of the Figure of the Earth. 343 



a homogeneous fluid mass, having the form of an oblate ellip- 

 tical spheroid, will be in equilibrio, when it revolves upon its 

 axis in a proper time. The attractive forces acting at every 

 point of the spheroid ; the rate of the diminution of gravity 

 from the pole to the equator ; and the relation between the 

 ellipticity and the centrifugal force ; are all determined with 

 great simplicity and elegance. It follows from the researches 

 of Maclaurin that, for every degree of ellipticity, there is 

 only time of revolution ; but D' Alembert, considering the 

 equation between the ellipticity and the velocity of rotation, 

 afterwards found that, when the latter quantity is given and 

 the former is sought, the problem admits of two different 

 solutions. 



In 1743, Clairaut published his Theorie de la Figure de la 

 Terre. This is a work of the greatest merit and elegance, 

 containing many new results, and treating every part of the 

 subject in a full and satisfactory manner. In the case of ho- 

 mogeneous spheroids Clairaut abandons the method followed 

 in his first researches, and adopts that of Maclaurin. But, 

 with respect to spheroids composed of strata of different den- 

 sities, he admits the hypothesis of a small ellipticity, which 

 simplifies the investigation, and is sufficiently exact lor deter- 

 mining the figure of the planets. 



In all these researches the oblate elliptical spheroid is alone 

 considered ; and the question is to prove that it will be in 

 equilibrio when it revolves upon its axis. Maclaurin solved 

 the problem generally and accurately in the case of the ho- 

 mogeneous spheroid. In all the other solutions the supposi- 

 tion of a very small ellipticity is admitted ; and therefore the 

 results are approximately, and not rigorously, proved. But 

 the theory was imperfect unless the investigation could be ex- 

 tended so as to take in all the possible figures of equilibrium, 

 or until it was shown that the elliptical spheroid alone fulfilled 

 the conditions. This brings us to the researches of Legendre 

 and Laplace ; but as the discoveries of these eminent geome- 

 ters were deduced from the hydrostatical theory of equili- 

 brium, it is necessary to notice briefly the progress made in 

 this part of the subject. 



4. When the effect of the centrifugal force to shorten the 

 seconds' pendulum in approaching the equator, was first dis- 

 covered, Huyghens immediately inferred that the terrestrial 

 meridians were not circular; and, in order to determine 

 the true figure, he assumed the principle that they must be 

 perpendicular to the direction of gravity. Newton investi- 

 gated the figure of a homogeneous fluid turning upon an axis, 



by 



