of decimals, and in part to the high dispersive powers of the prisms employed, which would render it necessary to employ more than two terms in Cauchy's formula to obtain a closer approximation. As the formulæ for interpolation would in this way be rendered extremely complicated, it is better, in the case of any series of observations embracing a particular part of the scale, simply to determine the mean of the errors, and to apply this mean with its proper sign to the computed values of the articular wave-length to be determined by the measurement of ices of refraction. If we apply such a correction in the cases four series of data and results given above, we find for the values of the wave-lengths the following numerical

These

the true values being respectively 656.56 and 438.56. results are, I think, sufficient to show that a valuable control for the accuracy of measurements of wave-lengths may be obtained even when prisms of high dispersive power are employed, provided that the intervals taken are not too large. It seems at least probable that a greater degree of precision is attainable in measuring indices in the case of substances of high than in those of low dispersive power, partly because the angular deviations to be measured are larger, and partly because the spectral lines are less crowded together.

The following example will serve to illustrate the advantage of taking shorter intervals:

The data are here also taken from Van der Willigen's measures

with the same prisms.

When the angular distances between three spectral lines are not too great, the angular deviation of the lines may, as I find, be substituted for the indices of refraction in formulæ (1), (2), and (3). The differences between the angular deviations are, of course, to be converted into seconds. The following results will show the degree of accuracy attainable by this method, the data

being taken from Ditscheiner's* measurements of the indices of a flint prism by Steinheil, of refracting angle 60° 4′ 59′′ :—

From this it appears that the error in the determination of the wave-length of the middle line C is only +0.02 when the angular deviations are employed, but amounts to -0.13 when the indices of refraction are taken as the elements of the calculation. Yet the interval between B and 877 is very large.

The following data are taken from another part of the scale, the measurements being made with the same prism :—

A

1655.6 516.58 49 4 44 516.56 1.62782 516·61 1693.8 514.08 49 6 47

1.62817

Hence the error in the determination of the wave-length of 1655 6 is, when the angular deviations are taken, only -0.02, and when the indices are taken +0.03. It must be borne in mind that in all the above-mentioned examples the angles are those of minimum deviation. As the numbers upon Kirchhoff's scale also represent angular (though not minimum) deviations, it seemed worth while to determine how far for a short interval these could be employed. Taking the three scale-numbers of the last example, the error in the wave-length of 1655-6 was found to be 0.38; and when the scale-numbers were taken as the sines or tangents of corresponding angles, +0·09.

The following data are taken from the more refrangible part of the spectrum, the measurements being also those of Ditscheiner, and made with the same prism :

...

2822-8 433-34 50 34 57 G. 2854.7 430.88 50 37 52 430.68 2969-7 429-90 50 38 47

...

1.64287

1-64334 430.83 1.64352

In this case the error in the wave-length of the middle line. (2854-7) is -0-20, as determined from the angular deviations, and -0.05 as determined from the indices. It must be borne in mind that, in this part of the spectrum, the determination

* Bestimmung der Wellenlängen der Fraunhoferschen Linien des Sonnenspectrums, p. 43.

both of wave-lengths and of indices of refraction is difficult, on account of the feeble intensity of the light.

Since only the differences between the angular deviations of the spectral lines are employed in the formula above given, it follows that, in determining wave-lengths by the method in question, it is not necessary to employ a spectrometer with a divided circle and appliances for the measurement of large angles. A common spectroscope will be sufficient if the observing-telescope be provided with a filar micrometer, by means of which the angular distances of any given line from two other lines of which the wave-lengths are known may be measured. The researches of Angström leave nothing to be desired as regards the wavelengths of standard lines; and the method given may prove a convenient means of determining with all requisite precision the wave-lengths of metallic lines.

XXII. On the Probable Character of Cometary Orbits. By A. S. DAVIS, B.A., Mathematical Master, Leeds Grammar School*.

THE

HE discovery by M. Hoek that certain comets may be so arranged in groups that all the members of the same group have directions nearly coincident and the planes of their orbits a common line of intersection, as well as the discovery by Professor Kirkwood, that there exists a connexion between the aphelion positions of comets and the direction of the sun's motion in space, tends to confirm the opinion that the parabolic comets are not permanent members of the solar system revolving in elliptic orbits of great length, but are casual visitors, coming from the interstellar regions of space, and, after passing through their perihelia, moving off, never perhaps to return again. On the other hand, the fact that the orbits of most of these comets do not sensibly differ from parabolas, has given rise to the opposite opinion that they are really ellipses of great length. No hyperbolic orbit of other than very small excentricity has been met with, contrary to what might be expected on the supposition that comets are non-periodic. For the parabolic character of a comet's orbit shows that its motion at a great distance from the sun is very nearly the same as that of the sun, and so is not independent of the sun's motion, as it might be expected to be on the hypothesis that it is not a permanent member of the solar system. Supposing that the motion of comets at a very great distance is independent of the motion of the sun, their average velocity at a great distance relatively to the sun must be at least as great as the velocity of the sun's motion in space.

* Communicated by the Author.

This follows also from another consideration. The average velocity of comets when at a great distance from any stars, with respect to any one star, must be as great as their average velocity with respect to any other star. As the average velocity of the stars is probably greater than, or at least as great as, the sun's velocity in space, the average velocity of comets at a great distance with respect to the sun must be at least as great as the velocity of the sun in space. This velocity is, according to Struve, 1.6 radius of the earth's orbit per annum; and this velocity at a very great distance gives for a comet whose perihelion distance is equal to the earth's distance from the sun an orbit whose excentricity is 1.06, an excentricity much greater than any which has yet been calculated. But the average velocity at a great distance of those comets which come near enough to the sun to be observed from the earth will not be the same as the average velocity of all comets at a great distance. This is easily seen from the consideration that, of two comets which have exactly the same direction at a great distance, that which has the smaller velocity will also have the smaller perihelion distance. On this account the average velocity at a great distance of those comets which come within a sufficiently small distance of the sun to be observed from the earth will be less than the average velocity of all comets at a great distance. On the other hand, supposing that there are as many comets in space moving with one velocity relatively to the sun as with another (which supposition will within certain limits be approximately true), the number of comets whose velocity at a great distance is V which within a given time come into the sun's sphere of attraction will be proportional to V, and thus the average velocity at a great distance of those comets which within a given time describe an orbit about the sun will on this account be greater than the average velocity of all comets. For these reasons the average excentricity will differ from 1·06.

It is my object in the present paper to calculate what is the probability that a comet which approaches near enough to the sun to be observed from the earth will have an excentricity differing by any given amount from unity, and, by comparing this theoretical probability with the facts derived from observation, to draw conclusions as to the probable character of cometary orbits. Let us suppose a sphere described about the sun with radius R of such a magnitude that the sun's attraction will not sensibly influence the directions and velocities of those comets which are near its surface. Leaving out of consideration for the present the existence of the fixed stars, this will be the case only when R is taken so large that the velocity which a comet has acquired from the sun's attraction in coming to the distance R is so small that it may be

neglected in comparison with the whole velocity of the comet. We will take R so large that this may be the case for all comets, except those whose velocities at R are very small indeed; and we will for the present consider that even these comets with very small velocities are as likely to have one direction as another.

Let us fix our attention upon those comets which enter the sphere through any small area. About the centre of this small area describe a sphere, and let r, 0, be the polar coordinates of any point on this sphere, being the angle which the radius through the point makes with the radius through the point opposite the sun. The probability that a comet will have a direction parallel to any of the radii which pass through the small element of the sphere 2. sin 0. A. Ap is proportional to this area. Hence the probable number of comets having directions parallel to radii through r2 sin 0. A0. Ap which pass within a given time through an area perpendicular to their direction is also proportional to sin 0.▲0. Ap; and therefore the probable number which will pass in a given time through the small area of the sphere about the sun, which area is inclined to their direction at

an angle, is proportional to r2 sin cos AÐ . Ap. 0

Hence the probable number of comets which will pass through the area in a given time and are inclined to the sun's direction at angles lying between 0 and 0 + A is proportional to

r2. sin . cos 0. A0.

We have not yet considered the velocity of the comets which enter. Supposing that there are as many comets with one velocity as another, the probable number of comets which in any time enter any area and have velocities lying between V and V+AV will be proportional to V. AV. Hence the probable number of comets which have velocities between V and V+AV and directions inclined to the sun's direction at angles between 0 and 0+0 which pass in any time into the sphere is proportional to

We proceed now to find the probable number of comets having perihelion distances lying between q and q+Aq, and excentricities lying between e and e+ Ae.

We have the following equations between V, 0, e, and q:

(see Tait and Steele's 'Dynamics,' p. 91), and

Phil. Mag. S. 4. Vol. 40. No. 266. Sept. 1870.

(2)

(3)