324* Mr. Galbraith on the Figure of the Earth. 



b be the polar semiaxis, b + c the radius of the equator, $ the 

 angle which a line drawn from the centre to a given point on 

 the surface makes with the axis of the spheroid, and f the force 

 with which the given point is attracted by the spheroid ; then it 

 has been shown by M. Chastellot, (in the French translation of 

 Newton's Principia, torn. ii. Paris edition of 1756, § xxxvii. 

 coroll. I. page 236, 237.) that 



/=— 0+-5T- IS sln W 



But if X be the latitude, and the angle of the vertical with 

 the radius, commonly called the reduction of the latitude, and 

 therefore $ = A. — - 0, or the reduced latitude, then 



there will result 



/= '%(* + f * * i^in'(X-fl)) (2)" 



This expression for the attraction, shows that the gravitating 

 force in proceeding from the equator to the pole, increases as 

 the square of the sine of the reduced latitude. From similar 

 principles, and the considerations formerly advanced, it fol- 

 lows that the increase of the force of gravity depending upon 

 the diminution of the centrifugal force, also increases as the 

 square of the sine of the reduced latitude. On combining these, 

 the absolute increase of the length of the pendulum from the 

 equator to the pole, is proportional to the square of the sine of 

 the reduced latitude, not of the observed latitude. I advance this 

 conclusion with diffidence, and with all due deference to those 

 who have preceded me, and am fully aware of the authority of 

 such names as those of Clairaut and Laplace : but I think my 

 conclusion is founded in truth ; and if it be so, it cannot be 

 shaken by the authority of any name, however great. No doubt 

 the difference between the square of the sine of the observed 

 and reduced latitude is not great; since at its maximum at 45°, 

 supposing the ellipticity to be about 7 £ n , making the reduction 

 11' 29" only, the square of the sine of the observed latitude, 

 or 45°, is 0*5, and the square of the sine of the reduced latitude, 

 or 44° 48' 31", is 0-4966596. Now if the excess of the polar 

 above the equatorial pendulum amount to 0*208 of an inch, 

 then 0*208 (0-5—0-4966596) = 0*000695 of an inch, the quan- 

 tity that the computed pendulum at the observed latitude of 

 45° would be too great : and hence, when the lengths of the 

 experimental and computed pendulums are compared in the 

 usual hypothesis, a disagreement at this parallel will always 

 occur, which may very easily be imputed to a wrong cause ; 



such 



