Where c' is an arbitrary constant and 2 indicates summation with regard to n for all integral values from 1 to infinity.

Where am cn and A are arbitrary constants and Sindicates summation with regard to m for a finite number of values of m, the largest of which is supposed to be small compared with p.

Since the velocities must be continuous on each side of AF, we must have along AF.

S(amcos my) = a − A + Σ(c„' − c„cos pny
-S(amsin my) + jay = Σ(cn' + cn )sin pny

}

(3)

These must hold from yo to у = πр. But if we expand the left-hand sides by Fourier's Theorem we get c, and c,' as functions of n. Since the left-hand side of the second equation of (3) must vanish when y = p, this furnishes one relation among the constants. We can then show that c, and c,' are small quantities at most of the order 1/p2.

It is easy to form from (2) the equation to the outer stream-line BEF. If the vessel be of finite breadth at infinity, A will be a small quantity of the order 1p.

=

Looking now at (2), we see that if OE be the surface of the liquid, u and v must when a -1 be small quantities at most of the order 1p. A and the term already satisfy that condition. In the S term m has several values. Suppose the particular m in (2) to be the smallest of these values, and suppose m = log p, then when a -1 the S term also satisfies the surface condition, and the more accurately the larger p is, since log p/p diminishes as p increases.

If FJ is a jet we must have, since AF is small, at every point of the jet, nearly

u2 + v2 = 2gr.

But we see at once from (1) that this condition is nearly fulfilled, the error being of the order 1/p2.

As a particular case, if we give to m in (2) the two values 8 and 9, p will be about 2980 and the maximum error (which will be at F) will be about +0000143.

If p be large enough, we can by taking a sufficient number of values of m, make the error of the order 1/p3 &c.

Roughly speaking, when the orifice is small compared with the depth of the liquid, the shape of the jet depends only on the orifice, being almost entirely independent of the shape of the vessel.

9. The Geometry of Confocal Conics. By Professor T. C. LEWIS.

1. If PP', QQ' be chords of contact of T in two confocal conics, then a conic can be described with Q, Q' as foci which shall touch the conic PP' at P and P'.

2. By taking T on the outer of the two ellipses, it follows that the ellipse with foci at Q, Q' which passes through T will have contact of the third order with the confocal through T; and the hyperbola with foci at Q, Q' which passes through T will have contact of the third order with the hyperbola whose foci are S and H which passes through T.

3. În fig. 3 PQ, PQ', P'Q, P'Q' are all tangents to one ellipse confocal with that through Q, Q'. Let this be 71 92 93 94 (fig. 3). The four tangents to an ellipse intersect in three pairs of points. If each of one pair is on a confocal conic, so also is each of the others. Q, Q' lie on one confocal; T, T' lie on another confocal. 4. As in 1, a conic with P, P' as foci can be drawn touching the confocal TT' at T and T'.. PT-TP' = PT' - T'P'.

.. a circle can be inscribed in the quadrilateral TPT'P'.

5. As a particular case, when P and Q coincide the two tangents from T meet the tangent at L in Q, Q' (fig. 4) which are on a confocal, and the circle inscribed in TQQ touches the ellipse at L.

6. In fig. 3, QP+ PQ'=QP' + P'Q'. Add to each Qq, + arc 9,94 +9Q'; and let PP' be consecutive points; Q being the intersection of consecutive tangents lies on the inner ellipse. Hence the string which must be placed round the inner ellipse to stretch to P is equal to the one which must be placed round it to stretch to P'.

Hence mode of describing confocal ellipses by placing strings round an ellipse and keeping them stretched by a pencil point.

And similarly if an endless string be placed round an ellipse whose circumference is shorter than the string, and if the point M' be fixed, and a loop of the string be passed through a small ring at L and pulled tight, then if a pencil be put through the ring and moved steadily away from the ellipse, a confocal hyperbola will be described (fig. 5).

Various other cases arise.

7. As long as T is on a confocal ellipse PT-PL is constant.

Also the tangents from T to the circle inscribed in TQQ' are of constant length. If the tangent at L meet the confocal ellipse through T in T and T", the confocal hyperbolas through T', T" will pass through P, P' respectively and T' T” -arc PP' is constant.

Eight equal tangents can be drawn from the outer to the inner of two confocal ellipses.

8. If in fig. 5 the tangent be drawn at M, M' or L' to meet the tangent from Tin Q and Q', then the point of contact of this tangent will be the point at which either an inscribed or an escribed circle of the triangle TQQ' will touch the ellipse.

9. If a triangle be drawn with each side touching an ellipse, then an infinite number of triangles of equal perimeter can be drawn whose sides touch the ellipse; their three angular points lying one on each of three confocal ellipses.

10. If the three confocals which are the loci of the angular points coincide, the triangles are inscribed in one and circumscribed about another of two confocal ellipses.

In this case their perimeter is a maximum for all triangles inscribed in the outer ellipse, and a minimum for all triangles circumscribing the inner ellipse. 11. So for a polygon of any number of sides.

Also each tangent is divided into two sections at the point of contact. If the polygon has n sides there are 2n sections; starting with any one of them and

numbering the sections in order from 1 to n, the remaining sections are equal in magnitude to those already passed, occurring in the same order. The sections of the sides of the triangle in fig. 6 with the same number attached are equal to one another.

10. Some Tangential Transformations, including Laguerre's Semi-Droites Réciproques. By Professor R. W. GENESE, M.A.

The equation to a straight line cutting off a length a from the positive axis of r and making an angle cot-'m with that axis, being x - a = my, a relation of the form

(1)

makes the line a tangent to the conic

Ay2+B+Ca2-2Fx-2Gxy + 2Hy = 0

・ ・ (2)

If now a second line x-a=m'y be obtained from the first by means of the relation

pm2 + 2r'mm' + qm22 + 2q′m + 2p'm'+r=0.

(3)

the envelope of this line will in general be of the fourth class. If, however, minors of the discriminants of (1) and (3) be connected by the relations

A CG
PRQ'

then the envelope degenerates into two of the second class.

(4)

the

The conditions (4) were obtained thus:-Considering a, m',m as the coordinates of a point referred to three orthogonal planes, (1) and (3) represent two cylinders, and (4) gives the conditions that these should have two common plane sections. The conditions may be geometrically interpreted thus; if two conic cylinders lie between two parallel planes (i.e. each cylinder touch both planes) their complete intersection consists of two conics.

here

Laguerre's transformation, analytically considered, gives the relation

m2 + 2kmm' + m22 = k2 −1

P = 1 − k2, R = 1 − k2, Q' = 0.

(3')

Then by (4), AC and G = 0, and the transformation is seen to be of simple use only for the case of circles. M. Laguerre's results, however, of which an account is given in his Géométrie de Direction,' are of exceptional elegance.

11. Note on the Normal to a Conic. By R. H. PINKERTON.

12. On the Importance of the Conception of Direction in Natural Philosophy. By E. T. DIXON.

This importance has already been recognised in the higher branches of science in the guise of Vector theories, and the chief reason it has not been made use of in elementary geometry is the want of a proper definition. Such proper definition might be deduced from the conception of direction as a relation between two positions which is independent of the distance between them and of the absolute position of either of them in space. The concept thus defined is independent of the conception of a straight line, and so may be used to define it, and is therefore distinct from the concept defined, by saying that two straight lines which have a common point have the same or different directions according as they coincide or not. That some notion of direction is necessary to elementary geometry is shown by the fact that without it right- and left-handed figures which are equal in every respect cannot be distinguished; and that the concept as defined is commonly entertained, is proved by the fact that it follows from Newton's Laws of Motion that absolutely fixed directions may be conceived in space, although absolutely fixed positions cannot.1

1 Vide The Foundations of Geometry (Deighton, Bell & Co.)

1. Report of the Committee on Researches on the Ultra-Violet Rays of the Solar Spectrum.-See Reports, p. 147.

2. Comparison of Eye and Hand Registration of Lines in the Violet and Ultra-Violet of the Solar Spectrum, against Photographic Records of the same, with the same Instrument, after a lapse of several years. By C. PIAZZI SMYTH, LL.D., F.R.S.E.

A comparison of the plates seems to lead to such practically useful conclusions as the following::

1. Two photographic representations are far more trustworthy than three or probably a much greater number of hand-drawn views of solar-spectrum lines, even when the eye imagines it sees them very clearly.

2. The photographic principle records with ease, and the utmost vigour of black, white and grey of various shades, a world of objects in certain spectral regions where the eye can see nothing whatever.

3. What photography depicts in such cases is what the human eye ought to see, and would see were it divinely perfect simply as an eye.

4. The ordinary spots, pin-holes and dust-marks, which too often abound in photography, never assume such shapes as might lead to their being mistaken by any experienced observer for a single one of Nature's solar-spectrum lines of light or shade.

5. The frequent errors, and then ail-pervading effects, fallen into by some photographers in the way of over or under exposure, and over or under development, may prevent the absolute intensity of any one line, on one plate alone, being usefully quoted as a scientific datum. But the relative intensities, and innumerable distinctions in hue and shape, of thick or thin, dark or light, closely arrayed or widely scattered, flutings gradating towards the violet or towards the red end, and regularly or irregularly spaced lines on the same plate, are full of most important and instructive particulars. While they mostly hold good also, from plate to plate of the same parts of spectral space, on all the proofs that may be taken both day after day, and through a very wide range of all the possibilities of perversion and misuse which may be humanly committed upon this most exquisite aid, viz. photography, to the noblest of the senses of man, vision.

6. In principle, all this has long been known to advanced workers in every civilised nation. But as it is not everywhere yet utilised to the extent it might well be, it is hoped that this further and rather multitudinous example, on an extreme scale too, of spectral separation, and capable of showing such a Titanic instance of a dark thunder-cloud-looking column as Great K, by pure photography only eight months ago, in a solar-spectrum telescopic field which was at the time absolute emptiness to the eye, may be useful in calling increased attention to similar and more extensive employments of photography in the future.

3. Note on Observing the Rotation of the Sun with the Spectroscope. By G. JOHNSTONE STONEY, M.A., D.Sc., F.R.S.

In this note the author described an arrangement for conspicuously exhibiting to the eye the rotation of the sun by the spectroscope. The sun's light, after reflection from the mirror of a heliostat, is received by a telescope lens which forms an image of the sun on the slit of the spectroscope. The lens is attached to a vertical board, and two screws are partly screwed side by side into the board and at some distance above the lens. The projecting heads of the screws rest on a