the complete solution of which is of the form.

x= = A cos (m1t + a) + B cos (m ̧t + ß). ..... (c),

where A, α, B, ẞ are arbitrary constants, and m2, m2 are the two roots, with signs changed, of the equation

From the above value of x that of y may be obtained by means of the formula

11. In problems connected with central forces particular forms of the following system of equations present themselves, viz.

d2x dR d'y dR d2z dR

=

=

(a),

where R is a given function of the quantity (x2 + y2+23)

or r. Multiplying the above equations by dx, dy, dz respectively, and integrating, we have

of which it is evident that two only are independent. Inte

grating these, we have

Squaring the last three equations and adding, we obtain a result which may be expressed in the form

or, by virtue of (b) and of the known value of r,

.(c),

(d),

whence

dt

=

rdr

√ {2r2 (R + B) — A2}

Again, it is evident that by means of (c) we can eliminate R from each equation of the system (a). For (c) gives

Substituting which in the first of the given equations, we

in which we must substitute for its value, viz.

·(g),

...

.(h),

To this expression it would be superfluous to annex an arbitrary constant before that substitution. For each of the second members of (ƒ), (g), (h) is expressible in the form C cos (+ C'), in which is already provided with an arbitrary constant.

The solution is therefore expressed by means of (e) and (i), which determine r and the auxiliary as functions of t, and by (f), (g), (h), which then enable us to express x, y, z as functions of t. As we have however made no attempt to preserve independence in the series of results, the constants will not be independent. If we add the squares of (f), (g), (h), we shall have

2

2

1 = (a ̧2 + a ̧2 + a ̧3) cos2 + 2 (a,b, +ɑb2+ɑ ̧b ̧) sin & cos

The six constants in (f), (g), (h), thus limited, supply the place of only three arbitrary constants, and there being three also involved in (e), the total number is six, as it ought to be. In the same way we may integrate the more general system

2

2

where R is a function of √(x+x...+x). The results, which have no application in our astronomy, are of the form which the above analysis would suggest. Binet, to whom the method is due, has applied it to the problem of elliptic motion. (Liouville, Tom. II. p. 457.) For all practical ends the employment of polar co-ordinates, as explained in treatises on dynamics, is to be preferred.

12. The following example presents itself in a discussion by M. Liouville*, of a very interesting case of the problem of three bodies.

* Sur un cas particulier du Problème des trois corps. Journal de Mathématiques, Tom. 1. 2nd series, p. 248.

where, for brevity, x' is put for cos (at +b), y' for sin (at + b).

If we transform the above equation by assuming

ux' + vy' = U, uy' - vx' = V,

we find, after all reductions are effected,

And these equations being linear and with constant coefficients, may be integrated by the process of the previous section.

dx dy

dx dy

dt

dt

3.

4 +9 +44x+49y=t, 3 +7+34x+38y=e'.

dt dt