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:

0451, 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),

{(P-3r)4,+(-3r+2)4,3+(~—2—3).2,-)},

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),

2o1 — 6μ3 +11 μ2 — 36r+24+9r2.2, +(r3+29r+60)2,-1

—0(r−4)2(r−7)3 — 2·0(r−5)3.

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.

T

idea naturally suggested itself of examining this substance in comparison with the products of artificial destructive distillation.

With this view, one of us was induced to procure, through the intervention of a friend, several tons of Rangoon tar, which was carefully collected at the source, and transmitted to Europe in well-secured vessels. Our experience in the course of this inquiry, has shown that this quantity, large as it may appear, was by no means too ample a supply. Burmese naphtha contains indeed so great a variety of substances, and some of them in so exceedingly minute a proportion, that even the large amount of material at our disposal was insufficient for the complete examination of several constituents, the presence of which we had succeeded in establishing beyond a doubt. As an example, we may state that Burmese naphtha contains small quantities of organic bases, the study of which we were compelled to postpone to a later period, when an additional quantity of material, which is now on its way to Europe, will have come to hand.

We have already mentioned that Rangoon tar is almost entirely volatile, and preliminary experiments proved to us that the distillation could be effected most conveniently, and with less danger of obtaining products of decomposition, in a current of steam; first of a temperature of 100° C. (212° F.), and subsequently of steam superheated by passing, before it entered the still, through a system of pipes the temperature of which could be regulated. Treated in this way, it furnishes 96 per cent. of volatile products, fluid and solid.

Steam of 100° C. (212° F.) carries over 11 per cent. of a volatile oil perfectly free from solid hydrocarbons, which at that temperature are entirely retained in the distillatory apparatus. Between the temperatures of 110° and 145° C. (230°–293° F.), 10 per cent. of a further distillate is obtained, which is almost free from solid hydrocarbons. The temperature may be raised to 160° C. (320° F.) without materially augmenting this per-centage; but on gradually increasing the temperature of the steam to the fusion-point of lead, the operation yields 20 additional per cent. of distillate, which retains its fluidity at 0° C. (32° F.), notwithstanding the presence in it of an appreciable quantity of solid matter. At this stage of the process the products of distillation begin to solidify on cooling, and about 31 per cent. of a crystalline material is obtained sufficiently consistent

*Warren De la Rue.

to be submitted to pressure. After this the consistence of the products of distillation begins to diminish; and whilst the temperature of the steam is considerably raised, 21 per cent. of a mixture of solids and liquids distil, the latter predominating especially as the operation proceeds.

In the last stage of the process the distillate completely changes its character, becoming very dark in colour, of a pitch-like consistence, and exhibiting scarcely an indication of the presence of crystalline matter. When this product, which amounts to about 3 per cent., has passed over, there remains in the still a coke-like mass, which contains a small quantity of earthy impurities.

Although there is a considerable difference between the specific gravities of the first and last fractions of the distillates, all the products of distillation, like the original oil, are lighter than water, and could be separated therefore by means of the well-known apparatus (called a Florentine flask) employed in the distillation of essential oils.

By exposing the distillates obtained beyond the temperature of 145° C. (293° F.) to a freezing mixture, nearly all the crystalline matter solidified, and became removeable by means of filtration and pressure. It was thus ascertained that Rangoon tar contains from 10 to 11 per cent. of solid constituents (paraffin).

Solid Constituents.

The solid product, when removed from the fluid hydrocarbons, still retains a portion of the latter with much obstinacy; in order to purify the solid, it has to be subjected to the action of boiling concentrated sulphuric acid, and to be subsequently washed, first with an alkaline solution, then with water. On redistillation, the paraffin is obtained quite white, but even now it still retains some fluid hydrocarbons which have resisted the action of the sulphuric acid; the greater part can be removed by pressure between folds of cloth in a powerful hydraulic press and subsequent exposure for some months. to the air, in which the fluids gradually disperse. By fractional crystallization from hot alcohol, we have been enabled to separate the solid product into at least two distinct compounds, which appear to have the same per-centage composition, agreeing either with C, II, or CH+1, but which differ from each other in their physical properties. By the action first of sulphuric acid saturated with anhydrous acid,