writings of no mathematician could shew a higher average. Great and irreparable as was the loss caused by his death, it was yet some consolation to feel that what he had left formed a worthy memorial of his powers and of the brilliance of his genius, and that his permanent place in the very highest rank of mathematicians was assured to him.

Professor CAYLEY endorsed every word Mr Glaisher had said: it was impossible to speak in too high terms of the value of Professor Smith's work. His wonderful knowledge of the processes of the higher parts of the Theory of Numbers shewed itself in everything he did. His work was of the very highest quality and excellence, and he could not too strongly express his sense of the great loss caused by his death.

The following communications were made to the Society:

(1) A new form of equation of the 16-nodal quartic surface. By Prof. CAYLEY*.

It was shewn that if x, y, z, E, n, 5, be homogeneous linear functions of four co-ordinates, subject to the identical relations

x+y+z++ n + 5 = 0,

ax+by+cz +Ĥ+gn + h5 = 0,

where af=bg=ch = 1, then √x + √yn + √z5= 0 is the equation of a quartic surface having the sixteen singular tangent planes (each touching it along a conic).

(2) On the mean radius of coils of insulated wire. By Lord RAYLEIGH.

In electrical work it is often necessary to use coils of such proportions that their constants cannot well be obtained from the data of construction, but must be determined by electrical comparison with other coils whose proportions are more favourable. A method for comparing the galvanometer-constants of two coils, i.e. of finding the ratio of magnetic forces at their centres when

This paper is published in full in the Journal für die reine und angewandte Mathematik, Bd. 94. "On the bitangents of a plane quartic."

traversed by the same current, is given in Maxwell's Treatise, Vol. II. § 753.

I have used a slight modification of Maxwell's arrangement which is perhaps an improvement, when the coils to be compared are of copper and therefore liable to change their resistance pretty quickly in sympathy with variations of temperature. The coils are placed as usual approximately in the plane of the meridian so that their centres and axes coincide, and a very short magnet with attached mirror is delicately suspended at the common centre. If the current from a battery be divided between the coils, connected in such a manner that the magnetic effects are opposed, it will be possible by adding resistance to one or other of the branches in multiple arc to annul the magnetic force at the centre, so that the same reading is obtained whichever way the battery current may circulate. The ratio of the galvanometer constants is then simply the ratio of the resistances in multiple arc.

To obtain this ratio in an accurate manner, the two branches already spoken of are combined with two other resistances of german silver, so as to form a Wheatstone's balance. Of these resistances both must be accurately known, and one at least must be adjustable. The electro-magnetic balance is first secured by variation of the resistance associated with one of the given coils which resistance does not require to be known. During this operation the galvanometer of the Wheatstone's bridge is shortcircuited. Afterwards the galvanometer is brought into action and the resistance-balance is adjusted. The ratio of the galvanometer-constants is thus equal to the ratio of the german silver resistances. The two adjustments may be so rapidly alternated as to eliminate any error due to changes of temperature in the copper wires. Indeed, if desired, the final tests of the electromagnetic and resistance-balances might be made simultaneously.

If the ratio of galvanometer-constants be the final object of the measurement, there is nothing more to be done; but if we desire to know the ratio of the mean radii of the coils we must introduce certain small corrections for the finite dimensions of the sections. In the first place, however, it will be desirable to consider a little more closely what should be understood by the mean radius of a coil.

In Maxwell's treatment of the subject (§ 700) the mean radius of a coil is considered to correspond with the geometrical centre of its rectangular section, that is to say, the windings are assumed to be uniformly distributed over the section. In practice absolute uniformity is not attainable, and it is therefore proper to take into account the effect of a small imperfection in this respect. The density of the windings, i. e. the number of windings per unit area,

may be denoted by p, and is to be regarded as approximately

constant.

P

The introduction of the factor o makes but little difference in the investigation of § 700. If we take the origin of co-ordinates x and y, no longer at the geometrical centre, but at what may be called the centre of density of the section, we shall have (as in the ordinary theory of the centre of gravity)

the integrations being extended over the area of the section. If P be any function of x and y, P the mean value of the function (with reference to p), P, the value at the origin, we have

the terms of the first order disappearing in consequence of the choice of origin. In the terms of the second order we may neglect the effect of variable density, and write

έ, ʼn being the breadths in the directions of x and y of the rectangular section.

Thus

+n2 dy.

dx*

The form of this expression is the same as when the windings are supposed to be distributed with absolute uniformity, but the mean radius and mean plane are to be reckoned with reference to the density of the windings.

In the application to the galvanometer-constant of a coil, we have, if A be the mean radius, the radial and n the axial dimension of the section,

η

A2

A2

by means of which, § and ʼn being approximately known, G, may be inferred from A, or conversely A may be inferred from G. If the ratio of galvanometer-constants of two coils has been determined by the electrical process, the ratio of mean radii can be accurately deduced by use of the above formula.

When the mean radius of a coil has been determined in this manner by comparison with another of proportions more favourable for calculation from the data of construction, other quantities relating to the coil may be deduced by mere calculation. For instance, the important constant g,, denoting the mean included by the windings, is connected with the mean radius A by the equation

A more direct process for determining g, electrically is given by Maxwell § 754, and has recently received an important application in the hands of Kohlrausch. In this method the quantity sought is proportional to the cube of a distance not very easy of precise measurement; and it is possible that the less direct method explained above may be the more accurate in practice.

(3) On the invisibility of small objects in a bad light. By Lord RAYLEIGH.

In a former communication to the Society (March 6, 1882) I made some remarks upon the extraordinary influence of apparent magnitude upon the visibility of objects whose apparent brightness' was given, and I hazarded the suggestion that in consequence of aberration (attending the large aperture of the pupil called into operation in a bad light) the focussing might be defective. Further experiment has proved that in my own case at any rate much of the effect is attributable to an even simpler cause. I have found that in a nearly dark room I am distinctly short-sighted. With concave spectacles of 36" negative focus my vision is rendered much sharper, and is attended with increased binocular effect. On a dark night small stars are much more evident with the aid of the spectacles than without them.

In a moderately good light I can detect no signs of shortsightedness. In trying to read large print at a distance I succeeded rather better without the glasses than with them. It seems therefore that the effect is not to be regarded as merely an aggravation of permanent short-sightedness by increase of aperture.

The use of spectacles does not however put the small and the large objects on a level of brightness when seen in a bad light, and the outstanding difference may still be plausibly attributed to aberration.

February 26, 1883.

Mr GLAISHER, PRESIDENT, IN THE CHAIR.

The following were balloted for and duly elected Fellows of the Society:

Rev. R. Appleton, M.A. Trinity College.
Mr R. F. Scott, M.A. St John's College.
Mr M. J. M. Hill, M.A. St Peter's College.
Mr. W. A. Bond, M.A. St John's College.
Mr J. Larmor, B.A. St John's College.

The following communications were made to the Society: (1) The original function of the canal of the central nervous system of Vertebrata. By A. SEDGWICK, M.A.

The central nervous system of all known animals with certain doubtful exceptions, arises from the epiblast. The region of the epiblast from which it arises may either persist in the adult as part of the superficial epidermis, or it may be pushed in so as to give rise to a tube, from the walls of which the central nervous system is developed. The last mentioned method is characteristic of the Vertebrata. The walls of this tube become differentiated into a superficial epithelial layer lining it, and an external mass of nervous matter. The tube persists as the canal of the nervous system; the epithelium lining it becomes the ciliated epithelium of this canal, which therefore corresponds to the external epithelium of the body-wall.

I may here draw attention to the fact that the vertebrate stock must have separated from that of other animals before the nervous system was separated from the external epithelium of the body; that in fact the vertebrate nervous system never is separated by any ingrowth of mesoblast from the superficial epiblast from which it arose; as is the case in all but the most primitive of the Invertebrata. This superficial epiblast in Vertebrata is involuted and gives rise to the ciliated epithelium just mentioned of the central canal. Three stages may be distinguished in the development of this canal, and I suppose that all three have had a functional counterpart in the evolution of the organ.

In the first stage a groove extended along the whole length of the middle dorsal (or ventral?) line of the body, the nervous system being placed in the deeper layers of the epidermis of this groove. This stage I propose to call the groove stage.

In the second stage, which may be called the siphon stage, the groove had become converted into a canal, open in front at or near the anterior end of the body, and open behind close to the anus.