Longitude deduced from two Stations. 107 



Taking any station on the surface of the solid, let y denote 

 the ordinate of the meridian perpendicular to the equator, 

 and X the distance of y from the centre : put 



/^dx^-\-dy'' xdx-\-ydy 



^ = y Tx 5 T = ~ ; 



then g is the normal, or the perpendicular to the surface of 

 the solid, limited by the equator, and t is the distance of the 

 foot of the normal from the centre. The latitude, or A, is the 

 angle which q makes with the equator ; and therefore, 



X ■= T ■\- q cos A,, y = q sin X. 

 For any other station we have siijiilarly, 



y = t' + g' cos x', y s= g' sin x' : 



And if w be the difference of longitude, the three coordinates 

 of the second station referred to the meridian of the first, will 

 be, x' cos w = (t' + q' cos X') cos w 



,r' sin w = (t' + g' cos x') sin w 

 y = q' sin x'. 

 Put m and vi' for the azimuths at the first and second stations : 

 then if we make two planes pass, one through q and the se- 

 cond station, and the other through g' and the first station, 

 we shall obtain these equations, 



Q = T sin X cos x' — t' sin x' cos X, 



sin u . I 1 Q. 



+ cos w sm X — COS X tan X' = ; . — :-, 



tan m cos X r 



(A') 



Bin &i . , , 1 Q 



: r + COS w sm X' — cos X' tan x = . — . 



tan m cos X x 



But in a spherical triangle, as described at p. S-t of the last 

 Number of this Journal, we have, 



+ cos w sin X — cos X tan x' = 0, 



tan /A 

 sin u 



+ COS CO sin x' — COS x' tan X = ; 



tan fi' 



and by subtracting these equations from the former, we get. 



Sin (|x — m) X 



sin /i sm m 

 sin u 



ni (m' — u!) X -. — —■ — : 

 ^ ' ' sin u sin m 



/i sin m cos X 



vT f>in " sin &' i sin u sin (i' i r 



Now, —. — = V, and —. — r = ; therefore, 



Bin /It cos X' sin^' cos X 



,.. , . sin m Q 



om (jU, — 7H) = ; X 



sin ;3' ' 



Q 



sill ^' 



P2 Let 



-,. , , ,, sin VI Q 



Sm(?M' — /x')= ; — X -: — r 



^ ' ' t' sill ft 



