AECHDEACON PEATT ON THE INDIAN AEG OE MEEIDIAN. 
587 
Let R, C, C' be the values of r at the mid-latitude and at the extremities of the arc ; 
R =a-\-a cos sin m—{a — h) sin^ m, 
C =a cos / 4'i3sin/ — — /»)sin^/, 
C'=a-\-a cos I' -|-|3 sin I' —(a — b) sin^ I'. 
Multiply by 1, M, N, add and make the coefficients of a and (3 vanish ; 
cosm +M cos /+N cos /'=0, sin w+M sin /+N sin l'=0 ; 
M= 
sin (m — l) 
= — p sec 2 i^=N ; 
sin [i' — l) 
R+MC+NC'=«(1+M+N)— (fl! — />)(siffim-l-M siffi /+N siffi /') 
=a(l-l-2M)— ^ [a—h){l — cos 2w+2M(l — cosX cos 2m)} 
(«+/»)(! 4-2M)+^(a — /i)(l+2M cos X) cos 2m 
(«+/») ^1 — sec (a— — cosX sec ^X^ cos 2m. 
Taking the variation with respect to the axes, 
SR= ^ (5a — sec ^ X^ 4-^ — cos X sec ^ X^ cos 2m. 
Put X=ll° 27' 11", 2m=47° 34' 25", 
gE=-0-0025078(5a4-55) + 0-0050586(5a-5J) 
= 0-00255085a-0-007566453. 
9. Third. The coordinates to the centre of the local ellipse from the centre of the 
ellipse equal to the mean ellipse drawn through the extremities of the arc. 
By eliminating (3 from the two equations which give C and C', we have 
(O' -a) sin I— (C — a) sin V— {a — b) sin I sin Z'(sin /' — sin /) 
^ sin {I— I') 
C'sin /— C sin /' sin/— sin/' , . 
r 7 ^{a+(a— ojsin l sm / } 
sin {I— I') 
sin (/ — /') 
Also 
C'sin /— C sin /' cosm 
sin (/— /') cos 2 X 
a-3-Q((^—^)(cos X — cos 2m) 
3= 
(C — a) cos /'— (C' — a) cos /+ (a — ^)(sin^/cos /' — sin^/'cos 
sin {I— I') 
C cos /' — C'cos / cos/' — cos/} 
sin {I— I') 
a— (a— ^)(l + cos I cos I' 
sin (/—/') ( 
C cos /'—C'cos / sinm f, 1 , 
=^to(7=7j— itali -2 (“-^Xcos X+cos 2m) j. 
4 L 
MDCCCLXI. 
