. / 



36 



Bowditch on the Oblateness of the Earth. 



b^^^ h(^> l^^^ 1 



We must then compute the series of quotients -j^j, -j^, -^, «c. ana 

 arrange the preceding equations according to the magnitude of 



quotients, beginning with the g 



The left hand sid 



of these equations arranged in this manner will be composed of 

 series of terms of the follovvins: form 



/iW.^ — cCi) 



/ 



A(2) . y 

 A(3). y 



c(2) 

 cC3) 



Cr) 



kc. 



I 



in which ¥'\ ¥'\ ¥'\ &c. are to be supposed positive, by changing 



the signs of the term when y has a negative coefficient. 



Then to 



find the value of y, which will render tlie sum of all the errors a 

 minimum, we must add the quantities W^^,h^^^, ¥'■>, &c. until their 

 sum begins to exceed the half sum of all these quantities ; and by 

 putting this sum equal to F we must determine r, so that 



hil) + h{2) +/i(3) .... +/i('-) ?^ A F, 



A(i)+/iC2)-f./iC5)..,. hQ--') <^F. 



m 



Then put y 



(r) 



hCO 



, and by substituting this in the expression 0.00865 



yf we shall obtain the ellipticity of the earth which will render 



.^ 



the sum of the 



^(1)^ ^(2)^ ^f.^ taken positively 



as 



La Place has proved in Book III, § 40 — 43 of his Mecani^ue 

 Celeste, 



La Place has applied the method to the observations made in 



-■■' 



the places mentioned in the note, page 33, and his system of equa- 



* In finding and arranging these quotients we must notice their signs, and if 

 anj of them are negative, they must be considered as less than the positive values, 



4, &c. This remark is made to prevent the mis- 



so that 1 > 



5 and 



2> 



takes we might fall into by neglecting the signs according to the directions of the 

 writer of the article of the Cyclopedia above mentioned. 



