v2 known. This velocity may be represented by k. Then rƒ(ø) For the value of ƒ (9), we consider the velocity to be that ka which a body would acquire in falling vertically through the height the pendulum has descended from the commencement of its motion, which (if a represent the limit of vibration) is (cos q-cosa). Then v2=2gl (cos -cos a), and If we suppose the impulse to continue to the distance ẞ on the other side of zero, and integrate between the limits a and -8, we shall have (employing the simpler symbols for the sake of convenience) dq2 d t2 = cos î—cosa+(m−r) (sina—sing)+r cos a (a—9)]. Replacing the trigonometrical functions by their values in terms of the arcs, and rejecting minute terms of higher orders than the second, we shall obtain, after reduction, In like manner, if t' express the time during which the impulse weights oppose the motion, Putting t+t', the total time of vibration, Developing these arcs in terms of their sines, according to the ordinary series, taking their difference, and expressing by 4t the difference between and the time of vibration of a free pendulum, τ [[(8+m) — (B — m) + 1 [(B+m) 3 — (3 —m) 3] + (a + m) 2.3(a+m) 3 1. 3 [(3 + m) 5 — (B = m) 3], &c.] ; which becomes, if we neglect insignificant terms, a + m2 √ [ 1 1 If as for the free pendulum beating seconds, then, put ting T for the total acceleration of the clock in making the number of beats made by the seconds pendulum in a day, and calling the entire remaining value of the second member of the last equation S 4T= 86400 S Π (2) The terms containing r having disappeared from these expressions, it would seem that the resistance of the air does not affect the time of vibration. These terms, however, have not been eliminated, but only neglected, in consequence of being connected with powers of the arc higher than the square. By preserving them, it may be shown that the resistance of the air produces an effect which is not altogether insensible; rather, however, by consuming some of the disturbing power of m, than by its direct influence. The reason of this is, that the resistance of the air opposes gravity during the descent of the pendulum, but favors it during the ascent. The value of the foregoing series depends upon the impelling power, and on the ratio between the arc of impulse and the are of vibration. The necessary impelling force itself, when the ratio just mentioned is fixed, depends upon the absolute magnitude of the arc a. If we assume this arc at two degrees, which is 035 of the radius taken as unity, and make the ratio of 8 to a=7071, as recommended by Mr. Denison, in his rudimentary treatise on clock and watch work, we may compute the value of m by the following process: Assume the pendulum to weigh twenty pounds, which is not far from the weight of that of the clock exhibited; and suppose it to expose a resisting surface to the air of thirty square inches. A column of air of equal base and weight would be about 1250 feet in height, and the velocity with which a fluid of this altitude would issue from a vessel in consequence of the superincumbent weight, is determined by the formula v2=2gh. Were the pendulum therefore to move with a velocity equal to the square root of 2 x 32 x 1250 (=80,000), the resistance would be equal to its weight. Hence, This being the coefficient of the variable resistance, the total effect of the resisting force may be found by integrating the expression, between the limits a and -a. Employing the symbol ", instead of the fractional coefficient, and taking the value of dt as given above, we have, calling the total resistance R, Since m is very minute in comparison with a, we may make =1, and also neglect -2ma. The small positive term at a+m the end becomes insensible, when multiplied by the general coefficient, in which 2 is a divisor-the term itself being insignificant compared with, with which it is connected by the sign. The errors thus introduced, besides being insensible, are in opposite directions, and nearly balanced. The simplified expression is then, rga2 g2 la2 R = = 2k2 This resistance extends over the whole arc of vibration; but the maintaining power acts effectively only between the limits This computation supposes the resisting surface to be plane. The actual value of k will vary with the form of the pendulum; and will ordinarily be considerably greater than it is here found to be. The disturbing effects upon the pendulum, deduced further on, will therefore be materially less than represented; since a less impelling force will be required to maintain the motion than the calculation exhibits. The actual value of k may be pretty nearly ascertained for bodies of regular shape, by considering the inclination of their surfaces to the direction of motion. and -8, or during the time found by integrating the expression. already given for d t between the same limits. Then, Or, putting S for the sum of the series within the brackets, The foregoing series rapidly converges, and if ẞ= '707 a, its sum is. Putting a, the length of the pendulum arm, measured at right angles to the pendulum rod from the centre of motion, 3 inches, and employing for the other symbols, the values heretofore given, we shall obtain for m and w the numerical values, m000001597. w= 2.914 grains, or 3 gr. nearly. Returning to the expressions (1) and (2), with the value of m thus determined, and still employing for ẞ the value 707 a, the sum of the series within the brackets in (1) will be found to be 1.384. And therefore 4 T, or the daily acceleration, will be, 2 X 86400 X •000001597 X 1.384 4T= 3.14159 035001597 =3.473 seconds. Whence it appears that this pendulum, in order to beat seconds, must be about three one thousandths of an inch longer than one entirely free. In order to investigate the liability of this pendulum to change of rate, we must observe that, at a constant temperature, it is impossible that there should be a change of rate without a change of the arc of vibration; and further, that there is no cause in operation to change the arc, except variations of density in the air. In expression (5) we observe that a2 varies as k2; but it is evident that 2 varies inversely as the density of the atmosphere. Or, putting D for the density, Putting the mean density of the air = 1, and substituting a finite difference for dD, we shall find that the corresponding finite difference of a will be but half as great in proportion to the entire arc, as the fractional change of density. If, therefore, under a constant temperature, the mercury in the barometer rise or fall one inch, or a change of density occur equal to one thirtieth of the mean, the arc of vibration will change one sixtieth part of the whole; that is to say, if the value of a is 2°, the arc will fall off, or increase to the amount of 2'. To compute the effect of such a change upon the quantity 4T, we may regard the series in equation (1) as being sensibly constant, and then, representing the whole expression, except the denominator of the coefficient fraction, by Q, and omitting the insignificant term m from the denominator, we shall have, Substituting a4 T for Q and reducing, we have, which, in the extreme case supposed above, gives a diminution of the daily acceleration equal to 058 sec. This change is, unfortunately, in the same direction as that of the circular error: but it is proportional to the quantity 4 T itself, which is directly as the maintaining power; which, again, as appears from equation (5), is as the square of the arc. Hence, therefore, the importance of reducing the arc of vibration, and the near approach to insensibility of the errors arising from its variations, when it is small. Were the arc only 1° on each side of the vertical, the error would be between one and two hundredths of a second per day. Were it half a degree, the clock, from this cause, would not be an entire second in error in nine months. The chief object had in view in the construction of the electric clock herewith exhibited, has been to secure the reduction of the arc of vibration. The work having just been completed, opportunity has not yet been allowed for experimentally deciding the question how great a reduction of arc is practicable; but the principle of the mechanism exacts no larger motion than |