producing in every lunar day a variation which is distinctly appreciable, in each of the three elements, by the instruments adopted and recommended in the Report of the Committee of Physics of the Royal Society, when due care is taken in conducting the observations, and suitable methods are employed in elaborating the results. 2. That the lunar diurnal variation in each of the three elements constitutes a double progression in each lunar day; the declination having two easterly and two westerly maxima, and the inclination and total force each two maxima and two minima between two successive passages of the moon over the astronomical meridian; the variation passing in every case four times through zero in the lunar day. The approximate range of the lunar-diurnal variation at Toronto is 38" in the declination, 4"-5 in the inclination, and ⚫000012 part of a total force. 3. That the lunar-diurnal variation thus obtained appears to be consistent with the hypothesis that the moon's magnetism is, in great part at least if not wholly, derived by induction from the magnetism of the earth. 4. That there is no appearance in the lunar-diurnal variation of the decennial period, which constitutes so marked a feature in the solar diurnal variations. "On Autopolar Polyedra." By the Rev. Thomas P. Kirkman, M.A. An autopolar polyedron is such, that any type or description that can be given of it remains unaltered, when summits are put for faces, and faces for summits. To every ẞ-gon B in it corresponds a B-ace b (or summit b of ẞ edges), which may be called the pole of that B-gon; and to every edge AB, between the a-gon A and the B-gon B, corresponds an edge ab, between the a-ace a and the B-ace b. Two such edges are called a gamic pair, or pair of gamics. The enumeration of autopolar p-edra is here entered upon as a step towards the determination of the number of p-edra. The theorems following are established, and shown to be of importance for the solution of the general problem. THEOREM I.-No polyedron, not a pyramid, has every edge both in a triangle and in a triace. Def. An edge of a polyedron is said to convanesce, when its two summits run into one; and it is said to evanesce, when its two faces revolve into one. An edge (AB) is said to be convanescible, when neither of the faces A and B is a triangle, and (AB) joins two summits which have not two collateral faces, one in either summit, besides A and B. An edge (ab) is said to be evanescible, when neither a nor b is a triace, and the two faces about (ab) are not, one in either, in two collateral summits, besides a and b. THEOREM II.-Every polyedron, not a pyramid, has either a convanescible or an evanescible edge. THEOREM III.-Any p-edral q-acron, not a pyramid, can be reduced by the vanishing of an edge, either to a (p-1)-edral q-acron, or to a p-edral (q-1)-acron. By such a reduction of a p-edral q-acron P to P', of P' to P", &c., P can be shown to be generable from a certain pyramid II; by which it is meant that I is the highest-ranked pyramid to which P can by such reduction be reduced. Hereby it is evident that the problem of enumeration of the x-edra is brought down to this: to determine how many (r+m)-edra are generable from the r-edral pyramid. The autopolars so generable are first considered, as the heteropolars are obtained by combination and selection of those operations with which the theory of the autopolars makes us acquainted. Autopolarity is of three kinds, nodal, enodal, and utral. Every even-based pyramid is nodally autopolar; i. e. it cannot but have two nodal summits. For example, the 5-edral and 7-edral pyramids have the signatures of their faces and summits thus arranged, the upper line showing the triangles, and the lower the triaces about the base, which as well as its pole the vertex, is signed zero. The two triaces in the triangle 5 are 3 and 2; the two triangles in the triace 1 are 6 and 1 in the 7-edron, and 4 and 1 in the 5-edron. The nodal summits and faces are 3 and 1 in the 5-edron, and 4 and 1 in the 7-edron. No other mode of autopolar signature is possible in these. Every odd-based pyramid is utrally autopolar. The 6-edral and 8-edral pyramids may receive either of the signatures following:- the first of which lines exhibits nodal faces and summits 31 and 41, while in the second every triangle is opposite its polar triace, and no face or summit is nodal. No pyramid is enodally autopolar, i. e. capable of only enodal signature. If we draw a 7-gon whose summits are 1234567, and then the dotted lines 73 and 75, and next taking three points in it, complete the 5-gon 34089, and join 93, 92, 81, 87, 06, 05, 04, we can sign the faces thus: 045=1, 506=2, 6087=3, 781=4, 1892=5, 293=6, 39804=7, 2371=0, 3754-8, 567-9. The type now represents an enodally autopolar 10-edron, in which no pair of gamics meet each other, or can by any autopolar arrangement be made to meet. The 18 edges of the solid are well represented thus, the odd places in a quadruplet showing summits, and the even, faces :— 1520 2630 3748 4158 5269 6379 7410 0783 8795 0251 0362 8473 8514 9625 9736 0147 3870 5978 The gamic pairs stand together, and no quadruplet exhibits fewer than four numbers. A nodally autopolar must always be, and a utrally autopolar may always be so signed, that two pairs of gamics shall exhibit in each quadruplet a duad of the form aa. In the above type it is observable that every duad, as 15, occurs four times. The same thing is to be seen in every autopolar type of edges. If we make use of the closed 10-gon 1239804567, as directed in a paper "On the Representation of Polyedra," in the 146th volume of the Transactions of the Royal Society, a paradigm of this 10-edron can be written out, exhibiting to the eye all the faces, summits, angles, and edges of the figure. The problems following are next proposed and solved. To find the number of autopolar (r+2)-edra generable from the (r+1)-edral pyramid. The answer is, (r>3), } { (r2−3r)4, +(12 −3r+2)4,-3+(r2—2r−3).2,-1}, where the circulator 8,1 or =0 as r is or is not =sm. To determine the number of autopolar (r+3)-edra generable from the (r+1)-edral pyramid. The solution is, (r>3), 24 { 2o1—6r2+11r2—36r+24+9r2.2, +(r3+29r+60)2,-1 } Hence it appears that there is one autopolar 6-edron, not a pyramid, and five autopolar 7-edra besides the 7-edral pyramid, viz. three generable from the 6-edral and two from the 5-edral pyramid. The problem of enumeration of the x-edra may, by a slight extension of the meaning of partition, be stated thus: to determine the k-partitions of a pyramid; and this depends on the problem, to find the k-partitions of a polygon, and on this, which is nearly the same question, to find the k-partitions of a pencil. By the k-partitions of a p-gon is meant the number of ways in which k lines can be drawn not one to cross another, and terminated either by the angles of the polygon, or by points assumed upon its sides or within its areas so as to break up the system of one face and p summits into a system of 1+h faces and p+i summits, where h+ik; it being understood that if a point be assumed within the area, three lines at least shall meet in it, and if on a side, one segment of it shall be counted among the k lines. The number of k-partitions proper, for which i=0, or of ways in which k-diagonals can be drawn none crossing another, is which is also the number of ways in which a pencil of p rays can be broken up into p+k pencils, by the addition of k lines, each one connecting two pencils. GEOLOGICAL SOCIETY. [Continued from p. 388.] April 8, 1857.-Colonel Portlock, R.E., President, in the Chair. The following communication was read :— "On the Species of Mastodon and Elephant occurring fossil in Great Britain. Part I. Mastodon." By H. Falconer, M.D., F.R.S., F.G.S. The object of this communication is to ascertain what are the species of the Proboscidea found fossil in Britain; what the specific names which ought to be applied to them; and what the principal formations and localities where they are elsewhere met with in Europe. The Mastodon of the Crag forms the subject of this first part of the memoir: the second part will treat of the Elephant-remains found in Britain. The author commenced by insisting on the importance to geology that every mammal found in the fossil state should be defined as regards, first, its specific distinctness; and, secondly, its range of existence geographically and in time, with as much severe exactitude as the available materials and the state of our knowledge will admit. He observed that with regard to the remains of the proboscidean genera, Dinotherium, Mastodon, and Elephant, some of which abound in the miocene and pliocene deposits of Europe, Asia, and America, the opinions respecting the species and their nomenclature, in all the standard palæontological works on the subject, are extremely unsettled and often contradictory. Dr. Falconer then proceeded to explain his views of the natural classification of the proboscidean Pachydermata, recent and fossil, according to dental characters. In the Dinothere, with its tapiroid molars, the last milk-molar and the antepenultimate (or first) true molar are invariably characterized by a "ternary-ridged-crownformula," or in other words, their crowns are divided into three transverse ridges. In the Mastodon not only the last milk-molar and the first true molar, but also the second or penultimate true molar (being three teeth in immediate contiguity), are invariably characterized in both jaws by an isomerous division of the crown into either three or four ridges; or, in other words, are severally characterized by either a "ternary-" or "quaternary-ridge-formula." These three isomerous-ridged teeth are referred to as "the intermediate molars." To the ternary-ridged species the author assigns the subgeneric name of Trilophodon; and Tetralophodon, to the quaternaryridged species. The molar in front, and that one behind these intermediate molars are also characteristically modified in these two subgenera. In Trilophodon the penultimate or second milk-molar is two-ridged, and the last true molar is four-ridged: in Tetralophodon, the former is three-ridged, and the latter five-ridged. The author considers it highly probable that a subgeneric group characterized by a quinary-ridge-formula (Pentelophodon) has existed in nature, but of which no remains have yet been discovered. The Elephants are distinguished from the Mastodons by the absence of an isomerous-ridge-formula, as regards the three intermediate molars, and by the ridges ranging from six up to an indefinite number in these teeth, in different groups of species. Dr. Falconer arranges the numerous fossil and recent forms in three natural subgenera, founded on the ridge-formula, in conjunction with other characters. In the Stegodon (comprising besides other forms the Mastodon elephantoides, Clift) the ridge-formula is hypisomerous; and the ridges number six or eight. The Loxodon (including the African Elephant) is also hypisomerous, and has from seven to nine plates or ridges. The Euelephas (including the Elephas indicus and six fossil species) is the largest and most important group, and comprises the typical Elephants having thin-plated molars. Here the ridge-formula is anisomerous, and regulated by progressive increments, as 8, 12, 16; the higher its numerical expression, the greater the liability to vary, within certain limits dependent upon the race, sex, and size of the individual; the lower molars often exhibiting an excess of plates over those in the upper molars. Reverting to the Mastodons, Dr. Falconer observed that the subgenera Trilophodon and Tetralophodon, as regards number of forms, are of nearly equal value; the former comprising seven, and the latter six well-marked species. Each group is divisible into two parallel subordinate groups. In the one series the ridges are broad, transverse, more or less compressed into an edge; with the intermediate valleys open throughout and entirely uninterrupted by subordinate tubercles. These are represented in Trilophodon by Triloph. ohioticus, and in Tetralophodon by Tetr. latidens. In the other series the ridges are composed of blunt conical points, which are fewer in number, flanked in front and behind by one or more subordinate outlying tubercles, which disturb the transverse direction of the ridges and block up the valleys. This series is represented by Trilophodon angustidens and by Tetralophodon arvernensis. In both subgenera the species with transverse compressed ridges may be compared with Dinotherium, as regards their molar crowns; and the other series with Hippopotamus. The European fossil species of Mastodon, according to the author, are the following:- Trilophodon Borsoni, I. Hays, Tril. tapiroides, Cuvier, Tril. angustidens, Cuvier (pro parte), Tril. pyrenaicus, Lartet MS., Tetralophodon longirostris, Kaup, and Tetr. arvernensis, Croizet and Jobert. With the exception of Triloph. Borsoni and Tetral. arvernensis, which are of Pliocene age, the above-named species are of Miocene age. Dr. Falconer proceeded to state that the remains of only one species of Mastodon have hitherto been discovered in the British Isles. They occur in what is called the Older Pliocene Red Crag, at Felixstow and Sutton, in Suffolk, and in the Newer Pliocene Fluvio-marine or Mammaliferous Crag at various localities near Norwich and in Suffolk. After remarking that Professor Owen had referred the teeth of the Crag Mastodon to M. angustidens, making M. longirostris and M. arvernensis to be synonyms of this species (as Cuvier had also done), Dr. Falconer gave in detail the history of the discovery and publication of the true M. angustidens (Cuvier), and of the M. arvernensis and M. longirostris. He then passed |