In all cases, as pointed out in the original paper, the increase is much greater for short regions than for longer ones, and finally, for genes which normally

give a crossing over percentage of 35% or more no increase at all results. A summary of these facts extracted from my earlier paper is given below. The percentage of increase over the control is given. Only those data which were collected at the same time are entirely comparable with each other.

As a whole this table shows conclusively that small percentages of crossing over are increased markedly by high or low temperatures, while larger percentages show a much less significant increase. For instance, as a result of exposure of the F1 females to 32°C. (no. 7 in the table) the crossing over value between black and purple was increased from 6 in the control to 15.7—an increase of 162%-while purple-curved in the same experiment showed an increase from 19.6 to 26.5 or only 35%. In no. 9 in the table it is shown that while the black-curved value was increased by 58% as a result of a temperature of 31.5°C., the value of the star to black region (38.4 units) not only did not increase, but actually showed a slight decrease. The amount of the increase is in all cases related to the crossover value involved. The last named case was explained in my former paper as follows (p. 157):

The first brood data show very clearly that while the blackcurved region of the chromosome shows an unquestionable increase of more than 50%, no increase at all is registered in the test between star and black. This can mean only that with such long distances any increase in the actual amount of single crossing over is compensated by a similar increase in double crossing over, and thus no increase at all appears in the percentage registered by the count.

The data show that the percentage of increase caused by high or low temperature is roughly in inverse proportion to the size of the crossover value involved. On Castle's three dimensional scheme these facts necessitate the view that long chromosomal 'distances' are less affected by temperature than are short ones. On the hypothesis of linear arrangement this relation is consistently explained by the assumption that the amount of double crossing over is increased by high and by low temperatures.

1 Plough, H. H., J. Exp. Zool., 24, 1917, (147).

* Bridges, C. B., Ibid., 19, 1915, (1).

THE SPATIAL RELATIONS OF GENES

BY A. H. STURTEVANT, C. B. BRIDGES, AND T. H. MORGAN
COLUMBIA UNIVERSITY AND CARNEGIE INSTITUTION OF WASHINGTON

Communicated April 11, 1919

Castle' has proposed an arrangement of linked genes in three dimensions, based on the assumption that the distance between any two loci is exactly proportional to the observed crossover value. He believes that this system gives a better agreement between observation and expectation than does the hypothesis of a strictly linear arrangement that we have developed.