stem-like base is a hypertrophy due to the irregular surface the colony was attached to, and that, therefore, this species cannot with certainty be separated from Clavelina.

Simon's Bay; 10 to 20 fathoms.

7. Description of New Astronomical Tables for the Computation of Anomalies. By Mr Edward Sang.

(Abstract.)

The planets move round the sun in ellipses, in such a manner that the radii vectores describe areas proportional to the times. Now, by means of parallel lines, we can always project an ellipse upon a plane surface so as to make the projection circular, and thus we have to consider the motion of a point in the circumference of a circle, describing round an excentric point areas proportional to the times. If we take S for the excentric point, that is for the projection of the sun, and suppose Q to be the projection of the planet's place, the area ASQ is proportional to the time elapsed since the perihelion passage. The angle AOQ is called, very inappropriately, the excentric anomaly; I prefer to call it the angle of position. If we suppose a point M to move uniformly along the circumference, with the periodic time of the planet, and to have reached M when

the actual projection of the planet is at Q, it is clear that the sector AOM must be equivalent to the area ASQ. The angle AOM is the mean anomaly.

Having drawn ESF perpendicular to the diameter ASOa, join QE and QF; then it is evident that the surface EQFA is halved by the compound line ASQ; wherefore

the area ASQ passed over by the radius vector is half the sum or half the difference of the circular segments QAF and QE, according as Q lies beyond AE or within it.

Denoting the arc AE by e, and the arc of position AQ by p, and

observing that the area AOM is equivalent to ASQ, we have, denoting the segment AOM by m,—

and thus the determination of m from p, or of p from m, is to be accomplished by help of a table of circular segments, which must be measured, not in parts of the square of the radius, but in degrees of the surface of the circle.

For the purpose of rendering this exceedingly simple formula available for actual calculation, a table was constructed of the sines for each minute of the quadrant, measured in degrees of arc; by its help the values of the circular segments for each minute of the whole circumference were written out, true to within one tenthousandth part of a second of the modern division.

When we have got a tolerable first approximation, this table enables us to compute the position corresponding to a given mean anomaly by a simple proportion.

In order to guide us to a first assumption, tables were constructed of the mean anomalies corresponding to each degree of position from 0° C. to 200° C., and for every value of e from 0° C. to 100° C., with their differences and variations, true to the nearest second; and thus, in every possible case, the solution of Kepler's problem is obtained in a few minutes, true to far within the hundredth part of a second of the new division.

For the construction of these tables, one million six hundred thousand figures were written, and of these the three volumes placed on the table contain about twelve hundred thousand. If the ancient division of the quadrant had been used, the labour would have been more than doubled.

8. The Discharge of Electricity through Olive Oil. By A. Macfarlane, D.Sc., and P. M. Playfair, M.A.

9. Note on the Colouring of Maps. By Frederick Guthrie.

From the Proceedings of the Royal Society of Edinburgh, No. 106, p. 501, it appears the colouring of maps is receiving attention. This note bears chiefly upon the history of the matter.

VOL. X.

4 8

Some thirty years ago, when I was attending Professor De Morgan's class, my brother, Francis Guthrie, who had recently ceased to attend them (and who is now professor of mathematics at the South African University, Cape Town), showed me the fact that the greatest necessary number of colours

2

1

4

3

to be used in colouring a map so as to avoid identity of colour in lineally contiguous districts is four. I should not be justified, after this lapse of time, in trying to give his proof, but the critical diagram was as in the margin. With my brother's permission I sub

mitted the theorem to Professor De Morgan, who expressed himself very pleased with it; accepted it as new; and, as I am informed by

those who subsequently at

tended his classes, was in the habit of acknowledging whence he had got his information.

If I remember rightly, the proof which my brother gave did not seem altogether satisfactory to himself; but I must refer to him those interested in the subject. I have at various intervals urged my brother to complete the theorem in three dimensions, but with little success.

It is clear that, at all events when unrestricted by continuity of curvature, the maximum number

of solids having superficial contact each with all is infinite. Thus, to take only one case, n straight rods, one edge of whose projections forms the tangent to successive points of a curve of one curvature, may so overlap one another that, when pressed and flattened at their points of contact, they give n-1 surfaces of contact.

How far the number is restricted when only one kind of superficial curvature is permitted must be left to be considered by those more apt than myself to think in three dimensions and knots.

10. Remarks on the previous Communication. By Prof. Tait.

I

(Abstract.)

In a paper read to the Society on 15th March last (ante, p. 501), gave a series of proofs of the theorem that four colours suffice for a map. All of these were long, and I felt that, while more than sufficient to prove the truth of the theorem, they gave little insight into its real nature and bearings. A somewhat similar remark may, I think, be made about Mr Kempe's proof.

But a remark incidentally made in the abstract of my former paper has led me to a totally different mode of attacking the question, which puts its nature in a clearer light. I have therefore withdrawn my former paper, as in great part superseded by the present one.

The remark referred to is to the effect that, if an even number of points be joined, so that three (and only three) lines meet in each, these lines may be coloured with three colours only, so that no two conterminous lines shall have the same colour. (When an odd number of the points forms a group, connected by one line only with the rest, the theorem is not true.)

This follows immediately from the main theorem when it is applied to a map in which the boundaries meet in threes (and the excepted case cannot then present itself). For we have only to colour such a map with the colours A, B, C, D. Then if the common boundaries of A and B, as also of C and D, be coloured a; those of A and C, and of B and D, ẞ; and those of A and D, and of B and C, y, it is clear that the three boundaries which meet in any one point will have the three colours a, B, y.

The proof of the elementary theorem is given easily by induction; and then the proof that four colours suffice for a map follows almost immediately from the theorem, by an inversion of the demonstration just given.

We escape the excepted case by taking the points as the summits of a polyhedron, all of which are trihedral; and when the figure is a pentagonal dodecahedron the theorem leads to Hamilton's Icosian Game.

11. Note on the Wire Telephone as a Transmitter.

By James Blyth, M.A.

It was shown some time ago by Dr Ferguson, and more recently by Professor Chrystal and Mr Preece, that a fine wire attached to a mechanical telephone can act very well as a receiver in a telephonic circuit, provided a make and break, or some form of microphone transmitter, be employed. None of these experimenters, however, have said anything about the action of such a wire as a transmitter. Being struck by the convertibility, in general, of all forms of telephone receivers into transmitters, and vice versa, it occurred to me to try how far this wire telephone, as it has been called, could be made to act as a transmitter to an ordinary Bell telephone as receiver. I was much interested to find that it could act in that capacity wonderfully well, as thereby a new element of some im portance is introduced into the discussion of the real cause or causes of the action of the wire telephone whether as receiver or as transmitter.

In my first experiments a battery of four Bunsen cells was included in a telephone circuit of small resistance. At the sending station, which we shall call A, an arrangement was made whereby different lengths of various kinds and thicknesses of wire could be inserted in the circuit. At first these wires were inserted by being soldered to the copper terminals, in order to keep clear of loose contacts; but it was afterwards found that all error arising from this source could be avoided by simply clamping the wires firmly between two binding screws. This method, from its greater convenience, was therefore afterwards adopted. To the middle of the inserted wire, and at right angles to it, was attached a fine iron wire about 15 inches long, the other end of which was connected to the centre of the parchment disc of a mechanical telephone. When this wire was stretched moderately tight the transmitting arrangement was complete. At the receiving station, which we shall call B, an ordinary double-ear Bell telephone of small resistance was employed.

When a fine iron wire about 9 inches long was inserted in the circuit at A, any musical sound uttered into the mechanical telephone was most distinctly reproduced at B. Speech could also be