368 Dr. Tiarks on the Longitudes of the 



, ,. cos X' . sin m' «* (* n *~* sin *') + 4" sin X . sin X') 

 Sin (rf-V) = sin /J « — V(l-^sin^) 



cos /' . sin m' „ / • . • , ;\ / • , C<1 • » • » /\ 



= r— : e 2 . (sm A.— sm X') (1 -f — sin X . sin A'). 



sin /J v / \ ' 2 ' 



But by the characteristic property of the geodetical line we 

 have cos I sin m = cos V sin m'; and it is therefore evident that 

 jtx,— m = m'—fjJ or m + m' = ju-4-jtx/, and the sum of the three 

 angles co + 7W+?w' = w+^H-^', independent of the ellipticity of 

 the meridians. If we now conceive any three points on the 

 spheroid connected by geodetical lines, and draw their meri- 

 dians, the comparison of the angles of the three triangles thus 

 formed, each by two meridians and a geodetical line, with the 

 analogous ones on the sphere, will immediately prove the ge- 

 neral proposition. 



Example. — Professor Bessel has accurately deduced from the 

 latitude of the observatory at Seeberg = 50° 56' 6"*7 (A'), the 

 length of the geodetical line from Seeberg to Dunkirk (whose lo- 

 garithm == 5*47830314) and the inclination of that line to the 

 meridian of Seeberg = 85° 38' 56"*S2 (m') (the last two as re- 

 sulting from General Muffling' s great measurement), the fol- 

 lowing results, supposing le = 8*9054355 and log. semi-polar- 

 axis= 6*51335464: 



Latitude of Dunkirk =51° 2' 12"*719 (A) 



Inclination rf the ^eti^ line to 1 ? &l IS . S23 {m) 



the meridian ot Dunkirk J v ' 



Difference of longitude between 1 & 9.\ 19*04 (co) 



Seeberg and Dunkirk f \ 



These quantities, therefore, certainly belong to the same sphe- 

 roid, whether right or wrong as to the places named. I find 

 from A, A' and m + m' in the spherical triangle 

 //, = 87°51'25"*78 

 IjJ = 85 38 46 -61 

 •3 = 5 15 28*44 

 w = 8 21 19 *09. And, 

 _ „ cos/, sin m . X— X X-fx' / e^ . % . .\ 



next : 2er. -— sm -~ — . cos - J — - ( 1 + — sm A sin A' ) 



sin /3 2 2 \ 2 / 



= 10"*26 and /x— m = 10"*257 as it ought to be. The equa- 

 tion for sin (p—m) is a complete check upon geodetical cal- 

 culations of this nature; for if the parts used do not strictly 

 belong to the same spheroid, the difference between the cal- 

 culated value of jx and the given value of m will vary consider- 

 ably from the value of the same difference by the formula. 



The operations of theTrigonometrical Survey alluded to may 

 be thus represented. — The quantities A, a', m, m! referring to 

 Beachy Head andDunnose, were assumed to be exactly known, 



and 



