394 M. BESSEL. ON THE ELLIPTIC SPHEROID OF REVOLUTION 



The problem requires that the observed latitudes <p, <f', $" . . ., 

 should be brought in correspondence with the measured di- 

 stances between the parallels by the application of alterations 

 X, of, x" . . ., of which the sum of the squares 



shall be a minimum : the values of g and a, a!, a" . . ., which 

 fulfil this condition, belong to the elliptic spheroid of revolution 

 sought. If we write <p + a^, and 41' + a/, for 4* and <{>', and if we 

 neglect the influence of the alterations on L as well as the squares 

 and products of x and a?', the above expression becomes 



3600 



(^s' — s) — l—2aia. sin I cos 2 L + co a' sin 2 I cos 4 L — . 



ff 



+ {x' — x) g, 



where § stands for 



1 — 2 « cos Z cos 2 L + 2 «' cos 2 / cos 4 L — . . . ; 



we have thus 



x'-a,'=~ {^^ (s'-s)-(Z-2c««sin/cos2L 



+ CO «' sin 2 / cos 4 L . . .) }, 

 and we must now so determine g and the compression as to 

 fulfil the above-named condition. 



If we take ^, and aj as the approximate values of g and a, 

 iTiake gi n , ia . 



5' = Y + i' « = «i(i + ^'); 



and if we neglect the squares and products of i and k, the ex- 

 pression for x' — X becomes 



1 rSGOO I ^(2«.sin/cos2L-«/sin2/cos4L + ...), 



1 3600,, .. co,^ . , „T ^""'i • nj AT . \j 



+ -. (s'— 5)^ + -(2a,smZcos2L— a, 3— isin2Zcos4L + ...)A-, 



? ffi ? ««i 



in which a\ and «! -5^' expressed by «i are respectively 

 5 o 25 ,. , 5 , 50 4 



Then, if we make 



1 r3600, , , ,\ , "^ / o -7 O T 



m = -—< (s' — s) — l Y + —< 2«, sni / cos 2 L 



-(l«''+f2<)^"^'^^"^'^}' r 



