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 parts of the 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.

XVII. "On Autopolar Polyedra." By the Rev. THOMAS P. KIRKMAN, M.A. Communicated by ARTHUR CAYLEY, Esq., F.R.S. Received June 19, 1856.

(Abstract.)

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 ẞ-ace b (or summit 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 ẞ-gon B, corresponds an edge ab, between the a-ace a and the ß-ace b. Two such edges are called a gamic pair, or pair of gamics.

The

The enumeration of autopolar p-edra is here entered upon as a step towards the determination of the number of p-edra. 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 II 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),

{(~_3r)4,+(P—3r+2)+,3+(~—2—3)-2,1},

where the circulator s=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),

r2 — 6r‍3 +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.

COMMUNICATIONS RECEIVED SINCE THE END OF THE SESSION.

By a Resolution of Council of the 26th of June, 1856, the President and Officers are henceforth authorized, at their discretion, to print in the 'Proceedings' abstracts of Papers received during the Recess, without waiting until such Papers shall have been read to the Society.

I. "Chemical Examination of Burmese Naphtha, or Rangoon Tar." By WARREN DE LA RUE, Ph.D., F.R.S., and HUGO MÜLLER, Ph.D. Received August 1, 1856.

In several localities of the kingdom of Burmah, there emanates from the soil in considerable quantity a peculiar oleaginous substance, which is employed for a variety of purposes, but chiefly as a lampfuel and as an unguent, by the natives, and exported in moderate quantities under the name of Burmese naphtha, or Rangoon tar.

It is obtained by sinking wells of about 60 feet in depth, in which the liquid is collected by the miner as it oozes from the soil..

At the common temperature this substance has the consistence of goose-fat; it is lighter than water, has usually a greenish-brown colour, and possesses a slight odour, peculiar, but not disagreeable. It consists almost entirely of volatile constituents.

Burmese naphtha has already attracted the attention of other chemists; at present we refrain from entering into a discussion of their results, since it is our intention to give a full history of this remarkable natural product when, after the completion of our experiments, we shall have the honour of submitting to the Royal Society a detailed account of our investigation. The object of the present communication is to trace a mere outline of the results at which we have arrived up to this moment.

The circumstances under which petroleum-for this is the collective term which comprehends a great variety of oily emanations similar to Burmese naphtha-occurs in nature, all tend to prove that these substances are the products of a slow destructive distillation of the residuary matter of a primeval creation: this being admitted, the

VOL. VIII.

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