EXPLANATION OF THE FIGURES.

Fig. 1.

Plate II.

Transverse section of Kaloxylon Hookeri, Will., with secondary vascular bundles.

a. Primary xylem of the stele or central cylinder with conjunctive parenchyma.

b. Xylem of secondary vascular bundles.

p. Phloem of secondary vascular bundles, including perhaps elements of the primary phloem.

m. Medullary rays separating the secondary vascular bundles. c. Inner zone of cortex with the characteristic elements with

special contents.

Fig. 2. Transverse section of Kaloxylon Hookeri, Will., in the primary condition.

a. Primary xylem of the stele or central cylinder, with conjunctive parenchyma.

p. Primary phloem of the stele.

d. Zone of parenchyma representing the pericycle and endodermis.

b. Inner zone of cortex.

c. Peripheral zone of cortex.

[Microscopical and Natural History Section.]

Ordinary Meeting, January 14th, 1895.

JOHN BOYD, Esq., President of the Section, in the Chair.

Mr. THOMAS HICK, B.Sc., B.A., was elected an Associate of the section.

Mr. OLDHAM exhibited a collection of stone implements, shells, beads, and seeds, arranged as ornaments, from the Sandwich Islands.

Mr. MARK STIRRUP, F.G.S., exhibited cones of pine trees planted along the west coast of France, from Bordeaux to Pau, to prevent the shifting of the sand dunes; specimens of Helichrysum stæchas, a plant growing in these pine forests; and the fruit of the Magnolia, from the same district.

Mr. THOS. ROGERS, exhibited three species of land shells from Lord Howe Island, Australia, one being named after Mr. Whitelegge, a naturalist, formerly of Manchester, viz. :-Helix Whiteleggii, H. How-insula, Nanina Sophia; also shells from New South Wales, viz. :-Rhytida confusa, Helix globosa, and a very rare Cypræa tesseleta, from the Sandwich Islands.

A Sketch of the Limitations which are enforced upon the Mathematical Forms of the Expressions for Physical Quantities in a Continuous Medium in consequence of the necessity for their Permanence of Form. By R. F. Gwyther, M.A.

(Received February 5th, 1895).

In all parts of Applied Mathematics we find the same forms of expressions occurring, and I propose in this sketch to show how this recurrence arises from the fact that the expression for a force, velocity or stress, must retain the properties of such quantities, however we may shift our axes of reference. The neglect of any consideration of this kind at one time led to erroneous physical views, since this similarity in form was connected with the physical quantities themselves, instead of being treated as a mere similarity of the mathematical expressions for their variations. The well known example is the ancient error of considering electricity to be a fluid, because in the measurement of its effects we obtain expressions which look like, and can be spoken of as if they were, those obtained by the motion of a fluid.

There is a mass of material which might be used in illustration of this subject, and I have to choose between the two plans of bringing illustrations from all sources, which might be the more interesting, and of selecting them from one subject so as to form a more continuous chain. As the latter seems the more instructive I have chosen it, and confine myself to questions of displacements and stresses in a continuous medium.

As an introduction, I take a case in which we deal with

co-ordinates only. The method is not quite the same as is developed in the rest of the paper, but it will, I hope, help to explain the more complex portion which follows. Except in connecting expressions on the basis of permanence of form, the paper contains no original results, unless it is in the introductory example.

I. As a first example of the condition of permanency of form of the expressions for a physical quantity in a continuous medium, I take one which is simple-in the sense that it deals with the mode in which the co-ordinates themselves enter the expression considered (whereas at a later stage the arguments will be functions of the co-ordinates)— and which is also definite in its statement, while later I shall consider quantities generically;-namely, the case of a plane polarised wave of light falling perpendicularly on and diffracted by a circular aperture of any size.

Consider the wave to travel in the positive direction along the axis of x, take the centre of the circular aperture as the origin and as axes of y and z two lines at right angles in the plane of the aperture, and let the displacement in the incident wave make an angle a with the axis of y.

Our object is to determine, as far as the permanency of form conditions will allow, the mathematical forms which must be taken to represent the displacement in the secondary wave on the positive side of the plane of yz. Consider the displacement at a point in the secondary wave which corresponds to the component cosa along the axis of y, and write its components

U = f(x, r, 0)cosa

V = p(x, r, 0)cosa

W=(x, r, 0)cosa

where x, and are the cylindrical co-ordinates of the point.

It follows at once that the components of the displace