It may here be asked, Is this lifting power of the surface currents sufficient to account for the vertical circulation which we find in the ocean? In all probability it is not. There seems to be no reason why this vertically rising current under the equator should draw its supplies from the furthest limits of the ocean, which it would require to do to explain the conditions we find existing. Yet there can be no doubt but that these horizontal surface currents really do assist in producing a vertical circulation.

3. On a New Investigation of the Series for the Sine and Cosine of an Arc. By Edward Sang.

The sines of the successive equidifferent arcs form a progression having for its general character the relation

and the properties of sines may be deduced from this general formula. Viewed in this light, the angular functions become cases only of more general ones.

If we suppose A, B, C to be three consecutive terms of such a progression we must have

from which, when three of the four quantities, A, B, C, v are given, the fourth may be found. Let then A and B, the first and second terms of the progression, and v the common coefficient, be given ; the succeeding terms may be computed thus:

from which it is obvious that the coefficient of A in the expression for pn, is a transcript of that of B in the preceding expression for (n-1). Hence, for the present, we may confine our attention to the latter.

The coefficients of B form the following progression:

in 40 0 in ø1 1

in 42 2+v

in 43 3+4v+v2

in 4 4+10v+6v2 + v3

in 45 5+20v+21v2 + 8v3 + v1

in 46 635v+56v2 + 36v3+10v4+v5

in 47 7+56v+126v2 + 120v3 +55v1 + 12v3 + vổ

in 48 8+84v+252v2 × 330v3 + 220v1 + 78v5 +14v3 + v2

in 49 9+120v+462v2+792v3 +715v1+ 364v5 +105v6 + 16v7 + &c.

and in general

in on +

n n 1 n n+1
υ+
1 1 2 3

n-2 n 1 n n + 1 n + 2
1 2 3 4 5

v2 + &c.

When u is positive the formulæ belong to the class of catenarian functions; when v is negative, to the circular ones.

If we put sin pa for 40, sin p+1 a for ø1, and chord a for v, we obtain

A

n2 - 11

1.2 3

n2 - 1 n2-41

sin A =wain Một - "iz (cho ) + " (cho )

n

n

1.2 3.4 5

Now when becomes indefinitely great, n sin

A

n

n

&c.}

becomes A, so also

In order to obtain the series for the cosine we must put pa= and therefore sin (p + 1)a = cos a, which gives

and if, in this, we substitute for cos a, its value 1 – § cho a2,

cho 2 + &c.

2 3

c. }

4. Note on the Bifilar Magnetometer. By J. A. Broun, F.R.S. Communicated by Professor Tait.

5. Addition to the paper "On the establishment of the Elementary Principles of Quaternions," by G. Plarr,— published in Vol. XXVII. of the Transactions of the Society. Communicated by Professor Tait.

6. Note on Mr Muir's Solution of a "Problem of Arrangement." By Professor Cayley.

This note has been printed along with Mr Muir's paper, ante, p. 382.

7. Preliminary Note on a New Method of Investigating the Properties of Knots. By Professor Tait.

As we cannot have knots in two dimensions, and as Prof. Klein has proved that they cannot exist in space of four dimensions, it would appear that the investigation of their properties belongs to that class of problems for which the methods of quaternions were specially devised. The equation

p = $(s),

where is a periodic function, of course represents any endless curve whatever. Now the only condition to which variations of this function (looked on as corresponding to deformations of the knot) is subject, is that no two values of p shall ever be equal even at a stage of the deformation. Subject to this proviso, may suffer any changes whatever-retaining of course its periodicity. Some of the simpler results of a study of this novel problem in the theory of equations were given, among others the complete representation of any knot whatever by three closed plane curves, nonautotomic and (if required) non-intersecting.

The following Gentlemen were elected Ordinary Fellows of the Society:

ROBEBT A. MACFIE, Dreghorn, Colinton.
WILLIAM STIRLING, Sc.D., M.D.

Monday, 21st May 1877.

PROFESSOR KELLAND, Vice-President, in the Chair. The following Communications were read:

1. On the Cranial Osteology of Rhizodopsis, and on some points in the Structure of Rhizodus. By Dr R. H. Traquair.

2. Notice of Recent Earthquake Shocks in Argyleshire in 1877. By David Stevenson, Civil Engineer.

Two earthquake shocks have lately occurred in Argyleshire of so decided a character that a description of their effects, as observed

at four of the Lighthouse stations on the west coas; of Scotland, will, it is thought, be interesting to the Society.

The first shock occurred on the 11th March, and was observed at the Lighthouse station of Hynish in the island of Tyree, and at Sound of Mul!, near Tobermory, the distance between the two places being about 34 statute miles.

The report from Tyree states :-"On the 11th current (March), at half-past 11 o'clock A.M., a smart shock of earthquake was felt all along the island; a great many people both heard the noise and at the same time felt the earth to tremble. It was heard and felt very distinctly at the station." Bar. 30.18 at 9 A.M.

That from the Sound of Mull says:-" On the 11th, at 11.30 A.M., this district was visited by a smart shock of earthquake. It began by a rumbling noise like distant thunder. When the noise was at its height the houses, and everything about them, shook, and the slates on the roof rattled. The shaking was not of long duration, but the noise was heard a considerable time before and after the trembling of the earth." Bar. 29.92 at 9 A.M.

The second shock, which seems to have been more severe, took place on the 23d April, and was observed at the island of Phladda, off Easdale, and at Lismore, at the eastern entrance to the Sound of Mull, the distance between the two stations being about 73 statute miles.

The report from Phladda states:-- "At 3.40 A.M. the Principal Keeper on the watch felt a severe shock of earthquake. The tower and dwelling-houses shook very much. All the neighbouring

islands felt it at the same time." Bar. 29.74 at 9 A.M.

At Lismore the lightkeeper describes the effect as follows (the lighthouse clock had been under repair):-"I beg leave to report that. on the morning of the 23d, at 3.30 A.M., while I was standing on the grating inside the lightroom I felt a heavy shock on the tower, with a strange rumbling sound of noise which lasted some seconds, and made everything in the lightroom shake at an alarming rate. It awoke all the inmates of the dwelling-houses. Mr M'Leod jumped out of bed, thinking the tower had fallen, but afterwards thought it was a peal of thunder. I do not think it was thunder. I saw no lightning, and the wind was