After the Royal Society, Edinburgh, had been pleased in 1880 to accept and print my paper on the general appearance of Gaseous spectra as seen on a very small scale, but complete on that scale from one end to the other of the visible spectrum, I was desirous to present them with some very highly Dispersed and much magnified views of the more interesting and probably crucial portions of the most important of those spectra.

If a row of n elements be taken, and closely following this array, but separated by a bar from it, we write n rows of n elements each; and closely following either outside column of this square array, but separated by a bar from it, we write n columns of n elements each; and closely following an outside row of this second square array, but separated by a bar from it, we write n rows of n elements each; and so on, passing from the rows or columns of one array to the columns or rows of the next, and ending not with a square array, but, as we began, with a single line of elements, we have the matrix representation of a bipartite function.

The 119 subsolids (marked ss) and the 244 unsolids (marked us), of these unifilars are here arranged in lists according to their flaps. Fe is the number of flaps of e loops upon a knot; and the headings of the lists, as, e.g.10I, F2 = 1, F1 = 3, describe so far all the knots in the lists. Thus in 10I each has one 2-ple flap and three single ones. After the number in the list comes always the base on which the knot is constructed by the rules of my paper, XVII. in vol. xxxii. part ii. of the Trans. R. S. E.; and the reader who has that paper before him will find it easy to draw any knot on its base, nearly always there figured, by the first given flap, which is the leading one of the knot described; thus in 10I the first written flap is the double one, and in 10P, F3 = 1, F1 = 2, it is the triple one. The leading flap of a subsolid is always followed by a colon.

The following additional remarks are the outcome of my study of the polyhedral data for tenfold knottiness, which I received from Mr Kirkman on the 26th of last January. My main object was, as in the first chapter of Part II., to determine the number of different types; as well as the number of essentially different forms which each type can assume, as distinguished from mere deformations due to the mode of projection.

This study has been a somewhat protracted one, in consequence (1) of the great number of tenfold knots; (2) of the very considerable number of distortions of several of the types, many of which are essentially distinct while others present themselves in pairs differing by mere reversion; and especially (3) of the fact that the polyhedral method often presents some of the distinct forms of one and the same type projected from essentially different points of view (of which, in the present case, there are sometimes twelve in all).

A mechanism may be defined as a combination of plates, bars, or flexible members jointed together, so that, while the parts may move relatively to each other, the relative positions of all the different parts are determinate for each given possible relative position of any two parts.

It follows immediately that the simultaneous relative displacements, velocities, and accelerations of velocity of all parts are also strictly determinate.

This determination, by accurate graphic means, forms the subject of this paper. In it those mechanisms alone are considered that are composed of rigid members, the motions of whose parts are all continually parallel to one plane, the constancy of the plane being defined relatively to one of the members of the mechanism itself.

Throughout the year 1884 the sun was seen over most countries under peculiar atmospheric circumstances; and these, instead of being confined to low altitudes, were never more conspicuous than during the summer of the year, and noon-tide of each day, and in the clear air on mountain heights of every country, whenever the sky was more or less free of actual clouds.

On such occasions then, the usual phenomenon to be noticed by the eye, was, that nothing like blue sky could be witnessed near the sun. But in place of that, there was a broad glare of whitish light extending for several degrees around the luminary; and beyond that range, or over 20 degrees distant from him, there spread a wide reddish haze, passing into purple, and at greater distances into blue, but nowhere a very decided and deep blue sky.

On account of the treacherous character of the natives of the Solomon Group, no extensive geological observations have ever been made in these islands since the period of their discovery by the Spaniards three centuries ago. For this reason my excursions in these regions were not free from personal risks; in many places they were considerably curtailed, and in some places they had to be abandoned altogether. Fortunately, however, I was able to make a detailed examination of several of the smaller islands, the natives of which had been first conciliated by the kindly tact of Lieut.-Commander C. F. Oldham, to whom I am indebted for much assistance in my pursuits. But from the circumstances in which I was placed, both on board and on shore, it was necessary, in order to accomplish much, to dare a little.

The following observations on atmospheric electricity were made on the top of Dodabetta, between 3rd and 12th January 1885. The position of the electrometer, when not otherwise stated, was at a height of about 5 feet above what remains of the walls of the bungalow formerly used as a meteorological observatory. Dodabetta is the highest hill in the Neilgherries, and the bungalow was built exactly on the top, which is in latitude 11° 24′ 5ʺ·40 N., longitude 76° 46′ 44ʺ·39 E., and at a height of 8642 feet above mean sea-level. The top is free from trees, and my tents were slightly below, and at a sufficient distance off to prevent them interfering in any way with the accuracy of the readings.

The group of the Phæosporeæ was first recognised by Thuret in 1850. It includes algæ that vary greatly in size, habit, external appearance, and mode of branching, some possessing a flat thalloid appearance with or without a long stalk (e.g., Laminariæ), others being filamentous and much branched (e.g., Ectocarpus, Sphacelaria, Chordaria, &c). The great majority of the genera included in the group are marine or brackish water forms, but recently E. Flahault has described, under the name of Lithoderma fontanum, a fresh water species from the neighbourhood of Montpellier.

The following research was undertaken chiefly with the object of furnishing a complete memoir on the morphology and physiology of Patella vulgata. A number of isolated observations on various organs are recorded, but no systematic account has as yet been written, so far as the writer is aware. He has endeavoured to incorporate these observations in such a general account, having first convinced himself of their accuracy so far as lay in his power.

1. If in a circulant the places (r, s) and (p, q), owing to the cyclical permutation, be occupied by the same element, then the complementary minor of this element in its first place is to its complementary minor in the second as (−1)r+s: (−1)p+q.

To prove that the complementaries are as (−1)r+s:(−1)p+q is the same as to prove that the cofactors are identical. And as the cofactor of an element of a determinant is not altered by the transposition of rows and columns provided the determinant itself is not thereby altered, it is evident that all we have to show is that the element in the place (r, s) may by transposition of rows and columns be made to take the place (p, q), and the determinant remain in outward form the same as before. Now it is a known property of the circulant that this can be done by first bringing the element in the place (r, s) into the pth row by cyclical transposition of rows, and thereafter making an exact similar set of transpositions of columns. Thus the theorem is established.

Let

be the equation to an algebraical curve of the nth degree, the co-ordinates of any point on which in a system of linear co-ordinates are (x, y, z), u0, u1, u2 …. being homogeneous functions of x and y of degrees indicated by the attached suffixes; then

is the equation to its Hessian, which is a curve of the 3(n − 2)th degree.

Every one of the 3n(n − 2) points of intersection of H and U is a point of inflexion on U if it be not a multiple point on U. In this last case the intersection may or may not be a point of inflexion on some one of the branches of U; but in any case where H passes through a multiple point the total number 3n(n − 2) of inflexions suffers a reduction.