Substituting for V2 in (2) from (3), we have

q2(1+e) — 2Rq sin2 - (e-1)R2 sin2 0=0..

(4)

To a perihelion distance q and an excentricity e correspond a velocity V and a direction at R given by (3) and (4). To q+Aq

and e correspond V+

dV

dq

de

. Ag and 0+ .Aq. To q and e+ Ae

do

dq

Ae and 0+ Ae; and to q+ Aq, e + Ae

dV
de

de

Aq+ . Ae and 0+

If we take V and as the coordinates of a point referred to rectangular coordinates, then AV. A represents the area of a small rectangle whose sides are AV, A0, and the expression (1) may be written

V sin cos x rectangle dV. de.

So if a be any small area about the point whose coordinates are V, 0, the expression V sin cos 0. a is proportional to the number of comets for which the values of 0 and V are represented by points within the small area a. Now all comets which have perihelion distances between q and g+Aq, whilst their excentricities lie between e and e+ Ae, have directions and velocities at a distance R represented by points lying in the parallelogram the coordinates of whose angular points are respectively

Differentiating (3) and (4) with respect to e and q, we find

To simplify the last two expressions, substitute for R sin its value

de

de

q2(1+e) 2g+ Re-1

q2(q-R)

R sin cos 0(2q+Re-1)2

g(e+1)(q+Re-1)

dq=R sin cos 0(2q+Re=1)2'

whence, by substitution and reduction,

and this, taken between the limits O and e, is

lying between q and q+Aq and excentricities lying between 1 and 1+e is proportional to log (1+ q)Aq.

This expression has been obtained on the assumption that at the distance R there are as many comets with one velocity as with another. Taking the average velocity of comets at a distance R to be 1.6, this assumption may be taken as roughly true for all velocities between 1.6 and 0, and for velocities not very much greater than 1.6. We will assume that it is true for all values between 3 and 0, and that there are no comets with velocities greater than 3. The fact that there are many comets which have their perihelion positions nearly in the direction of the sun's motion, shows that there are comets whose independent velocity in space before their attraction to the sun is greater than the velocity of the sun, or greater than 1.6. Hence there are comets whose relative velocities at a great distance with respect to the sun is greater than double the velocity of the sun, or greater than 3.2.

From the formula (3), the value of e for a velocity 3 at the distance R is, when R is very large,. With our present units

9

of time and space μ=42, and therefore e=

9

472

q= 23 xq; and the whole number of comets whose perihelion distances lie

between Չ and q+▲q is proportional to log (1+·23 ×

Ry
2

The proportion of the whole number of comets to the number of comets with excentricities less than e is

Ꭱ

log (1+28x): log(1+).

The greater R is assumed to be, the nearer is this ratio to unity, and consequently the smaller is the average excentricity of all the comets having perihelion distances between q and q + Aq.

Hence the average excentricity of cometary orbits with any given perihelion distance q depends upon the distance at which we assume that the directions and velocities of comets are altogether independent of the direction in which the sun lies with respect to them. The reason for this may be explained thus. Suppose at a distance R the motions of comets are independent of the position of the sun. Describe about the sun a sphere R

with radius R and another with radius

1000'

Of the comets

which enter the larger sphere, those which have a very small velocity will have their directions most deflected towards the sun.

Hence a greater number of comets with small velocities will enter the smaller sphere than of comets with large velocities; and, also, they will not have their directions of motion indiscriminately distributed, but the less the velocity of a comet the less will its probable direction of motion be inclined to the direction of the sun. Now, if there were no other bodies in the universe but the sun and comets, R would have to be taken infinite, and consequently there would be none but parabolic comets.

But the existence of the stars entirely alters the conditions of the problem. If we take the sphere (radius R) to enclose other stars besides the sun, the directions of the comets entering it will be altogether independent of the direction in which the sun lies with respect to them. The same will be the case if R be taken so small as to exclude all the stars; but yet not so small that the attraction of the sun will be so much superior to the attraction of the other stars as to cause the direction of the resultant of all the attractions acting on the comet to lie of necessity nearly in the direction of the sun.

We will assume that the annual parallax of the nearest stars is 1", so that their distance is 2 x 206265, or greater than 400000. We will take three values of R for our calculation. First, R=400000, which will exclude all the stars; secondly, R=200000, which will be about halfway between the sun and the nearest stars; and lastly, R=40000, or about ten times nearer the sun than any star.

The following Table exhibits the results of calculations on these three assumptions :

The assumption of the values 400000 and 200000 for the radius of the sphere which we suppose the comets to enter is open to the objection that at this distance the attraction of the stars will not be so small, compared with the attraction of the sun, as not to disturb the subsequent orbits of the comets which enter it. By taking R=40000, the attractions of the stars may be neglected; but the motions and directions of comets at this dis

tance cannot be supposed entirely independent of the position of the sun.

The truth will lie within the results of the calculation on these three assumptions. We see, then, that a large percentage of those comets which approach the sun in hyperbolic orbits will have orbits with excentricities greater than 1.02. Now out of the large number of comets whose orbits have been carefully calculated not one has an excentricity greater than 1·02; and we are therefore led to the conclusion that though several known comets are probably moving in hyperbolic orbits, yet by far the greater number of those comets whose orbits are undistinguishable from parabolas are moving in elliptic orbits of very great length.

XXIII. Experiments and Observations on the Adhesion between Solids and Liquids. By GIOVANNI LUVINI, Professor of Physics in the Royal Military Academy of Turin*.

M. PLATEAU, in the Eighth Series of his Theoretical and

Experimental Researches on Liquids†, describes the results of a very beautiful experiment from which he deduces some important consequences. In a cylindrical glass vessel, 11 centims. in diameter, he arranged a magnetic needle in the form of a rhomb 10 centims. in length, 7 millims. in width, and 0-3 millim. in thickness, turning in a horizontal plane on an axis coincident with that of the vessel. The needle being moved 90° out of its position of equilibrium and left to itself, returned back with a velocity depending on its length, on the magnetic intensity of the earth and of the needle, and on the passive resistance that the needle had to encounter during its return motion. M. Plateau arranges the experiment in such a manner as to be able to measure with great precision the time occupied by the needle, after it has been turned out of its plane, in traversing the first 85° towards its position of equilibrium. A liquid is poured into the vessel to such a height as to reach to the under surface of the needle, so that this may rest upon the liquid while the upper surface is exposed to the air. Under these conditions we have to determine the time occupied by the needle in traversing the first 85° upon the surface of the liquid. In another form of the

* Translated by Charles Tomlinson, F.R.S., from the Proceedings of the Royal Academy of Sciences of Turin of the 19th of June, 1870.

†Transactions of the Academy of Sciences of Brussels. Presented to the Academy July 4, 1868.