during the past twelve months-as during the preceding year; and I trust that my Presidency which closes to-day will not in future be thought to have been detrimental to the prosperity of the Society.

Mr BALFOUR having taken the chair the following communication was made to the Society:

Note on Abel's Theorem. By Professor CAYLEY.

Considering Abel's theorem in so far as it relates to the first kind of integrals, and as a differential instead of an integral theorem, the theorem may be stated as follows:

We have a fixed curve f(x, y, 1) = 0 of the order m; this implies a relation f'(x) dx +ƒ' (y) dy = 0, between the differentials dx, dy of the coordinates of a point on the curve; and we may therefore

write

and, instead of dx or dy, use do to denote the displacement of a point (x, y) on the curve.

Taking for greater simplicity the fixed curve to be a curve without nodes or cusps, and therefore of the deficiency (m −1)(m −2), we consider its mn intersections by a variable curve (x, y, 1) = 0 of the order n. And then, if (x, y, 1)"-3 denote an arbitrary rational and integral function of (x, y) of the order m-3, the theorem is that we have between the displacements do,, dw,...dw of the mn points of intersection, the relation

Σ (x, y, 1)m-3dw = 0,

where the left-hand side is the sum of the values of (x, y, 1)*dw, belonging to the mn points of intersection respectively.

For the proof, observe that, varying in any manner the curve 4, we obtain

where 8 is that part which depends on the variation of the coefficients, of the whole variation of ; viz. if = ax" + bx"1y+..., then 8px"da+x"1ydb + ..., dp is thus, in regard to the coordinates (x, y), a rational and integral function of the order n. Writing in this equation

=

df dw,

dx

and then multiplying each side by the arbitrary function (∞, y, 1)TM-3, we have

where 84 being of the order n in the variables, the numerator is a rational and integral function of (x, y) of the order m+n-3: hence by a theorem contained in Jacobi's paper Theoremata nova algebraica circa systema duarum æquationum inter duas variabiles, Crelle t. xiv. (1835) pp. 281-288, the sum on the right-hand side is = 0: hence the required result Σ (x, y, 1)TM-3dw = 0.

Observing that (x, y, 1) is an arbitrary function, the equation just obtained breaks up into the equations

Σdo=0, Exdw=0, Zydw=0,... ΣxTM*dw=0,... Σy"1dw= 0, viz. the number of equations is

1+2+...+(m2), (m-1) (m − 2),

= }

which is = p, the deficiency of the curve.

Suppose the fixed curve f(x, y, 1) = 0 is a cubic, m=3, and we have the single relation Edw=0, where the summation refers to the 3n points of intersection of the cubic and of the variable curve of the order n, p (x, y, 1) = 0.

In particular if this curve be a line, n = 1, and the equation is dw, + dw,+dw,=0; here the two points (x, y), (,, y,) taken at pleasure on the cubic, determine the line, and they consequently determine uniquely the third point of intersection (x, y); there should thus be a single equation giving the displacement do, in terms of the displacements do,, dw,; viz. this is the equation just dw1 + dw2+ dw1 = 0.

found

So if the variable curve be a conic, n=2; and we have between the displacements of the six points the relation

.dw, + dw, ... + dw1 = 0:

here five of the points determine the couic, and they therefore determine uniquely the sixth point; and there should be between the displacements a single relation as just found.

If the variable curve be a cubic, n = 3, and we have between the displacements of the nine points the relation

da, + dw,... + dw,= 0:

here eight of the points do not determine the cubic , but they nevertheless determine the ninth point, viz. (reproducing the reasoning which establishes this well-known and fundamental theorem as to cubic curves) if 4,0 be a particular cubic through the 8 points, then the general cubic is 4.+kf=0, and the intersections with f=0 are given by the equations 4 = 0, ƒ=0; whence the ninth point is independent of k, and is determined uniquely by the 8 points. There should thus be a single relation between the displacements, viz. this is the relation just found.

And so if the variable curve be a quartic, or curve of any higher order, it appears in like manner that there should be a single relation between the displacements; this relation being in fact the foregoing relation Σdw=0.

But take the fixed curve to be a quartic, m = 4: then we have between the displacements do the relation

Σ (x, y, 1) dw = 0,

that is the three equations

Σxdw=0, Zydw= 0, Edw=0.

If the variable curve is a conic, n=2, then there are 8 points of intersection; 5 of these taken at pleasure determine the conic, and they consequently determine the remaining 3 points of intersection: hence there should be 3 equations. And so if the variable curve be a curve of any higher order, then by considerations similar to those made use of in the case where the first curve is a cubic it appears that the number of equations between the displacements do should always be = 3.

=

But if the variable curve be a line, n = 1, then the number of the points of intersection is 4: 2 of these taken at pleasure determine the line, and they consequently determine the remaining 2 points of intersection; and the number of equations between the displacements de should thus be 2. But by what precedes we have the 3 equations

dw1+ dw,+ dw,+ dw ̧= (),

x ̧dw ̧ + x ̧dw ̧ + x ̧dw ̧+x ̧dw ̧= 0,

y1dw ̧ + y ̧dw ̧+y,dw ̧+y ̧dw ̧ = 0;

=

here the 4 points of intersection are on a line y=ax+b; we have therefore y-ax1 + b,... y1 = ax,+b; the equations between the

dw's give (y,-ax1 - b1) dw, + ... + (y-ax - b) dw, = 0, that is, is a single relation 0=0; or the 3 equations thus reduce themselves to 2 independent equations.

Again, if the fixed curve be a quintic, m = 5, there are here between the displacements the 6 equations

Σx3dw=0, Σxydw=0, Σy'dw = 0,

the two cases in which the number of independent equations is less than 6 are 1o when the variable curve is a line, and 2o when the variable curve is a conic. For the line n=1, and the number should be=3. We have the above 6 equations; but the equation of the line is ax+by+c=0, that is, we have ax1+by1+c=0, &c.; we deduce the 3 identical equations

Σx (ax+by+c) = 0, Σy (ax+by+c) = 0, Σ (ax+by+c) = 0, and the number of independent equations is thus 6-3, as it should be.

3

conic, n = 2; the number of The points of intersection 1)=0; we have therefore

So when the variable curve is a independent equations should be=5. lie on a conic (a, b, c, f, g, h(x, y, the several equations (a, b, c, f, g, h)x1, y1, 1)2=0, &c.: we have therefore the single identical equation

Σ (a, b, c, f, g, h(x, y, 1)' dw = 0,

and the number of independent equations is 6-1, 5 as it should be.

Obviously the like considerations apply to the case where the fixed curve is a curve of any given order whatever.

November 14, 1881.

Mr F. M. BALFOUR, PRESIDENT, IN THE CHAIR.

The following communication was made to the Society: On the rocks of the Channel Islands, No II. By Professor LIVEING.

In a former communication on this subject I drew from my observations of the rocks of the Channel Islands, principally those of Guernsey, the conclusion that the granitic structure is a metamorphic character which may be imparted either to stratified or to igneous rocks, which is not due to igneous fusion, but is rather the result of continued variations of temperature always far short

of fusion, assisted by the action of water or steam and perhaps other gases. Further observations made during last summer, chiefly on the rocks of Serk and Jersey, tend to confirm the conclusions previously arrived at, and at the same time bring out some fresh points which appear to be of sufficient interest to lay before the Society.

On Guernsey I have little to add to what I said before, except to point out that in the section which I gave, the strata should have been represented as bending downwards on the S. side far more abruptly so as to become nearly vertical in that part of the island.

These

The geology of Serk is very simple, at the same time very characteristic, so that it is an excellent commentary on that of the other Channel Islands. Serk is a table-land elevated about 400 feet above the sea-level. On the East and West sides, that is the sides which form the longer dimensions of the island, the table ends in precipitous cliffs, but at the North and South ends it falls more gradually to the sea, ending in long promontories continued for some distance by detached rocks. The island of Brecqhou forms a similar prolongation with a gradual slope to the sea on the West side, and may really be considered as part of Serk, from which it is separated by the narrow Gonliot pass only about 80 yards wide. The cliffs are on all sides intersected by deep ravines, sometimes by chasms with sides absolutely vertical, and are perforated at the sea-level by numerous, often deep, caves. features are, of course, the natural consequences of its geological structure. The mass of the island is formed of a crystalline hornblende schist cleaving readily parallel to the bedding, and making an excellent building material in everything but its colour which is coal-black. This schist rests, as seen at one place only, namely on the shore at the Port du Moulin on the N.W. side, on a stratum of grey syenite or syenitic gneiss. Upwards the schist passes into a more compact rock consisting of alternate laminæ of hornblende and of felspar mixed with a little quartz. again passes rather abruptly into a grey syenite in which the bedding can be traced for a short distance upwards and is then lost for a considerable thickness. The greater part of little Serk is composed of this granitic rock closely resembling that of the extreme north of Guernsey, and not unlike that of Herm which supplied the steps at the foot of the Duke of York's column in London. The originally stratified character of this syenite is however noway doubtful, for the bedding can be traced in it in many places, and at Port Goury in the South end of little Serk a thin bed in it crops out which has a slightly different mineral character, is distinctly bedded, and being pervious to water has undergone more decomposition than the rock above and below it, and

This