(11.)

k = A1 R1 + A2 — α1,

m = — B1 R1 + B2 — b1,

n = C1 R1 + C2—C1.

Then the first of equations (C') becomes

(§ — R1)2 —

k2 + m2 + n2

2 (k A1 + m B1 + n C1) (p — R1) = 1— (4‚2 + B‚2 + C'2) *

1 − (A‚ ̧3 + B2 + C ̧2)

Thus the problem is solved analytically by means of equations not transcending the second degree.

The number of cases of a sphere touching four given spheres may be determined as follows:

The required sphere may touch the four given spheres all externally, 1 case; or all internally, 1 case; or two internally, two externally, 6 cases; or three externally, one internally, 4 cases; or three internally, one externally, 4 cases; making sixteen cases, as found by Major ALVORD, all included in equations (C').

=

I must not omit to notice what would otherwise seem to be a defect in the preceding solutions. In both problems there seem to be exceptional cases to which the formulæ are inapplicable. For example, in the tangencies of circles, if b1 = b2 = bg, then it appears that A1 A2 =, and it is impossible to determine o̟, a, ß, by the proposed method. The reason of this is obvious; for the assumed equations (10), in the first problem, were only employed because the two equations (11) contained three unknown quantities, a, ß, g, and were, therefore, indeterminate. But when b1 = b2 = b2,

0'

one of the unknown quantities, ẞ, disappears from the equations (11); hence the assumed equations (10) are inapplicable, and, at the same time, they are not wanted; for a and g are then determined by the two equations (11), each of the first degree, and then ẞ is found from either of the equations (A') of the second degree. The same considerations apply to the corresponding case of the second general problem, when, for example, c1 = c2 = C3 C4, in which case P1 = 0, Q. Qa = 0; P2 P1 = 1, Q hence A1, B1, A2, and B2

a

Ꭴ

0

=

0

are indeterminate. But under these conditions the unknown quantity y disappears from the three equations of the first degree obtained in the early part of the process; hence the remaining quantities, a, ß, g, are easily found, and thence y from one of the equations (C), of the second degree. Thus in every case the problems are brought within the resolution of equations not surpassing the second degree. I might now give various applications of the second problem which I have computed, but omit them for fear of occupying too much space in the Monthly unnecessarily.

INSTANCES OF NEARLY COMMENSURABLE PERIODS IN THE SOLAR SYSTEM.

BY DANIEL KIRKWOOD, Professor of Mathematics, Indiana University.

THE following instances of nearly commensurable periods in the solar system have not, I believe, been previously noticed:

For the 3d and 4th satellites of Jupiter we have

Neptune = 13

Remarks. The periods used in (a) are taken from HERSCHEL'S Outlines; those in (b) and (c) from the Proceedings of the American Association for the Advancement of Science, 1854, p. 55. The periods of the asteroids are taken from the Mathematical Monthly for February, 1859. The period of TUTTLE'S Comet is adopted from GOULD'S Astronomical Journal, No. 118. That of HALLEY'S is the mean of the six periods from 1378 to 1835. The epochs at which Neptune must have great perturbing influence on the motion of HALLEY'S Comet are separated by intervals of about 988 years. The last occurred in the early part of the fifteenth century; the next will be about the beginning of the twenty-fifth. It may also be remarked that the planets Jupiter and Uranus present a case of almost exact commensurability. Eighty-five times the period of the former, minus twelve times that of the latter, forms a quantity which amounts to only of the period of Uranus. Hence in about 1008 years from the epoch of any conjunction of these planets, another conjunction will occur only twenty-three minutes from the place of the former.

19. If <q=6=0°,

β 0°, or=

α

180°, then ẞ has either the same direction as a or the opposite, and q degenerates into either a positive or negative real number, and is then called a Scalar. Like tensors, scalars may be applied to any line in space without regard to its direction; and like them, they are commutative in combination with any quaternion. The product of scalars is evidently a scalar; and the conjugate of a scalar does not differ from the scalar itself.

20. If q=90°, then q is called a Vector. Vectors will be denoted by v, v', &c. It is evident from (4) and (3′) that

that is, the square of a vector is a negative scalar. The square root of (17), v = m √ — 1, may be regarded as the expression for a vector of indeterminate axis, whose tensor is m. When m= 1, v = √ — 1

is called a unit vector. √—1, may be used to denote this vector, when its axis is fixed in the direction of Ax. q. The geometrical signification then of V-1 in this system, is the operation of turning any line to which it may be applied through an angle of 90° around any axis perpendicular to that line. Any quaternion may be expressed as a power of a vector, since, by (11), if, m being a positive number,

(18) q = [m √ — 1,]",

then Tq=m",

<q=n 90°,

and the independent real quantities m and n are sufficient to determine Tq and <q.

γ

21. The lines a and ẞ being given, let the directions of the lines

and & be respectively parallel and perpendicular to that of a, and

let their lengths be such that y+B. These conditions completely determine y and 8, both of which will be in the plane of a and B. Then, by § 8,

9 B÷α = (y + d) ÷ a = y ÷ a + 8 ÷ a;

but ya is a scalar, and 8÷a is a vector. Any given quaternion then may be resolved into a sum of two other determinate quaternions, one of which is a scalar, and the other a vector; in this case the former is called the scalar of the given quaternion, and the latter its vector. The scalar of a quaternion, q, is written Sq, and its vector Vq; thus

The sum and difference of (19) and (19 a) gives

If two or more quaternions are equal, their scalars must be equal, and also their vectors; and conversely if their scalars and vectors are equal each to each, the quaternions must be equal.

22. Let the planes of the two vectors, v and v', intersect in the line a; and let v a= ẞ, and v' a = = B'; then both ẞ and B' will be in the plane of Ax. v and Ax. v, and, if y = ẞ+ ß', (v + v') = γ + α. But ya is a vector, whose axis, in the plane of Ax. v and Ax. v, makes the same angles with these axes that the line y does with the lines ẞ and B' respectively, and whose tensor bears the same ratios to Tv and Tv' that Ty does to Tẞ and Tẞ' respectively. If then we set off on Ax.v and Ax. v lines whose tensors are equal 17

VOL. II.