N 



JBowditch on the Oblateness of the Earth, 83 



used by La Place in computing the most probable oblateness from 

 any number of observations of the lengths of a pendulum, vibrat- 

 ing in a second of time iu different latitudes. 



The principles he assumes lor computing the most probable 

 figure of an elliptical meridian of the earth, is that the sum of all 

 the errors of the observed lengths of the pendulums noticing their 

 signs should be nothing, and that the sum of all these errors taken 

 positively should be a minimum. 



Now if a^'\ a% a^'', &c. be the observed lengths of the pendu- 

 lums, p^^\ p% p^^^, &c. the squares of the sines of the corresponding 

 latitudes, and z -h py the general expression of the length of the 

 pendulums, also x^^\ 0?% a?^%&c. the errors of observation, we shall 

 have the following system of equations, in which we shall suppose 

 ji^^\ p^^^ p^\ &c. to be an increasing progression. 



a{3) — a _ jj(3). y =r x{o) (A) 



n being the number of observations. If we add all these equa- 

 tions together the sum of the errors on the right hand aide of the 



sum will by hypothesis be equal to nothing ; and if we divide this 



sum by n, we shall obtain an equation of this form 



A — «— P2/ = o. (B) 



which being subtracted from each of the equations (A), we shall 

 get the following system ol equations 



5(2) _ q{2\ y = xm (0) 



&(3) — 5(3). 2/ = arC3) 



kc. 



