BIBLIOGRAPHY.

The following works are some of those consulted in the preparation of this report, and will be found of use to students studying the Irish Phaophyceae :—

HOLMES AND BATTERS,

"A Revised List of the British Marine Algae, with an Appendix." Ann. Bot., v. 5, 1890.

J. B. FARMER AND J. LL. WILLIAMS,

Phil. Trans. Roy. Soc., v. 190, 1898.
"The Antherozoids of Dictyota and
Taonia," Ann. Bot. xi., 1897.
"Reproduction of Dictyota dichotoma."
Ann. Bot. xii., 1898.

"Sur quelques Algues Phéosporées para-
sites." Journal de Botanique. 1892.

Phycologia Britannica, 1846-51.

"Sur quelques Myrionimacées." Annales des Sciences Naturelles. 8 sér. (Bot.).

"Algological Notes." Jour. Mar. Biol. Assoc., vol. iv., 1896.

Sylloge Algarum, vol. iii., "Fucoidea," 1895.

Études Phycologiques."

"Atlas d. Deutschen Meeresalgen," 1889. "Plurilocular Zoosporangia of Aspero

coccus Bullosus and Myriotrichia clavaformis." J. of Bot., Nov., 1891. "Tabula Phycologica."

"Bemerkungen zur marinen Algenvegetation von Helgoland." 1894.

Ber. d. deutsch. Bot. Gesellsch., xvi.,

pp. 35, 37.

KJELLMAN, F. R.,

"Ueber die Laminarien Norwegens."

(Christiania vidensk-selsk. Forhandl. 1884, No. 14).

"Observations relatives à la sexualité des Phéosporées." Jour. de Bot., 1896-97. "La copulation Isogamique de l' Ectocarpus siliculosus. Est-elle apparente ou réelle ?" Mem. d. sci. nat. et math. de Cherbourg, t. xxx.

"Phæophyceae."

Engler & Prantl, D. Nat. Pflanzenfamilien, 1891-96.

CROUAN, P. L., ET H. M., . "Florule du Finistère." Brest, 1867.

"Irish Naturalist" has had notes on Irish Algæ in its pages from

time to time.

XXVII.

GEOMETRY OF SURFACES DERIVED FROM CUBICS. BY ROBERT RUSSELL, M.A., F.T.C.D.

[Read JUNE 26, 1899.]

1. It is well known that the locus of a point P, whose polar quadric with regard to a cubic surface is a curve having its vertex at P', is a surface of the fourth degree-the Hessian, and that the polar quadric of P' is a cone having its vertex at P. Such points are called corresponding points on the Hessian, and several elegant properties of this surface are to be found in Salmon's "Geometry of Three Dimensions." If the equation of the cubic surface be written in Sylvester's canonical form

and if the coordinates of P are x, y, z, v, w, those of P' are

It is easy to see that the line joining PP' belongs to a congruency, that is, moves in space subject to two conditions; and it has been shown by Sir William Rowan Hamilton that such lines are in general bitangents to a surface. Several of the properties of this surface which we shall denote by the symbol C are discussed in the following pages. 2. Points on the surface.-In order to determine points on this surface, we have to consider where PP' is met by consecutive lines of the congruency.

These are the points of contact of PP' with C.

Let έ, n, k, v, w be the coordinates of one these points T, then

in passing to points near to PP', §, n, Š, v, w remain unchanged; therefore,

with four similar equations in y, z, v, w, from which, by putting for έ, n, . . ., their values from (1), we easily get

x + y + 8 + v + w = 0, dx + dy + dz + dv + dw = 0,

we have for the equation

It contains the irrelevant factor 0; dividing by this, there remains a quadratic.

Denoting the roots of this quadratic by 01, 02, we see that the line PP' touches C in two points, T and T', whose coordinates are given by putting 0, and 0, for 0 in (1).

The geometrical interpretation of this quadratic leads to the following property of the surface C.

Suppose the line PP' meets the Hessian in two other points, U, U', the coordinates of either of these points are proportional to

and in order that this point may lie on the Hessian, we must have

It follows, therefore, that TU, and T'U' are divided harmonically by PP' Hence we derive a construction for the surface C. Take a pair of corresponding points P, P' in the Hessian, produce the line joining them to meet it again in U, U'; the harmonic conjugates of UU' with respect to PP' are the points on the bitangent surface in which it is touched by PP'. This is exactly analogous to the property of the Cayleyan of a plane cubic.

3. The directions of the points near P and P', so tha consecutive lines of the congruency may intersect, can be readily found.

From (3) we have, for the point consecutive to P,

hence the directions are on the lines joining P to the points V, V', whose coordinates are

These points V, V' are on the Hessian, and they are the points corresponding to U, U', respectively. The tangent plane to the Hessian at Vis