when λ< λ'.

Making λ= ∞o the space becomes that bounded by the cone -a and the sphere ra; and the potential inside an uninsulated hollow conductor of this form under the influence of a charge q at the point rae on the axis is given by

when r>r. To obtain the potential in Mehler's case when the cone extends to infinity put a = ∞ and then

when r>r', where the summations extend to all the positive values of n which make Pn (cos a) vanish. When a = π/2 α=π/2

when r'>r, where the summation extends to all the positive odd integers, that is

which agrees as it ought to with the expression for the potential due to a charge q at a point distant from an infinite conducting plane at potential zero.

(2) To find the potential at any point due to the spindle formed by the revolution of a segment of a circle about its chord, when its surface is freely charged.

This is immediately obtained by inversion from the above case. Let & be the angle in the segment of the circle whose revolution describes the spindle, the angle in any other segment of a circle on the same chord, log, where r1, 1⁄2 (~1>r1) are the distances of a point on a segment from the extremities of the chord; then putting q=-Vor and observing that the cone of angle in the dielectric inverts into the spindle the generating segment of which contains an angle 。, the potential at any point due to the spindle when charged to potential V, is given by

where = cos

Ho

0

and the summation extends to all the positive values of n which make Pn (cos) vanish. The case of the sphere is that when =π/2. It may be verified that the density of the distribution on the spindle near one of the conical points agrees with that found § 1. For the density at any point on it is given by

where r' is the length of the axis

of the spindle, r the distance of the point on the

surface from the conical point and n is the least zero of Pn (cos §). Now when 。, equal and the values of n which occur are k+no, where

toy, is nearly π, is

sin nπ k is any positive integer and 2n, log

ӘР JP n дп

* Loc. cit., Proc. Lond. Math. Soc. 1899.

VOL. XVIII.

38

XIV. On
On the Effects of Dilution, Temperature, and other circumstances,
on the Absorption Spectra of Solutions of Didymium and Erbium Salts.
By G. D. LIVEING, M.A., Professor of Chemistry.

[Received 15 October 1899.]

IN November 1898 I made a preliminary communication to the Society giving results of observations on the absorption spectra of aqueous solutions of salts of didymium and erbium in various degrees of dilution. Since then most of the observations have been repeated with improved apparatus, whereby several anomalies in the photographs have been removed, and a great many additional observations made, so that it will probably be best to make this communication quite independent of the preliminary one, and, at the risk of a little repetition, complete in itself so far as it goes.

APPARATUS.

The observations were made in part directly by the eye with an ordinary spectroscope, and partly by photography. On the former I rely only for the part of the spectrum below the indigo, on the latter for the more refrangible part. The spectroscope chiefly used for the former had two whole prisms of 60° and two half-prisms, all of white flint glass, telescopes with achromatic object glasses of 12 inches focal length, and eye-piece of very low magnifying power. It was useless to employ higher dispersion or magnification, because the absorption bands, even the sharpest of them which is that of didymium at about 427, are all diffuse, and higher dispersion or magnification renders some details invisible. In comparing by eye the spectra produced by two solutions, one was thrown in by reflexion in the usual way, and, after making the comparison, the positions of the solutions were interchanged and the observation repeated, in order to correct any error arising from a difference of intensity between the light entering directly and that coming in by reflexion.

For photography the spectrum was formed by one prism of 60° and two halfprisms, all of calcite, the object glasses of the telescopes were quartz lenses of 18.5 inches focal length for the sodium yellow light. The photographic plate was of course

inclined to the axis of the telescope so that, as far as the doubly refracting character of the calcite prisms allows, the image might be in tolerably good focus across the whole width of the plate, two and a half inches.

To concentrate the light, and make it, for the parts of the spectrum not subject to absorption, nearly uniform whatever the thickness of the absorbent stratum of liquid, a quartz lens of three inches focal length was fixed at that distance in front of the slit, and a similar lens fifteen inches further off, and three inches beyond the second lens was fixed a screen with a circular hole in it about one-eighth of an inch in diameter, and beyond that was of course the source of light. The centres of the hole in the screen and of the two lenses were aligned with the axis of the collimator. The distance between the lenses was fixed so as to allow of the interposition of the longest trough, used as a water bath for maintaining the temperature of the tubes containing the solutions. These troughs were of brass fitted with a plate of quartz at each end, and each had in it two V-shaped septa on which the tube with solution rested, and thereby took up at once its right position in the course of the pencil of light between the lenses. The tubes holding the solutions were of glass, fitted at the ends with quartz plates. These plates were held in position by outer brass plates with central circular perforations, connected by three wires passing along the outside of the tube and furnished with screw nuts by which the plates could be firmly pressed against the ends of the tube. The joint between the quartz plate and the end of the tube was made water-tight by a washer of thin rubber. The washers all had the same sized circular opening which determined the cross section of the pencil of rays falling on the slit. This seemingly complicated arrangement was adopted because it was necessary to have joints which would not be affected by a temperature of 100°, or by dilute acids, or by alcohol, and could be easily taken to pieces for cleaning the tube or plates.

Each tube had a branch on its upper side which was left open for the purpose of filling the tube, and to allow of expansion of the liquid when it was heated. Tubes of four lengths in geometrical progression, namely of 38 mm., 76 mm., 152.5 mm., and 305 mm., and a cell with quartz faces having an interval of 67 mm. between them, were used to hold the solutions; and for a few observations a cell of only 5 mm. thickness was used.

For observations on the effects of temperature, the trough containing the tube with solution was filled with water and a photograph of the spectrum taken at the temperature of the room; the trough was then heated by one or more gas lamps until the water boiled, the gas lamps were then lowered so as to maintain the bath 3 or 4 degrees below the boiling point, bubbles adhering to the quartz plates swept off with a feather, and when the whole appeared to be in a steady condition another photograph was taken. Unless the solution in the tube were a very dilute one there was not much trouble with bubbles in the solution, but bubbles in the bath were very troublesome, and had to be removed because they impeded the passage of the light, and thereby