For this purpose the series of curves shown in fig. 3 were obtained by Mr. Wilson in the following manner:-To the head from which the fibre is suspended he attached a light arm movable between two stops through an angle of about 5°. When the instrument was in equilibrium he then moved the arm rapidly up to one of the stops and back, thus giving the couple an impulse, in consequence of which it continued to oscillate for some twenty seconds or so, while the spot of light traced out the left-hand curve in fig. 3. When equilibrium had been restored he moved the arm in a similar manner up to the other stop and set the couple swinging in the opposite direction, and thus obtained the second curve in the same figure. In this way, by giving alternately right-handed and left-handed impulses to the couple, the five curves in fig. 3 were obtained, from which it is possible to derive the constants a and b.

Now, if p, and 4, are the first two maximum values of the ordinate, and 2 the first minimum value during the free motion of the couple after an impulse has been imparted to it as above described, and if is the final value of when the position of equilibrium has been attained (which will be of the nature of an index correction to the readings of 4), we have clearly

Reading the five curves in fig. 3, we obtain the following values of 1, 2, and 3, expressed in millimetres :

The mean of these five determinations gives us 0.3534 as the

We have also determined half the period of a free oscillation from each of the five curves separately with the following results :

:

We have thus obtained, from these five curves representing the free motion of the system, the following values:

In order to determine the value of the quantity e, the instrument was suddenly exposed to the radiation from a Leslie's cube filled with boiling water until it came to rest, and then the heat was as suddenly cut off, the instant at which the exposure began and ended being indicated by a flash of light which traced a line on the falling plate. The result of this procedure is shown in fig. 4.

1 The curves in figures 3, 4, 5, and 7 are reduced copies of the original photographs.

Now, equation (5) shows that when the instrument is exposed to

کم

FIG. 4.

a constant source of heat the motion is represented by the equation

When t = ∞, or when thermal equilibrium has been established, if represent the final value of the ordinate, K.

=

Hence if 0, 1, P2, &c., denote the values of the ordinates at the

Also, for the motion, when the heat is cut off we shall have an exactly similar series of equations, except that will represent the final or zero reading of the ordinate.

In the case before us the values of are so small, in consequence of the rapid damping down of the oscillations, that we had better restrict ourselves to the first four of these equations. If we denote

by x, and e 8 by y,

and if we eliminate ▲ cos e and C, we obtain from each of the curves

the two equations

4042-412

From the four curves in fig. 4 we find the following values of 4,41, &c., with the resulting values of x

y and xy:-
:-

41

The resulting values of x and y from each of the four curves are,

accordingly,

Mean,

These figures seem to show two things:-Firstly, that the damping effect is less when the instrument is heated than when cold, since the value of x just found is greater than that found from the curves of

free oscillation.

Secondly, it may be noticed that the values of both x and y are greater as derived from the 1st and 3rd curves than those deduced from the other two curves. The materials are, however, hardly sufficient to decide, whether this represents a real difference depending on the direction of the motion or is merely an accidental discrepancy.

ап

We have thus the following results, taking fore the value found from the heat curves (the system being then in circumstances

more closely analogous to those which take place during a transit of the sun than when cold):

8 = 0·9838 or 56°.37,

a = 0.2558,

7 = 0·4054;

From these we find

whence b2 = 1.0333.

We thus find the coefficients of the differential equation on p. 303 :—

2a + 1 = 0·9170,

2al + b2 = 1.2407,

bal = 0.4189.

If the curve were a smooth curve without points of inflexion, it