r 



48 - 



Bowditch on the Oblateness of the Earth. 



that part, we shall have to determine a- the following expression 

 given by La Place in Vol. S, p. H7« 



(11) 



■4^ 



■^ 



du' 



\ 



\il<p ' 



g-^/ 



/ dd 



From this he finds, by neglecting 



term u^ (7), and putting 



u 



f 



sin. ^^ 1 1 + h. COS. 2{<p + S). J 



7f 



h 



■^ 



^ 



COS. 4" 



Q 



(12) 



The 



which is easily deduced from (11) by using his value of w'. 



_ t 



term n''' (7) which he has neglected would produce in (11) the term 



«£. tan. "4^2 



COS. ■^ 





COS V ^ 



,1 



By adding this to the expression (13) we obtain the corrected val- 

 ue of ^, namely 



^ , tan, '4'2. COS. -^^2 . _ , , ^x 



%ahi. — -, — ^—, Sin. 2(f-^ C) 



COS. y \i / 



(13) 



which is easily reduced to the more simple form 



2oLhi. sin. %I/. tan. ^^. sin. S (9 i C) (14) 



J ■ 



This corrected value is less than half of that given by La Place 5 

 for the ratio of the quantities (13), (13) is expressed by '-r^r- or 



COS. il' 



i + sec. 4/^, which always exceeds 2. 



The length of an arch of one degree, measured upon the earth's 

 surface in a direction perpendicular to the meridian, is found by 

 mnltiplying the radius of curvative R given by La Place in Vol. 



S, p. 123 of his (^ Mec. CelJ' by 1°; hence it becomes 



s. 



I 



l°.fl+att/-a.('^.).tan.|> 



a* f ' 



uuu, \ 



cos.-<^^ 



(15) 



uj 4. 



/> 



being the 



r 



values of m', ^, at the first point of the arch. 

 Itt this we must substitute the value of u' (7). La Place neglect- 

 ed u^f and thence he found for this expression the following value 



t 



fi 



r 



