i 



- -/ 



46 



Bowditcli on the Ohlateness of the Earth. 



Now 



general equatioa of 



oiJ 



semi-axes 



k 



1c 



k 



\'m V « 



resented bv 



3 by Pag. 8 of the above mentioned vol 

 m f -f n%^ = ¥ OT x" +yM- «' = P + (i 



+ (1 



). z% and 



m, 1 



f the ord 



may 



ill the second member of this equation put ^ 



Ic 



\ 



% 



k. sin. 4/, as is evident from Pag. 113; of the same woik^ ob 



k differs from r. oulv bv terms of 



order a. S 



atituting these values, and those of af + y^ + z^ = r-^y it becomes 

 r^ = Jif, {1 + (1 — m). cos. \J/^ sin 9^ + (1 — n). sin. \|/^}, whose 



square root, neglecting a^ becomes, 



r 



k+Jc, 



(1 



2 



~. cos 4/^ sin p'. + &._ 



n) 



2 



. sin. x(/ 



(3) 



Putting cos. \I/^ 



1 



sm 



sm. 9 





COS. 2f^ we 



A^ 



shall get 



r 



k-hk. 



1 



m 



4 



fc, sin. 4/^ I 



1 



m 



(I 



4 



2 



11) m 



*T~ — 



1 



4 



.cos 



.29} 



+ 



m 



I 



4 



^. cos. g 9, and if we suppose k + k 



1 



m 



4 



1 



> 



we ma^ 



subst 



k 



1 in the terms multiplied by 1 



m 



i 



n 



9 



and then putting 



m 



7 



4 



cch 



1 



m 



{ I ~») 



} 



4 



S 



a, the preceding ex- 



pression will become 



r 



1 



a sin. ^|/^ {,1 + A. cos. S 9} +ah, cos. 29. 



(4) 



If we change the origin of the angle 9, so as to write f+€, instead 

 of 9, it becomes 



1 



a. sin. 4^K[i+h.coa.2{f + C)l +«A. cos. 2(9 +C). 



Comparing this with the expression (1) assumed by La Place 

 find that he has nei 



(5) 



lected the last term ah. cos. 2 (<p -h C). When 



90°, the expression 

 axes, and when ■vL 



1 



becomes 1 +aA 



for 



polar semi- 



ponding to the variable radius of the equator of an ellipsoid which 



m 



