20 On Practiced Geodesy. 



^ ' (cos I' — COS L") sin L" 



sin 8,/ 



(1 — e^) — e^ (cos I' — cos L") cos L" 



which we may write in the forms — 



. _ 2 • e^ • sin I {I" + L') sin l (l" — L^) sin L^ 



^^^ ' ~ (I — e^) + 2 • e^ • sin i (^" + L') sin ^ (Z" — L') cos 17 



(ss) 



_ 2 • e^ . sin 1 (L" + I') sin i (L" — Z') sin L" 



^^^ " ~ (l —e') — '2 -e'^' sin J (L" + r) sin J (L" — /') cos L" 



(to be used when extreme accuracy is desired.) 



Hence evidently (since S, or S,, is always less than 20 seconds) 

 we have — 



sin 8, = 2 ( :j ^j sin L' sin J (^ + L') sin | (^" — L') 



sin 8,, = 2 ( ^ _^ ^2 )sin L'' sin J (L" + ^') sin i (L" — Z') 



giving 8, in excess, and 8,, too small. However, in all 

 ordinary cases, they give values of 8,, 8,,, correct to toV o part 

 of one second. And since — 



sin J (Z"+ L') sin J (^"— L') = sin (D, — A,) • ^^^li^ 



= 1 • sin (D^ — A,) tan J ^,, • ^^^ 



sin 1 (L" + I') sin i (L" — I') = sin (A, — D J • ^^^Hl' 

 ^ ^ '' sm <o 



= A • sm (A, — J) J tan ^ 2, • -; — r- 

 Therefore we have the equally approximate relations — 



sin S, = 2 (^^ sin L' • "° ("' " ^-> • sin» ^ .„ 



^1 — e / sin o) -^ " 



I e^ \ . ,,, sin(D, — AJ 



= \t ^/ sm^ L • ^ — 7 tan l z,, , . 



^1 — e-/ sm A^, 2 // (^g) 



o ^ e' A . ;,, sin A,, sin (D, — A,,) . „ ^ 



= 2 I .j -J sin I" ' ". ^ ^ '^ • sin^ 1 z,, 



^1 — e^ sin D, sm w ^ // 



/ ^ \ sin A,, sin (D, — A..) . ^ 



= 1 1 -i I — =-2 • Sin- z.. ' tan A z„ 



Vl — 67 sm^ (0 // 2 // 



Z' ^^ ^ • 2 7„ sin A,, sin (D, — . A,,) , 



= Vl -/ ^1^ ^ * • 2 >> ^ * tan 1 z,, 



^1 — 6;-/ sin'' D, 2 // 



