S77 



PENDULUM. 



PENDULUM. 



373 



mouvement du pendule." ('Acad. Roy. des Sciences/ 1735, p. 507.) 

 vs its motion was sensible for eighteen hours. It seems that 

 this pendulum, the vibrations of which were to be counted by a clock, 

 was also intended to measure the actual length of the pendulum. 

 Messrs. Bouguer and La Condamme both had detached pendulums 

 made after Graham's idea. Bouguer (same volume, p. 526) describes 

 this pendulum as an invention of his own ; La Condamine (' Journal 

 du Voyage/ p. 143) is more open, and says he took the idea from a 

 copy which Hugo made after Graham's. This is almost exactly Kater's 

 invariable pendulum. Mairan's measurement of the length of the 

 seconds pendulum (' Acad. Roy. des Sciences/ 1735, p. 153) is a good 

 specimen of the old method of measuring the length of the pendulum : 

 and the measures of Godin, Bouguer, and La Condamine, in the 

 same volume, are worthy of notice. For references to various 

 pendulum experiments, see Laknde, 'Astronomie/ 3rd edit., s. 2710, 

 c( .- 7. 



In the first volume of the ' Transactions of the Society of Arts/ 

 p. 23S, Mr. Hatton proposed, as a mode of fixing a permanent standard 

 of length, to suspend a weight from a fine hair to a clip in an upright 

 bar, sliding up and down in a vertical frame. The hair passed through 

 a fixed clip. The weight was to be swung, and the vibrations counted, 

 in two positions of the bar, and from the difference in the times of 

 vibration and the space through which the bar was moved, the length 

 of the seconds pendulum was to be computed. 



In 1787 Mr. John Whitehurst published 'An attempt towards 

 obtaining invariable Measures of Length/ &c., which is remarkable for 

 its ingenuity. He suspended a leaden ball with a flat steel wire in 

 front of a straight upright frame, the wire being long enough to make 

 forty-two oscillations in a minute. A clock with dead-beat escapement 

 and a clip to hold the wire was slid up and down the frame, and 

 secured and adjustable at two points where the clip made the free 

 oscillations respectively forty-two and eighty-four in a minute. The 

 crutch of the clock, being continued upwards in a screw, carried a 

 weight, by moving which the oscillations of the crutch alone could be 

 regulated to forty-two and eighty -four oscillations, and therefore would 

 not interfere u-ith (lie free oscillation of the bail and icire, but only keep 

 up their motion. The going weight of the clock was in each case such 

 as sustained an oscillation of 3. It is clear that if all were properly 

 executed, the clock-frame with its clip must have been shifted between 

 the two positions through a space equal to the difference between the 

 simple pendulums which correspond to forty-two and eighty-four 

 oscillations per minute. A line was drawn in each position along the 

 upper edge of the clock frame upon a brass rule fixed to the upright 

 support, and this space was afterwards accurately measured, and the 

 length of the simple seconds penduluinjthence computed. Whitehurst's 

 length of the seconds pendulum is 39*1196 inches of Troughton's 

 standard, but the corrections for the buoyancy of the air and for 

 temperature are not introduced. It is probable that he introduced 

 greater errors than those he wished to get rid of in Hatton's method, 

 for the real difficulty is not that of counting the vibrations, but of 

 measuring the length between the two clips, in avoiding the errors of 

 temperature, and the uncertainty as to the effective point of suspen- 

 sion. The /irinrijjf of Hatton's method, that of measuring the 

 difference between two pendulums, has been adopted, as we shall see, 

 liy !!--.!. 



The foregoing account is merely a sketch of the history of this 

 mechanical problem, which in the hands of Borda, and more recently 

 of Kater and Bessel. has received a more accurate solution. There arc 

 still anomalies and imperfections in some parts of the processes which 

 require clearing up, but the errors have been reduced within compara- 

 tively moderate limits. Before describing these experiments we shall 

 give a brief account of the formul;e which they require. 



The expression which connects the time of one oscillation of a simple 

 pendulum in an infinitesmal arc, with its length I, at a place where the 



tt 



force of gravity is represented by g, ia t * -77, being 3'141596, or 



circumference to diameter 1 ; the measure of gravity g, being twice the 

 space through which a body would fall freely in 1', or, what is the 

 name thing, the space through which a body would move in 1', with 

 the Telocity which it acquires in falling freely for !' 



Hence if I be the length of the simple seconds pendulum, 

 r/ = r"-l; therefore g ia known when I can be measured. The process 

 therefore of finding the effective force of gravity at any place is 

 reduced to finding the length of the simple pendulum which vibrates 

 seconds. 



The French astronomers, in their great survey of the arc of the 

 meridian, determined the abtolule length of the pendulum at different 

 stations between Dunkerque and Fonnentera, and also in the con- 

 tinuation to Unst in the Shetland Isles, which is included in the 

 English trigonometrical survey. It is, however, an operation of great 

 delicacy, and when only the variation of gravity between different 

 places ia required, as is the case in researches into the figure of the 

 earth, the observation may be more easily performed by swinging the 

 tame pendulum in different places, and ascertaining the number of 

 vibrations whir-h it makes in a day. Thus if n and ' be the number 

 of vibrations made in a day by the tame pendulum I, at two different 

 places at which the forces of gravity are >j and >J , and the duration 



of one vibration at each place be t and t', then since the time 

 of one vibration = a day divided by the number of vibrations, we 

 shall have 



1 1 

 -:-, 



H*..!.J_ 

 W gk-g'k' 



or g 



That is, the force of gravity varies as the square of the number of 

 vibrations of a given pendulum in the same time, which is usually 

 taken to be a mean solar day. If, therefore, the number of vibrations 

 of a pendulum in one day at a given place, London for instance, be 

 known, and it is then transported to different places, and the number 

 of vibrations in a day counted, a simple proportion will connect the 

 forces of gravity at London and ^every place at which the observation 

 has been made. 



If the length I' of the simple pendulum at any station be required 

 from these observations, 



In'* 



since I : I' : : g : g 1 : : " : n", 



I = ~^T, which gives the length of the pendulum at any place in 



terms of the length at London, and the number of vibrations per diem 

 at that place and London. 



Though it scarcely belongs to our subject, we will give the expres- 

 sion by which the ellipticity of the earth is determined from pendulum . 

 observations. The length of the seconds pendulum at any latitude, 

 may be supposed i=A + B.sin. 2 A, where A and B are constant 

 quantities. Now from all the good observations, either of the 

 actual length or the number of vibrations per day of the samo 

 pendulum, determine the values of A and B; then, by Clairaut's 

 theorem, 



B 

 the ellipticity of the earth = * '008668 ; 



whence the ellipticity is found. By ellipticity is meant the excess of 

 the equatorial over the polar radius of the earth, divided by the polar 

 radius. 



The apparatus of Borda will be generally intelligible from the follow- 

 ing description and figures. 



.' 



Borda's Pendulum Apparatus. 



The plumb-line is suspended from a knife-edge piece, A B, and is 

 attached below to a cup, E, which is ground to fit very exactly the 

 platinum ball below. A little grease is rubbed on the inside of , the 

 cup, making the contact perfect enough to exclude the air and to sus- 

 pend the balL The knife-edge rests on agate planes, a, b, which are 

 carefully levelled, and the frame o D, which carries the planes, is fixed 

 immoveably in a horizontal position. The plumb-line is in front of the 

 comparing clock, which has a small cross drawn on the bob. Wheu both 



tm 



* This in Mr. Airy's value for j- where m is the ratio of the centrifugal 



force at the equator to the force of gravity there. (' Encyclopaedia Metropoli- 

 tana,' Figure of the Earth, sect. 2.) Biot, ' Astronomic,' vol. 3, additions, 

 p. 169, givea 0-00865; the centrifugal force m being supposed j|j of the force 

 of gravity at the equator. 



