On the Figure of the Earth. 371 



mass, and diameter of a fl_v, on the other part ; so that one rota- 

 tion of the fly, with its initial velocity, shall produce a dynamic 

 effect equal to that of the piston in n successive strokes. 



N. B. The mass of the fly is supposed equally distributed over 

 its rim, and the diameter of the crank handle equal to the course 

 of the piston. 



15. The major and minor axes of an elliptical billiard-table are 

 2a and 2h. Suppose an elastic ball to be propelled through one 

 of the foci perpendicularly to the major axis, what will be the 

 rectangular co-ordinates which indicate its position at the tenth 

 reflection? and will it, after any finite number of reflections, move 

 to and fro in the direction of the major axis ? 



LXIII. On the Figure of the Earth. By M. De Laplace'*'. 



JLt has been proved by numerous experiments made with the pen- 

 dulum, that the increase of gravity follows a very regular progres- 

 sion, and is nearly as the square of the sine of the latitude. This 

 force being the result of attractions of all the terrestrial molecules, 

 observations thereon, compared with the theory of the attraction 

 of spheroids, offer the only means that can enable us to penetrate 

 into the internal constitution of the earth ; and the result is, that 

 the earth is formed of strata, of which the density increases from 

 the surface to the centre, round which thev are regularly arranged. 

 In the Connoissance des Terns for 1821, I published the follow- 

 ing theorem, which I demonstrated in vol. ii. of the Nouveaux 

 Mimoires de V Acadcmie des Sciences. 



'■' If we take the length of the seconds pendulum at the equa- 

 tor as unity, and if to the length of this pendulum, observed at 

 any point on the surface of the terrestrial spheroid, be added, half 

 tiie height of this point above the level of the sea, divided by half 

 the polar axis, a height which is given by barometrical observa- 

 tion, the increase of this length, thus corrected, will be, on the 

 hypothesis of a constant density below a small depth, equal to 

 the product of the square of the sine of the latitude by five-fourths 

 of the ratio of the centrifugal force to the gravity at the equator, 

 or by 43 ten-thousandths." 



The above theorem is generally true, whatever may be the den- 

 sity of the sea, and the manner in which it covers tiie eartli. Ex- 

 periments made with the pendulum in both hemispheres agree in 

 giving to the square of the sine of latitude, a cocUicient somewhat 

 larger — nearly e(|ual to 54 ten-thousandths. These exjieriments, 

 therefore, prove sufficiently that the earth is not homogeneous in 



* From Annules de Chimie ct Vhijs. tome xi. 



A a 2 the 



