of geodesy was ( Tiiraut's treatise. /'.»/• /• 



de la terre. 1 * On tin' hypothesis that tin- earth is a 



spheroid of equilibrium, thai is, su< li that a layer of 



water would spread .ill over it. and tli.it the internal 



density varies so tli.it layers of equal density are 



coaxial spheroids, Clairaul derived .1 historic 



theorem: If ■>,. y P are tin- values ol gravity .it the 



e(|iiatoi and pole, respei ti\ ely , and 1 the Cent ifugal 



force at the equator divided hy -,,. then the ellip- 



5 7 1" 7k 

 ticity a - ( •— 



- 7b 



Laplace showed that the surfaces ol equal density 

 might have any nearly spherical form, and Stokes 

 showed that it is unnecessary to assume any law of 

 densityaslongastheextern.il surface is a spheroid of 

 equilibrium. 18 It follows from Clairaut's theorem 

 that if the earth is an oblate spheroid, its ellipticity 

 can be determined from relative values of gravity 

 and the absolute value at the equator involved in c. 

 Observations with nonreversible, invariable com- 

 pound pendulums have contributed to the applii ation 

 ol ( lairaut's theorem in its original and contemporary 

 extended form for the determination of the figure and 



gravity field of the eai th. 



Earlv Types of Pendulums 



The pendulum employed in observations of gravity 

 prior to the 19th century usually consisted of a small 

 weight suspended by a filament (l\*j,a. 4 6). The 

 pioneer experimenters with "simple" pendulums 

 changed the length of the suspension until the pen- 

 dulum beat seconds. I'ieard in 1669 determined the 

 length of the seconds pendulum at Paris with a 

 "simple" pendulum which consisted of a copper ball 

 an inch in diameter suspended by a fiber of pite 

 from jaws (pite was a preparation of the leaf of a 

 species of aloe and was not affected appreciably l>v 

 moisture). 



A celebrated set of experiments with a "simple" 

 pendulum was conducted by Bouguer 1 * in 1737 in 

 the Andes, as part of the expedition to measure the 

 Peruvian arc. The bob of the pendulum was a double 



' Paris, 1743. 



15 George Gabriel Stokes, "On Attraction and on Clair- 

 iui\ I heorem," Cambridge and Dublin Mathematical journal 

 (1849), vol. 4, p. 194. 



'• Sec Collection de memoires, vol. 4, p. B 34, and J. H. PoYNTWQ 

 and Sir J. J. Thomson, Properties of Matter (London, 1927), 

 p. 24. 



truncated cone, and the length was measured from 

 the jaw suspension to the center ol oa illation of the 

 thread and bob. Bouguer allowed for change of 

 length of his measuring rod with temperature and 

 ii the buoyant j oi the air. I fe determined the 

 time of swing by an elementary form of the method 

 of coincidences. The thread of the pendulum was 

 swung in front of a scale and Bouguer observed how 

 long it took the pendulum to lose .1 number ol vibra- 

 tions on the seconds clock. F01 this purpose, he 

 noted the time when the beat ol the clock was heard 

 and, simultaneously, (he thread moved past the 

 center of the scale. A historic aspect ol Bouguer's 

 method was that he employed an "invariable" 

 pendulum, thai is, the length was maintained the 

 same al the various stations of observation, a pro- 

 cedure thai has been described as having been 



invented l>\ Bouguer. 



Since 7 x-jtjg, it follows that T\\T\ g a ,»,. 

 I litis, ii the absolute value of gravity is known .it one 

 station, the value al any other station can be deter- 

 mined from the ratio oi the Squares of limes of swim; 

 of an invariable pendulum at the two stations. I 

 the above equation, if 7, is the tune of swing 

 station where the intensity of gravity is g, and /.- 

 is the time at a station when- the intensity is g | Ag, 

 then A*/*= 71/71-1. 



Bouguer's investigations with his invariable pendu- 

 lum yielded methods for the determination of the in- 

 ternal structure of the earth. On the Peruvian 

 expedition, he determined the length of the seconds 

 pendulum at three stations, including one at Quito, 

 at varying distances above sea level. If values of 

 gravity at stations of differenl elevation are to be 

 compared, they must be reduced to the same level. 

 usually to sea level. Since gravity decreases with 

 height above sea level m accordance with the law of 

 gravitation, a free-air reduction must be applied to 

 values of gravity determined above the level 1 

 sea. Bouguer originated the additional reduction for 

 the increase in gravity on a mountain or plateau 

 caused by the attraction of the matter in a plate. 

 From the relative values of gravity at elevated sta- 

 tions in Peru and it sea level, Bouguer calculated 

 thai the mean density of the earth was 4.7 times 

 greater than that of the cordi/leras. 1 " For greater 

 accuracy in the study of the internal structure of the 

 earth, in the 19th century the Bouguer plate reduction 



17 Poyntdjo and Thomson, ibid., p. 22. 



PAPER 41: DEVELOPMF.NT OF GRAVITY I'lNDl'IIMS IN 1'IIK T'Tll CKNTfKV 



309 



