These openings, which are also present in numbers on the region of the cheek, indicate the ducts of numerous mucous glands which lie immediately underneath the mucous membrane. The tongue of Beluga does not appear to differ much from that of other toothed whales. In Globiocephalus intermedius, according to Macalister, not only the tip, but also the margins of the organ are free, whereas in Beluga the latter are attached along nearly their whole length. Dr Murie refers to the presence of a distinct froenum in Globiocephalus melas, but this structure is not recognisable in Beluga.

In connection with the problem of Map-colouring, I incidentally gave (Proc. K.S.E. 1880, p. 502) a theorem which may be stated as follows : — If 2n points be joined by 3n lines, so that three lines, and three only, meet at each point, these lines can he divided (usually in many different ways) into three groups of n each, such that one of each group ends at each of the points.

The geographical area embraced in the present memoir forms a well-marked basin traversed along its centre by the estuary of the Forth. It is bounded on the north by the chain of the Ochil Hills, on the south by the range of the Pentland and Lammermuir uplands. Towards the west it joins along a low watershed the basin of the Clyde, while eastwards it dips under the waters of the North Sea. Within this defined space the Carboniferous rocks occupy what may be described as one great synclinal trough, varied by innumerable smaller synclines and anticlines. Save where cut out by powerful dislocations, their lower members rise up along the margins of the basin, while their highest portions cover a smaller area in the centre. The older formations forming the northern and southern boundaries of the area belong chiefly to the Lower Old Red Sandstone, in the Lammermuir district to the Lower Silurian. The Carboniferous rocks everywhere rest upon them unconformably.

Although, from the careful descriptions which have been given by several competent naturalists, we may now consider ourselves tolerably well acquainted with the structure of the dainty siliceous skeletons of this and several allied Hexactinellid Sponges, this is by no means the case with their soft tissues ; and the great cause of our imperfect knowledge of these interesting structures is that no observer has hitherto succeeded in procuring a really well-preserved Sponge of this group. It was with pleasure, therefore, that I accepted the offer of the Director of the Challenger Expedition to place some well-preserved examples belonging to different genera in my hands for investigation. Of all the specimens which have been sent to me, some fragments of Euplectella aspergillum in absolute alcohol are much the best preserved, and therefore the best suited for thorough examination. I commence with the description of the soft parts of this well-known and beautiful form, following the classification which has been adopted by Sir Wyville Thomson in his descriptions of the species.

The following paper contains a short digest of investigations communicated to the Society on several occasions during the past, and the present, session. The work had been for some months laid aside, but my attention was recalled to it by Professor Chrystal's valuable paper, in which he treats Minding's Theorem as an example of Plücker's methods, and also by the help of Rodrigues' co-ordinates. I am induced to publish a few of my results in full, as I think that a comparison of the analysis employed by Chrystal, with the very different analysis employed by myself, may be useful as well as interesting, especially from the point of view of the simplicity of the quaternion method. Even when the quaternion processes are written out at full length, they are in general shorter than the most condensed forms of ordinary analysis; and there can be no doubt that they are much more easily interpretable into the corresponding geometrical ideas.