But for the purpose of calculation it is best to integrate by a series the differential equation for Q: assume

We have thus for 93, 94, 95... the values 1, 2, 14, 82, 593, 4820, and thence

u = (1 − x2)(1 + 2x + 14x2 + 82x3 +593x1 +4820x5 + .),

In the more simple problem, where the arrangements of the n things are such that no one of them occupies its original place, if u, be the number of arrangements, we have

but the calculation is most easily performed by means of the foregoing equation of differences, itself obtained from the differential equation written in the foregoing form,

( − 1 + 2x + 3x2)u + (x2 + x3)u' = −1.

6. On Amphicheiral Forms and their Relations.
By Professor Tait.

(Abstract.)

If a cord be knotted, any number of times, according to the pattern below

it is obviously perverted by simple inversion. Hence, when the free ends are joined it is an amphicheiral knot. Its simplest form is that of 4-fold knottiness. All its forms have knottiness expressible as 4n.

VOL. IX.

3 F

The following pattern gives amphicheiral knots of knottiness 2+6n.

And on the following pattern may be formed amphicheiral knots of all the orders included in 6n and 4+ 6n.

Among them these forms contain all the even numbers, so that there is at least one amphicheiral form of every even order.

Many more complex forms are given in the paper, several of which are closely connected with knitting, &c.

In one of my former papers I gave examples of type-symbol which individually represent two perfectly different knots.

I now give examples of the same knot represented by typesymbols which have neither right nor left-handed parts in common. One of the most remarkable of these is

which can be analysed (but not separated) into a combination of the two forms of the 4-fold amphicheiral knot.

The following Gentlemen were elected Fellows of the Society :

WILLIAM JOLLY, H.M. Inspector of Schools, Inverness.

ROB. MILNER MORRISON, 21 Manor Place.

JOHN GIBSON, Ph.D., 12 Greenhill Gardens.

CHAS. E. UNDERHILL, B.A., M.B., 8 Coates Crescent.

CHARLES EDWARD WILSON, M.A., LL.D., 19 Palmerston Place.
GEORGE CARR ROBINSON, East Preston Street.

Monday, 16th April 1877.

PROFESSOR KELLAND, Vice-President, in the Chair.

The following Communications were read:—

1. On the Toothing of Un-round Discs which are intended to roll upon each other. By Edward Sang.

(Abstract.)

This paper contained an extension to discs of any shape whatever of the principles explained in the "New General Theory of the Teeth of Wheels," as applicable to circular discs.

2. On the Mineralogy of Scotland.—Chapter II.
By Professor Heddle.

In this chapter Professor Heddle submitted the results of the analyses of Orthoclase from fifteen localities; of Albite from four; of Oligoclase from eight; of Labradorite from eleven; of Andesiel from five; of Anorthite from three; and of Latrobite from two. The three last minerals being now, for the first time, recognised as British species.

Dr Heddle also described a peculiar association of Orthoclase with Oligoclase, in crystals nearly of the form of the former, from certain localities; he drew the conclusion from his researches that the above felspars are all well individualised, if not by physical, at least by chemical characters; while they are probably more or less special to certain rocks.

3. Least Roots of Equations. By J. D. Hamilton Dickson.

Monday, 7th May 1877.

DAVID STEVENSON, Esq., Vice-President, in the
Chair.

The following Communications were read:

1. On new and little-known Fossil Fishes from the Edinburgh District. No. III. By Dr R. H. Traquair.

2. On Ocean Circulation. By John Aitken.

It is with extreme reluctance that I venture to disturb the present repose of the much-contested field of ocean circulation. My object is not, however, to provoke discussion on the general theory of ocean circulation, as I am sure all will agree in thinking that the subject has already been discussed far beyond the point at which it is likely to be benefited by discussion. My object is simply to call attention to certain influences at work in the ocean, the effects of which seem to have been totally overlooked.

The first of them to which I wish to refer is the influence of the winds on the ocean. The extreme holders of the wind theory of ocean circulation consider that the action of the wind is quite sufficient to account for all the currents which we find in the ocean. That the wind is a cause of ocean currents no one can doubt. If we examine a lake when the wind is blowing over it, we shall find that the plants growing in the shallow water near the surface are all bending in the direction of the wind, indicating that there is a current at the surface flowing in the direction of the wind,-the appearance of the bending plants in the lake reminding one of a slow-running river. To supply the water for this surface current there must, of course, be another current, flowing in the opposite direction underneath. The lake and the ocean are not, however, parallel cases. In the case of the lake, the wind is blowing in the same direction all over it, so that the return current is forced to flow underneath the surface, as it cannot get back any other way; whereas, in the ocean, the wind blows in one direction at one part,