Emil Fischer—the distinguished chemist—was born at Euskirchen on October 9, 1852. He was educated at the Bonn Gymnasium and afterwards at the Universities of Bonn and Strassburg. After occupying the Chairs of Chemistry at Munich, Erlangen, and Würzburg, he finally settled in Berlin where the greater part of his far-reaching work in chemistry was done. He received the Davy Medal in 1890 from the Royal Society of London, and was awarded the Nobel Prize in 1902.

In my previous paper to the Society on the same subject I came to the conclusion that the temperature of the surface of open cultivated soil fell rapidly at the beginning of a calm clear night until it was such a number of degrees below the temperature at the 4-in. depth as to make the upward conduction from that depth to the surface balance the radiation. After this stage was reached the surface and 4-in. temperatures fell at the same rate.

If, therefore, the temperatures of the surface and 4-in. depth and the conductivity of the layer of soil between the 4-in. depth and the surface were known from readings of electrical resistance thermometers, and the rate of radiation was calculated from the value of the relative humidity, I suggested that it might be possible to forecast the minimum soil temperature for a calm clear night as early as the previous afternoon.

Perhaps the most interesting theorem as yet known in regard to circulants concerns the determinant whose matrix is the sum of two circulant matrices, one of the latter being taken symmetric with respect to the primary diagonal and the other with respect to the secondary diagonal: for example, the determinant

whose matrix is the sum of the matrices

that is to say, the matrices of

In a paper published in the Proceedings of the Royal Society of Edinburgh (vol. xxxviii, pp. 166–168), Mr Alexander G. Ramage described cases of mirage as seen on the Queensferry Road, near Edinburgh. Similar appearances have been since then observed and reported from different parts of the country, and the photographs of these by Mr G. F. Quilter, Ingatestone, are of some interest. Prints of five photographs taken by Mr Quilter were shown at the meeting of the Society held on 3rd November 1919; four were taken at one place about 100 yards’ distance from where the dry road appeared to be a pool. The camera looked along the road in a direction E.N.E., the road rising slightly towards the position where the mirage was seen. The best photographs, which were taken on 15th June 1919, at 1.30 p.m., showed certain posts on the north side of the road appearing distinctly reflected in the mirage pool. In the fifth photograph, taken at a different locality in August 1919, the shaded side of a telegraph post with neighbouring trees and a white-walled house beyond were distinctly seen as if reflected from a pool in the middle of the road. The main interest lies in the fact that the photographs bring out the phenomenon quite clearly.

§ 1. Let us suppose that a light wave is being propagated and absorbed in a homogeneous medium. Take OY as the direction of propagation, and consider a slice of the medium bounded by two planes at right angles to the direction of propagation and distant dy apart; dy is small in comparison with the wave-length. Let x=f(t) denote the mean displacement of the electrons, and let X = A cos gt denote the electric intensity of the light wave in the slice, measured in electrostatic units. Then the average rate at which work is being done on an electron is

§ 1. The scattering of X-rays is one of the most promising methods of approaching the constitution of the atom. Sir J. J. Thomson's theory undoubtedly gives the main features of the phenomenon. But there are certain facts, namely:

(i) the increase of scattering with heavy elements for soft rays,

(ii) the extra-radiation, and

(iii) the low value of the scattering coefficient for γ-rays,

for which it affords no explanation. It seems worth while trying to extend the theory so as to cover these facts also.

The consideration of the passage of light waves through three media elucidates an interesting point in the Theory of Newton's Rings.

Let the three media be denoted by the numerals 1, 2, 3, and let z = 0 be the plane of separation between media 1 and 2 and z = −a the plane of separation between media 2 and 3. Also let the axis of x in the plane z = 0 be parallel to the wave front of the incident plane waves.

The temperature of the air depends on so many varying factors that its prediction is a matter of considerable difficulty, and can only be made with any degree of certainty when the minimum number of these factors is at work. Underground temperatures are dependent not only on the same factors as affect the air temperature, but, in addition, are much more affected by conductivity, rainfall, evaporation, latent heat, etc., and thus the changes in temperature beneath the surface of the soil constitute a much more complex problem.

In a recently published paper (Proc. R.S., Dec. 1919) Mr W. J. Johnston makes a brilliant use of Clifford's extension of Hamilton's quaternions to the case of four (or more) variables. He proceeds, by adding a fourth independent and real unit vector and making the corresponding tensor imaginary, to modify the quaternionic ∇ so as to make it applicable to physical problems on the basis of relativity. In this respect his vectors are simply the direct extension of Hamilton's bivectors.

So far back as March 1855 Brioschi in effect formulated the theorem that any even-ordered determinant is expressible as a Pfaffian; for example in the case of the fourth order he gave the equality

where (h, k) stands for

this expression being obtained by multiplying the hth column

of the given determinant by the kth column

of an equivalent determinant.

(1) Outline of Investigation.—During the last half-century the temperature of the air has been recorded by thermographs at a number of places, and the registered graphs have been measured and mean daily temperature curves have been calculated from them for each month of the year. The amount of labour entailed in these operations is enormous, and it may well be asked whether the figures cannot be put to some further use. It appears to me that a comparison of the daily curves obtained at various places would furnish important information in regard to climate. To effect this the curves may be represented by Fourier's Series, and the values of the constants at the various places may be compared. Another method of comparison, however, seems to me more promising.

Experiments on the specific heats of gases and on their infra-red emission and absorption have gone a good way towards showing how the molecules take up energy when a gas is heated, and have raised questions as to how far the observed facts may be explained on the basis of “Newtonian dynamics,” and what particulars suggest or necessitate a resort to the Quantum Theory. It may be useful to state some of these questions without attempting categorical answers, which indeed cannot be offered until the physical concepts underlying the Quantum Theory have become more definite.

The problem of Graduation or Adjustment, with which the present paper is concerned, may be defined as follows. Let a number u be a function of a number x: and suppose that, corresponding to the values … −3, − 2, −1, 0, 1, 2, 3 … of x, we have obtained, as a result of observation, values … u−3, u−2, u−1, u0, u1, u2, u3 … for u. Owing to errors of observation, these observed values when plotted against the corresponding values of x do not lie on a smooth curve, although for theoretical reasons we believe that they would do so if freed from errors. The problem is to determine the most probable set of “graduated” or “adjusted” values

which differ only slightly from the above observed values, and which lie on a smooth curve.

Among the most useful results of quaternion or vectorial algebra we may count the identities and transformations worked out by Hamilton and Tait and their all too few followers. It is not merely that a vector identity is equivalent to three scalar identities—a fact which aids us greatly to condense our calculations in respect to bulk. Yet more important is the greater fruitfulness of a vectorial relation in giving rise to derived relations which we would be less likely to perceive from a purely scalar analysis.

In two former contributions to this Society I described the pharmacological actions of harmaline and harmine, the two alkaloids found in the seeds of Peganum harmala. In regard to harmaline I came to the following conclusions:— “It differs from most alkaloids in that it does not exert, to the same extent as they do, a selective action on one kind of tissue. It attacks not only highly specialised tissues such as voluntary muscle, muscle of the heart, blood-vessels and uterus, and cells of the central nervous system, but also less highly differentiated cells, such as pigment cells, protozoa (Raab), and ciliated epithelium (Jacobson).

There is a large accumulation of evidence in support of Sir Ernest Rutherford's conception of atomic structure. According to this conception an uncharged atom consists of certain groupings of negative electrons rotating about a positively charged nucleus. Dr Bohr's theory of line spectra, which is based on this structure, frankly discards dynamical conditions of stability. There is no doubt, however, that this theory has been very successful in accounting for the numerical relationships between the frequencies of spectral lines.

In 1888 Tammann, applying a form of the gas current saturation method to the measurement of the dissociation pressures of salt hydrates, obtained results which were uniformly higher by from 2 to 5 per cent, than the results obtained by Frowein with the tensimeter. This anomalous behaviour was confirmed, in Nernst's laboratory, by Schottky, who reported that the initial dissociation pressures developed in tensimetric measurements were, in cited cases, higher than the equilibrium values. In 1911 Partington, re-studying certain of the facts, added further confirmation by the use of the gas current saturation method in a somewhat altered form.

I. The Tonal Order of the Chromatic Scale.

The exact ratios of the chromatic scale were first made known by Claudius Ptolemy of Alexandria about A.D. 150. We give them here as he found them, unaltered and unalterable.

It is a most remarkable fact that while the ratios of the chromatic scale have thus been known for nearly two thousand years, its mathematical symmetry and origin have remained unrecognised down to the present day.

For the concrete musical faculty is compounded of two distinct elements, physical and mathematical.

In its physical aspect (with which we are familiar), the scale is a series of notes arranged in order of pitch upwards from the tonic (or keynote).

In its mathematical aspect, the scale is a system of intervals measured symmetrically upwards and downwards from tonic and dominant as central interval (or tonal centre).

The physical order of the scale is that which we most easily appreciate, but the mathematical order is that in which it originates. The chromatic scale therefore is, primarily, not a series of notes, but a system of intervals (or tonal system).

Diagram I shows the scale in its familiar order of pitch, and it is remarkable how effectually this physical order conceals the mathematical from view.

It is now thirty-two years since I last brought this subject before the Royal Society of Edinburgh. Since then nothing has been done to remedy the imperfections which were then shown to characterise the thermometer screens used throughout the country. In some respects these imperfections may seem to be of small importance. Yet in these days when accuracy of measurement is being more and more insisted on, and particularly now when reconstruction is everywhere in vogue, it does seem desirable that apparatus largely used by thousands of observers should be as perfect as it is possible to make it.