32 Mr. Ivory on the shortest Distance 



the first may be supposed the less. But we may proceed in 

 a different manner. If we suppose that e 2 varies, while the 

 extremities of the curvilineal arc continue upon the same 

 meridians, and retain constantly the same latitudes, \f/ will 

 remain unchanged, but s will have different values as e z as- 

 sumes different magnitudes. When e 2 = 0, s will become 

 equal to an arc ir of a great circle on the surface of the cir- 

 cumscribing sphere, the extremities of <r having the same la- 

 titudes, and the same difference of longitude, as the two points 

 on the surface of the spheroid ; and at the same time b will 

 become equal to /3, the sine of the inclination of the same 

 great circle to the equator. Now the values of s and b, when 

 e 2 = 0, being <r and |3, we may assume in general, 



s — <t + A e 2 + B <? 4 + &c. ; 

 then, -if- = 2 A > - ^Y 7 = 8 B > &c -> 



ede ede 



the values of the differentials, which must be deduced from 

 the foregoing expressions of s and \{/, being estimated on the 



supposition that e 2 = 0. 



For the sake of simplicity, let Q be written for \/ b 2 — sin 2 \ ; 

 then, by differentiating the two expressions of s and \J/ with 

 respect to e 2 , \J/ being constant and b 2 a function of e 2 , we shall 

 get, 



bdb fdXcosX 1 1—6* PdXcosX 1 _ ,.v 



"" ede J Q3 * "a \-e*J Q ' "ZT — » * ' 



ds bdb ^l-e* fd X cos X 1 



vf' 



ede ede ^/l — e 2 ft 2 ^ Q 3 A 



1 -}-6 2 — 2e 2 6« fdXcosX 1 



. ~ • "; 5 /"; ttst /*d A. cos X sin 2 X 1 



+ 3*/l -<r. ^/ 1 - e>& .J . - : 



and by combining the two differential expressions so as to ex- 

 terminate — ; — from the latter, there will result, 



ede 



ds o jv^f 1 ~ e * pdx cos A. 1 



ede ' 4/\—&b* J Q A 3 



d X cos X sin * X 



+ 3 */ 1 - e 2 . */ 1 - e 2 b 2 .f- 

 Further it will be found that 



Q 



. j 6 2 A 2 - Q 2 



Sin - Xsas __ 



and 



