Let us now establish in a clearer light the fact that the commensurability of mean motions does not necessarily produce instability. The solution of the problem of three bodies can be reduced to the integration of a system of eight differential equations of the first order; and by a suitable selection of variables, these may be made to take the canonical shape; that is the differential coefficient of each varying element, with respect to the time, will be equal to the positive or the negative of the partial differential coefficient, with respect to the conjugate element, of a function R, analogous to, but not identical with, the disturbing function in perturbations. Thus the eight elements are divided into two classes; four being functions of the mean distances and eccentricities, relate to the dimensions of the two orbits; while the other four, their conjugates, are simply the elementary arguments of the periodic terms contained by Rin its developed form. The selection of these last may be made arbitrarily. If we take one of them, as 0, to coincide with an argument of R, whose mean motion nearly or exactly vanishes, and call the element, conjugate to this, , we shall have the two differential equations

Let us now suppose that R is reduced to its terms which have only in their arguments; then

where B, A, &c., are functions of and the three other elements of its class. As R thus limited does not contain the three elements which are in the class of 0, its partial differential coefficients, with respect to these quantities, vanish. Then the three elements accompaning in its class are constant, and R, as we have limited it, contains no other variables than and 0. Thus if the differential equations determining and are multiplied, the first by do, and the second byd, and the results added, we have an exact differential, which being integrated gives R = a constant, or as it may be written,

CB+ A cos 0 + A' cos 20+ A" cos 30+

....

In order to obtain the values of and 0 in terms of t, we should have to make another integration, but this integral suffices to show whether the element, on which depend the dimensions of the orbits, is confined between finite limits. The value of the constant is readily obtained by substituting in the right member of the equation the values of 0, ℗ and the three other elements of its class, which have place at a determinate

time, as for instance the epoch from which t is counted. Then this equation may be regarded as the polar equation of a curve, upon which the values of and are always found together. Let us suppose that, being taken as the radius and as the angle, the equation is represented by a curve having one or more of such branches as those in the figure. Pis the pole from which is measured, and PQ the line from which is measured. Now

if the values of and 0,

at a determinate time, are found on the closed branch which envelops the pole P, it is plain will always be comprised between certain finite limits. And in this case the mean motion of the argument 0 cannot vanish, as moves through the entire circumference. Here it is always possible to develope ahd in converging infinite series consisting of periodic terms, such as

00 + 91 cos [0 。 (t + c) ] + @2 cos 2 [0。 (t + c)] +

0

0。 (t+c) + 01 sin [0。 (t+c)] + 02 sin 2 [0。 (t+c)] +

2

.....

....

where 90, 91, 92, &c., 00, 01, 02, &c., and c are constants. These are the series of which Delaunay has made such constant use in his Theory of the Motion of the Moon.

But if the values of and at a determinate time, are found upon the closed branch holding the middle place in the figure, will always be contained within finite limits, while 0, its mean motion vanishing, will make oscillations forth and back between definite limits. Hence although the mean motions are here exactly commensurable, no instability results. This case obtains in the three inner satellites of Jupiter, and it also has place in the system of the sun, earth and moon, when the last, as well as the earth, is regarded as a planet circulating about the sun.

In all this we must remember that the values of and the three other elements of its class, both at the origin of time and ever before and after, must be such that they allow the development of R in periodic series to be convergent; else any conclusions derived from this series are not legitimately established.

From this we see that the commensurability or incommensurability of mean motions has no marked connection with stability. This last may be said to depend rather on whether the elements, such as the mean distances and eccentricities, which determine the dimensions of the orbits, have at

a given time such values as make the orbits decidedly elliptic, and permit to them the vibrations caused by the action of the members of the system, without inteference or in other words intersections. There can hardly be a doubt that our solar system as composed of the sun and eight principal planets fulfills these conditions.

SOLUTION OF A PROBLEM.

BY E. W. HYDE, PROF. OF MATH. IN PA. MILITARY ACADEMY, CHESTER, PA.

Locus of a right line moving so that two fixed points of it remain constantly in two other right lines, which are perpendicular to each other but do not intersect.

Let the axis of y and line A B be the directrices.

A line in XY through two points is

x-x1 X1-X2

..(a)

or, (y-y1)(x1-x2)=(x—x1)(y,—Y 2).

For the line c we have

X1

x2 = cc s 4, Y2

=tan 0 =

c cos &

... cos

=

=

sin V1-cos2 = 1

Substituting these values in equation (a)

c2 (ò z — 22)2 — d3 x2 (ò — 2 z) — d2 z2 (x2 + y2) = 0........ ..(b)

the equation of the surface.

........

If we intersect the surface by a plane y= tx we have in the equation only the square of x, hence the curve of section is symmetrical about 2, and this is the axis of the surface. If we move the origin to a point on z half way between the two directrices we obtain a more symmetrical form of the equation.

Intersect by the plane zna, and we have after reduction

4

— (n−1)2 x2 + — (n+1)2 y2 = (n2—1)2, ............................(d)

an ellipse whose semi-axes are

4 22

If n=0, A =B, and the curve is a circle. If no, we have likewise,

hence there is a constant difference between the transverse and conjugate axes. When however, n becomes less than unity, c is the sum of the semi-axes instead of their difference. For when n = 1, B = 0, and the section is a right line, and when n < 1, B is negative, hence the algebraic difference is the sum. If n = 1, A = 0.

The orthographic projection of this surface on any plane parallel to the axis is an hyperbola.

To prove this we shall obtain the equation of the tangent cylinder whose elements are perpendicular to the axis.

The equation of a section perpendicular to z is (from (d) above)

A line tangent to an ellipse is y - m x = √ m2 a2 + b2. But

4 a2 m2 x2- 8 a2 m x y + 4 a2 y2 — c2 (m2 + 1) 22

-2 ac2 (m2 — 1) ≈ — a2 c2 (m2 + 1) = 0. ......(e)

This is the tangent cylinder, and its sections are hyperbolas; as for instance that by the plane y = 0, for which we have

4 a2 m2 x2- c2 (m2 + 1) z2 2 ac2 (m2

1) ≈ — a2 c2 (m2 + 1) = 0.

z

If m = 0 in equation (e) it reduces to 2 aya c ± c z = 0,

MOTION OF A SPHERE ON AN INCLINED PLANE.-BY WALTER SIVERLY, OIL CITY, PA.-Let m = the mass of the sphere, a its radius, k its radius of gyration, ẞ the inclination of the plane, u the coefficient of the dynamical friction between the plane and sphere, R the reaction of the plane on the sphere, s the space passed over by the center of the sphere in the time t, the angle through which it has revolved about its center, and let it be projected down the plain with the velocity v, and at the same time impressed with an angular velocity w. For its motion,