The present memoir is a continuation of those already published by one of us, on the physiology of the Invertebrata.

The following problem was sent me by Mr Kirkman, F.R.S.; its solution illustrates the Calculus of Relationship, Proc. Roy. Soc. Edinb., vol. xi. pp, 5 and 162:—

Two boys, Smith and Jones, of the same age, are each the nephew of the other; how many legal solutions?

The statements are that Smith is the nephew of Jones, and Jones the nephew of Smith. Let c denote child, then denotes parent, and the equations are

Hence .

In a paper communicated to this Society last Session, I gave an empirical formula representing the relation between current-strength and time when platinum plates are used for the passage of a current through acidulated -water, the electromotive force being that of a single Daniell cell. As is well known, we can look upon the electrodes as condensers of very great capacity.

As in all the thermoelectric diagrams hitherto published the lines of the metals stop short at the melting points, it seemed interesting to the writer to trace the change in the thermoelectric properties of a metal as the temperature is raised to, and beyond, its melting point. The fusible metals containing bismuth or antimony were rejected, because their large expansion at the point of solidification rendered them very unmanageable; ordinary block tin was accordingly chosen.

For some time Signor Battelli has been engaged, with remarkable success, in measuring directly the amount of the ‘Thomson effect’ in various metals.

With the exception of iron, the common metals have given him results coinciding as closely as could be expected with those I found in 1872 by an indirect method. Among other particularly satisfactory things, he has directly verified the first of the two changes of sign of the Thomson effect in nickel. And I think it will be allowed that what I introduced long ago as a mere working hypothesis,–that the Thomson effect is directly proportional to the absolute temperature,–if it was not completely established as a fact by my own experiments, has been made absolutely certain by the recent work of Campbell and of Battelli.

In this paper the author pointed out that the experiments detailed in his preceding paper on Transition-resistance and Polarisation, show that, whatever be the ultimate nature of Dielectrics, any insulator upon which a film of gas is condensed must exhibit the phenomena of electric absorption and residual discharge.

The author remarked that one of the constituents of the double fog-bow described in some of Mr Omond's recent observations, is obviously the ordinary primary rainbow, diminished in consequence of the very small size of the water drops. But the other, having nearly the same radius but with its colours in the opposite order, appears to be due to ice-crystals in the fog. This is quite consistent with the record of temperatures. Just as small drops of water may remain unfrozen in air below 0° C, small ice crystals may remain unmelted at temperatures above that point.

By permission of the Meeting, a letter was read from the Astronomer-Royal for Scotland on the fall of a part of the cliff below Nelson's Monument.

A grant was made by the British Medical Association, on the recommendation of the Scientific Grants Committee of the Association, towards the expenses of a research, a part of which appears in this communication.

As the alkaloid morphine, discovered in 1804, was the first substance of the kind known, it naturally received a good deal of attention. Some years later the new base was analysed by several distinguished chemists, but their results did not lead them to a formula accurately indicating the composition of the alkaloid. To Laurent belongs the honour of having given the formula C17H19NO3 (using the modern notation), the accuracy of which formula has been confirmed by subsequent analyses, particularly by those of Matthiessen and Wright. Nothing was ever observed to suggest that the formula is a multiple of these members until Wright, after an elaborate investigation of the derivatives of morphine and codeine, came to the conclusion that the true formula must be at least double the empirical, and that we ought therefore to write C34H38N2O6.

A monochromatic rainbow looks like a contradiction in terms. As a rainbow of this kind was, however, seen lately, its occurrence seems worth putting on record. On the afternoon of Christmas day I went for a walk in the direction of the high ground to the south of Falkirk. Shortly after starting I observed in the east what appeared to be a peculiar pillar-like cloud, lit up with the light of the setting sun. What specially attracted my attention was that the streak of illumination was vertical, and not the usual horizontal band-form we are accustomed to. I looked in the direction of the sun to see if I could trace any peculiar opening in the clouds through which the light passed, but failed to do so.

Many, who have kept dogs, are aware that if the skin covering the side of the body be scratched, a dog will move the leg of that side as if itself to scratch the part touched. This fact is known to the physiologist, and a, so-called, scratching centre, to which the sensory impulses are carried, and from which motor impulses to the muscles pass, has been shown to exist in the spinal cord.

Hearing, again by accident, that Professor Boltzmann has in the Vienna Sitzungsberichte published a new attack on my papers about the Kinetic Theory, I at once ordered a copy, which has at length arrived. As my papers appeared in our Transactions, I think my answer to this fresh attack should be communicated in the first place to this Society. The time I can spare for such work at this period of the year is very scant, and Prof. Boltzmann has raised a multitude of questions. I will take them in order. But I must commence by saying, with reference to Prof. Boltzmann's peculiar remarks on my behaviour as a critic, that, while leaving them to the judgment of readers, I shall have to bring before the same readers several instances in which Prof. Boltzmann has completely misstated the contents or the objects of my papers. This is not a new departure. In his first attack on me he said that I had nowhere stated that my investigations were confined to hard spherical particles; whereas I had been particularly explicit on that very point. But fresh cases of a similar character abound in this new attack.

The problem is to contrive an apparatus in which, by means of inertia, acceleration of angular velocity about any axis in either sense may be observed and approximately measured.