Mr. Galbraith on the Figure of the Earth, 323 



titude; and consequently the diminution of gravity from this 

 cause, in proceeding from the pole to the equator, will also be as 

 the square of the cosine of the latitude. But the square of the 

 sine increases as the square of the cosine diminishes ; therefore 

 the increase of gravity in proceeding from the equator to the 

 pole, will be as the square of the sine of the observed latitude 

 nearly. A little consideration will readily show, however, that 

 this is correct, on the supposition of the earth being a perfect 

 sphere, and is only an approximation not far from the truth in 

 the case where the compression is small. The centrifugal force, 

 therefore, is strictly proportional to the ordinate to the polar 

 axis which involves the compression, — the very thing we are in 

 quest of. Now as this is supposed to be the unknown quantity, 

 it must be assumed equal to about ^j, what it is already known 

 to be nearly; we may infer that the centrifugal force decreases 

 as the square of the sine of the reduced latitude exactly. The 

 difference arising from these two suppositions is undoubtedly 

 small ; but in very nice disquisitions, minute quantities ought 

 not to be entirely disregarded. 



The length of the pendulum is as the force of gravity; and 

 it has generally been inferred, from suppositions not far from 

 the truth, that the length of the pendulum from the equator to 

 the pole, increases as the square of the sine of the latitude. 

 The latitude hitherto used is that derived directly from obser- 

 vation, supposing the plumb-line to hang perpendicularly to 

 the surface, or to the tangent plane to that surface, at the place 

 of observation. Now there is no doubt that, from the equili- 

 brium of the fluid part of the earth, the direction of gravity at 

 the surface is exactly, or very nearly, in this vertical line. But 

 the force of gravity at any point on the surface of a spheroid 

 is (Schubert, Astronomie Physique, vol. iii. § 125.) inversely 

 proportional to the radius of the spheroid, or the line drawn 

 from the place of observation on the surface to the centre. 

 This is conformable to experience ; for it is known that the 

 length of the pendulum, independently of centrifugal force, in- 

 creases as we proceed from the equator to the pole where it is 

 nearest the centre ; and consequently, on that very account, its 

 length there is greatest. I am well aware that the length of 

 the pendulum has, so far as I know, always been affirmed to 

 increase from the equator to the pole, nearly as the square of 

 the sine of the observed latitude ; though from what I have al- 

 ready advanced, it appears that it increases as the square of 

 the sine of the reduced latitude, the compression being assumed 

 by estimation, as has already been observed. 



" For in an oblate spheroid (Plavfair's Outlines of Natural 

 Philosophy, vol. ii. § 296.) differing little from a sphere, if 



2 T 2 h be 



