it will swing to an equal height on the other side, and continue to oscillate until friction with the air brings it to rest in the vertical position.

At every point of the arc ACZ the force of gravity g acts vertically; resolving it in the direction of the wire

and at right-angles thereto, the component m is neutralized by the point O, and ʼn alone tends to give the pellet motion. The arc in Fig. 386 is greatly enlarged for clearness of illustration, but in reality the demonstration that follows is applicable only to a small arc, which ACZ must be considered, so that a portion of it, as AC, may not differ appreciably from its chord.

In its descent, the pellet gradually increases its velocity until the point C is reached; in its ascent toward Z, the velocity gradually diminishes: at any point B, the velocity is the same that it would be at F, if-unattached to any wire-the pellet should fall freely from A to F, the vertical distance between A and B. By Mechanics, this vertical velocity is

the semi-diameter of the arc through which the pellet

the half amplitude of oscillation. By Geom., the chord is a mean proportional between the diameter and its projection on this diameter: whence

Whence, from (14) and (15), by means of (11), (12), and (13),

the latter becomes

$2

Substituting (17) and (18) in (16), and then this in (10),

2.1

2.1

If x=o, we get the maximum velocity (at C) that is,

If x=s, we obtain the velocity (v''') at the limits of oscil

lation (A and Z), that is,

To find the time of swing, or interval from departure of the pellet at A to its arrival at Z: construct Fig. 387 so

that the line A'C'Z' shall be the same length as the arc ACZ of Fig. 386 when straightened out, that is,

Upon AZ' as a diameter, draw the semi-circle A'UZ'; and along this (A'UZ′), suppose a body to move uniformly with the velocity

the successive projections of this uniform motion upon A'C'Z' will form unequal spaces along this diameter, and the velocity at any point P' will be identical with the velocity for a corresponding length AB of the arc ACB, Fig. 386. To prove this, let P be any position of the moving body, and P' its projection upon A'Z': while the body moves uniformly from P to Q with the velocity v', its projection moves with a variable velocity from P' to Qdenote it by v"; if the spaces and the time required to traverse them be very small, the velocity along P'Q' will be proportional to its space, and we should have

v :v"=PQ:PQ=PQ:PH.

v'

P'

(23)

The triangles PQH and PP'C' are similar, and there

But PC' = A'C' = s, and let PC' =x, whence PP' =√√/s2 — x2,

and introducing the value of (20) in (27), this last becomes

The second members of (19) and (29) are the same, whence v=v, and thus it is proven that the variable velocity along the arc ACZ of Fig. 386 is identical with the variable velocity along the straight line A'C'Z' of Fig. 387.

The time t, then, that the pellet takes to make the oscillation on the arc ACZ, Fig. 386, is the same as that required for the projected motion to traverse the straight line A'C'Z', Fig. 387, and also the same that the body moves with uniform velocity v' over the semicircle A'PUZ'. This

latter time, by Mechanics, is contained in the general equation for uniform motion,

in this case, d is a semi-circumference (A'PUZ'), which, by Geom., is equal to (z.A'C'), or, {by (22)}, (≈.s), where 3.1416. Substituting this value of d and that of v from (20) in (31), this becomes

In this, it will be seen, as stated at the outset, that the period of oscillation depends on two quantities - the weight and form of the moving body represented by land the force g acting upon it.

Theoretically, t is not the same for all arcs of swing, but increases slowly with the semi-arc.

In June, 1898, I made some experiments with the needle used for the ordinary intensity observations on board ship: it is 3 inches long, mounted on a pivot in a circular brass box covered with glass. Friction between cap and pivot as well as (probably) electric currents excited in the material of the box by the needles' motion put a strong check upon its freedom-it came to rest under the influence of the Earth's field alone after making twenty oscillations, beginning with a semi-arc of 15°; and after making thirty oscillations, with a semi-arc of 25°.