192 Putnam — Results of Pendulum Observations. 



reliable a value can be obtained from such meager data, derived 

 entirely within the borders of the United States. The values 

 reduced to sea level by Faye's reduction were used, and to 

 simplify the computation were averaged together for each 

 degree of latitude. Conditional equations were formed of the 

 form g ( p= x + y sin 2 <p, representing the variation of gravity 

 with latitude, where x is gravity at the equator, y is the differ- 

 ence between gravity at the pole and the equator, and g^ 

 is observed gravity at latitude <p ; x and y were then computed 

 by the method of least squares, and substituted in Clairant's 

 theorem. Using Helmert's expansion of Clairant's theorem,* 

 the value g-^y.o" was obtained for the amount of flattening, or 

 the difference between the earth's polar and equatorial axes 

 divided by the equatorial axis.f Helmert in 1884 obtained the 

 value 2-9V.3- from a discussion of pendulum observations in 

 various parts of the world, and Clarke from a similar discussion 

 in 1880 obtained 3--9V.2- Two of the most important values 

 obtained from arc measurements are those of Bessel (1841) 

 Wt.^' an d Clarke (1880) g^V^-. A variation of one unit in the 

 denominator of these ratios corresponds to a change in the dif- 

 ference between the earth's major and minor axes, of about 481 

 feet (147 meters), and from the discrepancies in the above and 

 other results one may judge that the actual uncertainty may 

 be nearly a mile.J The value derived from the pendulum 

 observations in the United States is not very different from 

 Helmert's and Bessel's, but it is of significance only in showing 

 that a fairly accordant result may be obtained from so few 

 observations very narrowly distributed in latitude, and in fur- 

 ther confirming the validity of the reduction to the sea level 

 tentatively employed, and the theory of the condition of the 

 earth's crust on which that reduction is based, the equilibrium 

 or isostatic theory.g This result has been obtained by combin- 

 ing observations made at altitudes above sea level ranging 

 from 6 ft. to 14085 ft. (2 m. to 4293 m.), and in a great variety 

 of continental locations. 



*"Geodasie," by F. R. Helmert, vol. ii, p. 83. 



f A result for the flattening of ¥ ^g was obtained by comparing the four 

 southern stations, Austin, Laredo, Galveston and New Orleans, with four stations 

 in nearly the same longitude near the 39th parallel in the central plains, St. Louis, 

 Kansas City, Ellsworth and Wallace. 



^ Prof. Harkness says : " Indeed the facts thus far advanced scarcely warrant 

 any conclusion more definite than that the flattening probably lies between ^^ 

 and 3-1^, but we shall see presently that there is some further evidence which 

 tends in the direction of the smaller limit " [3-jjol — " The Solar Parallax and its 

 Related Constants," p. 103. 



§For discussions of the relation of the Coast and Geodetic Survey pendu- 

 lum observations to these theories, see papers by Mr. G. K. Gilbert, Bulletin 

 Phil. Society of Washington, vol. xiii, p. 61 ; by Rev. 0. Fisher, Nature, vol. 

 lii, p. 433, Sept. 5, 1895 ; and by M. Faye, Comptes Rendus de TAcademie des 

 Sciences, 20 May, 1895. 



