washed with water, dried, digested with pure, hot hydrochloric acid, washed with boiling water, and dried at 100° C. The chloride was then reduced with pure sodium carbonate and wood charcoal, the ingot being treated as before. It was hammered into a rod with a polished steel hammer, on a similar anvil, and drawn through steel and agate draw-plates

and annealed.

دو

Antimony. This was also prepared by Mr. Wilkinson, being reduced by him from pure antimony oxide, and considered by him to be "very pure.' I melted it and cast it into a long rod in an iron mould, and broke off the ends and soldered them on at right angles so as to dip into the oil baths, and also soldered copper wires on to the ends to form the junction. (This was the only case in which I used solder; in all others I merely twisted the wires together.)

Gold. This was an exceedingly pure specimen supplied by the Melbourne branch of the Royal Mint. The percentage of gold being 99.994 according to the most delicate tests that could be made at the Mint. The Mint has also supplied the Physical Laboratory with a specimen of purity 99.998, but I was content to use the inferior of the two. The wire was drawn through three holes of the draw-plate after the last annealing, and may therefore be considered hard-drawn.

Zinc. The zinc I used was supplied in rods by Messrs. Baird and Tatlock as pure redistilled zinc for making Clark cells. I could not draw it, and fused several rods end to end and bent it into the required shape.

Thallium.-This was given me by Mr. Wilkinson, who obtained it from Schuchardt as pure. Professor Masson analyzed it for me and found it contained 97.9 per cent of thallium, 1.5 per cent. lead, and traces of arsenic and copper. Cadmium was also Schuchardt's pure, given me by Mr. Wilkinson. It was in form of a thick rod, which I hammered into a finer one and drew it through steel draw-plates.

Tin.-From same source and treated in same way as

cadmium.

Copper." Pure Swedish Copper," as sold in the city for gold-refining processes. I used it in the hard-drawn state. Aluminium.-Commercial Neuchâtel aluminium.

7

i

XIX. On Ridge-Lines and Lines connected with them. By J. McCOWAN, M.A., D.Sc., University College, Dundee*.

THE HE topography of mountainous regions, districts of upland and valley, was discussed by Cayley, in 1859, in a paper "On Contour and Slope Lines"†, and again by Maxwell, in 1870, in a paper "On Hills and Dales". So far as contour- and slope-lines are immediately concerned the discussion may therefore be regarded as complete; but I desire to define and call attention to certain lines connected with these which I shall discuss in some detail in the following paper, and which are especially characteristic of the general configuration of a region of mountain and valley. These lines I have called ridge-lines, but they are not to be confounded with those to which Cayley gave that name. word "ridge" as he employs it seems in general usage to have given place to the term " watershed," used in its stead by Maxwell; so perhaps I may be permitted to transfer it to the lines I wish to discuss, as being specially descriptive of them. It may be noted that in general there will be a ridge, as I define it very near to, and in some cases coincident with, the particular line of slope to which Cayley gave the name.

§ 1. Contour-Lines and Lines of Slope.

The

Consider the configuration of a surface, of any form, relatively to a plane fixed with respect to it. This plane will be called the base and will be regarded as horizontal, so that planes parallel and perpendicular to it may be described as horizontal and vertical planes respectively. The surface may be, for example, that of any portion of land, and the base the sea-level, provided that the part considered is not so large as to require the curvature of the earth's surface to be taken into account.

:

The curves in which the surface is intersected by horizontal planes are called contour-lines, or simply contours. Lines on the surface which cut the contours orthogonally are called lines of slope the inclination at any point of a line of slope to the base is therefore equal to the inclination to the base of the tangent plane at the same point and is a measure of the slope there. Points at a maximum or minimum height above the base at which horizontal tangent planes touch the surface are

* Communicated by the Author. Read before the Edinburgh Mathematical Society, December 8, 1893.

+ Phil. Mag. [4] vol. xviii.; or 'Collected Papers,' vol. iv.

Phil. Mag. [4] vol. xl. ; or Collected Papers,' vol. ii.

called summits and immits respectively: all lines of slope run from summits to immits, and obviously cannot cross each other at any intermediate point, though a line may branch at any point where there is a horizontal tangent plane. At summits and immits the surface is synclastic and the contours are reduced to points; but a horizontal plane may touch the surface at a point, called a col, where the surface is anticlastic, and where consequently two branches of a contour-line will in general intersect, the point of contact being a double point of the curve of section. Other varieties of contact need not be specially noted here as their occurrence is very exceptional. In the case of a land surface the summits are the tops of the hills, and the immits the bottoms of lakes or basins, while the cols are such points as the heads of passes between hills or places where streams flow out from lakes. The two lines of slope from a col to the two adjacent summits form together what is called a watershed, and the line of slope from a col to an immit is called a watercourse*.

§ 2. Definition of Ridge-Lines.

The section of a surface by a plane parallel and very near to the tangent plane at any point, or, as it is generally called, the indicatrix for that point, is in general a conic. The directions of the axes of this conic, depending only on the form of the surface and not at all on its relation to the base, are independent of the directions of the slope- and contourlines, and therefore the surface in the immediate neighbourhood of any point is in general unsymmetrical about the line of slope through the point. There are, however, points on the surface at which the directions of the principal axes of the indicatrix coincide with the directions of the slope- and contour-lines, and the locus of such points is a line which I propose to call a ridge-line. The ridge-line, then, is defined by this property:-A surface, in the immediate neighbourhood of any point on a ridge-line, is symmetrical about the line of slope through that point.

The topographical significance of the line is to be noted. The features of a hilly country which are, perhaps, most characteristic are the ridges and spurs of the hills and the glens or valleys between them. The general trend of these features is given by the ridge-lines, each ridge-line being, so to speak, the central or axial line, the line of (infinitesimal) symmetry of a ridge or valley-the valleys may be regarded as negative

For more detailed treatment of the matters of this section reference may be made to the papers of Cayley and Maxwell, loc. cit. ante.

or inverted ridges. The ridge-lines run down the bottoms of the glens or along the tops of the ridges and spurs of the hills, or again down the lines where the descent is most precipitous.

Many of the most interesting properties of the ridge-lines may be derived almost immediately from the definition. For instance, it may be noted that wherever a line of slope or contour is crossed by a ridge-line, the lines of slope or contour touch the lines of curvature of the surface, for the directions of principal curvature are given by the axes of the indicatrix. It is most convenient, however, to proceed at once to the equation to the ridge-lines from which all such properties almost immediately follow.

§ 3. The Equation to the Ridge-Lines.

Take the base as the plane z=0, z being measured vertically upwards. Let §, n, be the coordinates of a point on the surface near to the point x, y, z also on the surface. Suppose, for the present, the surface to be determined by the equation

Then, employing the usual notation for the partial differential coefficients of, the surface may, by Taylor's Theorem, be represented in the immediate neighbourhood of the point x, y, z by

?=z+p§+qn+}{r§2+2s&n+tn2} + &c.

(2)

The indicatrix for the point x, y, z, that is, the section of the surface by the plane, parallel and very near to the tangent plane,

will therefore be determined by

25' =r¿2+2s&n+tn2,

(3)

(4)

which is the equation to the projection of the indicatrix on the base. It will be convenient in future to use the word "projection" simply, whenever a projection on the base is to be understood.

At any point on a ridge-line the line of slope touches a principal axis of the indicatrix: the projections of these lines therefore touch also; and so, if o be the angle the projection of that axis of the indicatrix which touches the line of slope makes with the axis of x,

Phil. Mag. S. 5. Vol. 37. No. 225. Feb. 1894.

(5)

Ꭱ

The directions of the axis of the conic (4) are, however, given by

(cos2-sin2 )s=cos &. sin &. (r−t) ;

and on eliminating & by (5) this gives

(p2-q2)s=pq(r−t);

(6)

(7)

which must hold at all points on a ridge-line, and is therefore the equation to its projection. Equation (7) may be regarded as that of a cylinder whose intersection with the surface, given by (1), is the ridge-line. It will, however, be convenient to regard parts of the general locus as separate ridge-lines; and in general each of the branches which passes through a multiple point, such as a summit or immit (v. infra), will be spoken of as a separate ridge or ridge-line.

It may be remarked that if the surface is given by an equation such as (1) of the nth degree, the equation (7) to the projections of the ridge-lines is of degree 3n—4.

It is interesting to note, and the fact constitutes a not unimportant claim of the ridge-lines to attention, that whereas in many cases lines of slope and contour-lines cannot be found at all, being determined by differential equations which are in general not integrable in finite terms, the ridge-lines can always be deduced directly from the equation to the surface by mere differentiation.

§ 4. The Relation of the Summits, Immits, &c. to the

Ridge-Lines.

The equation (7) which determines the ridge-lines is satisfied by p=0, 9=0, and therefore-as is also geometrically obvious-all points at which the tangent plane is horizontal, that is to say, all summits, immits, cols, &c. lie on the ridge-lines: a result in accordance with the popular notion of a ridge, namely, a line running along the summits of a range of hills.

At such points equation (2), which gives the form of the adjacent surface, reduces, on neglecting terms of the third degree, to

(=z+1{rç2+2s&n +tn2};

and therefore by (7), the projections of the ridge-lines are given by

{(r§+sn)2 — (s§+tn)2}s = (r§+sn) (sĘ +tn) (r−t), which reduces to