in wet sand was kept open till, by gentle infiltration, the hollow was filled, and an exact likeness of the interior of the tunnel realised on the infiltrated matter! A very good example of the forms figured by Mr Hancock occurs on one of the slabs on the table. At a meeting of the Dublin Geological Society in 1858, Professor Haughton, the president, read a paper "On the Occurrence of some New and Rare Forms of Annelidoid Tracks in the Coal Measures, Lugacurren, Queen's County."* A year later, and again in 1860, he returned to the subject, indicating his belief that the Lugacurren tracks resembled those described by Mr Hancock, and ultimately accepting his theory of their crustacean origin. As Professor Haughton's papers were illustrated by lithographic plates ad naturam, they, like Mr Hancock's, are available for comparison with the specimen before us. The Lugacurren specimens differ from the Northumberland forms in having at one part four, in one case, and in another five circular depressions in the median line. They differ from that on the table in having circular depressions confined to a small part of the form, and a median groove, instead of, as here, oval depressions running the whole length of the body, and a median ridge. In neither the English nor the Irish specimens is the characteristic striation so conspicuous as in this. These, however, seem all to be specimens of closely related species. Both Mr Hancock and Professor Haughton think that the depressions in the median groove may have been made by the pygidium of a carboniferous trilobite, the print of the tail being the only trace the animal has left of its having had a place in this deposit! Now such marks as those on the Lugacurren specimers could only have been made by the pygidium being set down vertically, then lifted for a time, and so placed at regular intervals, their being no evidence of dragging, most unlikely, if not impossible conditions. Fortunately, a good representation of the form figured by Professor Haughton occurs also on these slabs. From this it is evident that the characteristic circular punctures run the whole length of the body. It seems to me that the reference to the track of Sulcator arenarius has been misleading. It can, indeed, have no bearing on any * Journal of the Geological Society of Dublin, vol. viii. 1857-1860. of these closely related forms, being unisulcus, destitute of striæ, and not more than three-eighths of an inch wide, while all these are striated and seven-eighths of an inch wide; that under notice being moreover distinctly bisulcate. It is acknowledged that the relations of this to recent forms are obscure. The features which chiefly claim attention are First, The outline of the animal.-A glance at the specimen is sufficient to convince us that we have here not a track merely, but the representation of an annulose form. (See Plate IV. fig. 1.) On one of the slabs this is associated with several inches of the track over which it has passed. The median dorsal line is fully exposed above, while the median ridge, which makes the track bisulcate, is precisely what would be formed by the ventral groove of a nereis-Alitta virens (Sars), for example. The stria which pass round the body leave no traces of their outline in this track; but in another, from which the representation of the animal was removed, these striæ are well marked. Again, on the bulged sides of the tortuous outline, the stria are wider than on the opposite side, while in the comparatively straight parts they are symmetrical. So far as I know, similar markings do not occur on any recent annelid. They are, however, represented, though not so distinctly as here, on a small annelidEpitrachys rugosis-figured by Ehler of Erlangen in his paper on the "Fossil Worms of the Lithographic States of Bavaria."* Second, The tracks. They differ widely from the tracks both of mollusca and crustacea-those of the former being, for the most part, sharp in their turns, and those of the latter consist generally of lines more or less straight, not tortuous. In addition, they have two furrows divided by a distinct median ridge. I am sure that had the able observers named above seen such specimens of the tracks, and also of the intaglios of the rounded dorsal surface as are now on the table, they would not have questioned the true annulose character of this organism. Third, The median dorsal line.-This is exceedingly well represented, not only on the outline of the animal itself, but also in one of the intaglios referred to. It consists of small, shallow, oval * Ueber fossile Würmer aus dem lithographischen Schiefer in Bayern Cassel, 1869. depressions, deepest in the centre, and narrowing at each end, where they meet a slight ridge which stretches between the depressions, giving to the line, looked at from a short distance, a chainWere the branchial tufts of some recent annelids like appearance. Fourth, The characteristic striation. This is most distinctly and even sharply marked on the form in the gritstone slab. It is also, though less definitely, marked on some of the softer micaceous slabs. Mr Hancock says, with reference to his specimen which has most resemblance to this-"The transverse striæ on the surface of the grooved form certainly gives it much the appearance of some organism;" but the value of this acknowledgment is lost by the supposition that the striæ might have been "produced by the intermitting progress of the animal." Now it is simply impossible that such striation as is seen here could have been produced in this manner. The expression "ondeleusement et symmétriquement," used by Cuvier in describing the striæ on the shell of a cephalopode, very well indicates a leading feature of this striation. Indeed, the symmetry of these beautifully regular undulating striæ may be best understood by comparing them with the striæ on the shells both of recent and fossil nautilidæ. Fig 3 is intended for a representation, ad naturam, of the characteristic striation, but the striæ are sharper and better defined than shown on the figure. In conclusion, it will be seen that the distinctive features of the specimen now brought under the notice of the Society are the median dorsal line and the waved striation. In the generic features set down at the head of this paper, I have described the former thus, linea dorsualis continua, alternasque cavaturas ovatas et vertices ortendens. As, however, uncertainty attaches to the nature of this line, the latter-the striation-may be taken as the outstanding generic feature,-striæ tenues, in ordinem undulatæ, et ubique corpus cingentes, ita ut cutis subrugosa videatur. The term CYMADERMA, or wave-skin, is proposed for a genus, whose true zoological position is as yet uncertain. Should the examination of other specimens show that the oval depressions in the median dorsal line are only specific marks, not points of insertion of organs, and the stria mere lines formed by the contraction of the cutis-a most unlikely circumstance—the organism would have closer nemertean than annelid relations. But, if proof be ultimately obtained that the branchia-like organs referred to above were connected with the oval depressions, and that the transverse markings are really not striæ but annuli, the zoological position of the animal will be among true annelids, characterised, however, by structural features widely divergent from recent forms. 5. On Eisenstein's Continued Fraction. By Thomas Muir, M.A. 6. Note on an Infinitude of Operations. By Thomas Muir, M.A. n=∞ The assumption that there is a limit in Professor Tait's problem regarding the interpretation of Lim, (cos") means that if we start with an angle x, and find the number which is its cosine, then the number which is the cosine of this number, and so on, we shall at last come to a limiting result w, such that cos w = w. The problem is thus transformed into the solution of the equation w = cos ∞, which may be accomplished as follows:— Taking therefore 733 as a first approximation, we find that with it W COS @= - ⚫01 and taking 75 or 2 as another approximation, which is readily seen to differ from the former by erring in excess, we find that with it from which two results by the regula falsi we derive a better a result correct to at least the fifth place. This peculiar constant we should expect to find connected in some way with π, and that such a connection exists is easily seen An explicit expression of the same relation is obtained from the identity |