On Practical Geodesy. 19 



normals E,^, R,^, we easily find the following expressions for 

 8, and 8,,— 



^ _ e^ (K, sin l^ — R,^ sin l^) cos I, 



' R, — ^ (R, sin ^ — R,, sin /,,) sin I, , x 



5j _ e^ (R, sin l^ — R,^ sin I,}) cos l„ 

 \ ~ R,, + e"- (R, sin Z, — R,, sin Z J sin I,, 



And from the plane triangles whose bases are Z^Z^^, and ver- 

 tices S^, S^^, we have — 



5> ^ (R, cos V — R,, cos l') sin L' 



sin 8, = — ^^ — '- '- 



• 2, ^ (R, cos V — R,, cos I") sin L" 



sm 8,, = —^-^ -^ 1 



Again, from the triangles S^S^^Z^, ^0^00-^00' ^^ ^^ evident 



that — 



R, cos (z. — a.) R,, cos (2,, — a,,) / N 



Q,^ COS a^ Q^ cos a,^ 



and, to 8 places of decimals in their logarithms, we have — 

 R/ R/, -I / \ 



Hence, from the triana^les Z Z S , Z Z S„ , we have the 



^ ' o o 00 o' o 00 00' 



relations — 



sin U _ R^ sin L" _ R,, 



sin l^ ~ E,^, ' sin I" ~ R, . 



snch that their logs, are the same to 7 places of decimals. 

 And if in the first and second of (81) we substitute for 



^, and ^ the above equivalents, we have with an accuracy 



to at least 7 places of decimals in their logs. — 



sin 8^ = e^ (sin L' cos r — cos I" sin l') , x 



sin 8,, = e^ (cos I' sin I" — sin L" cos I") ^^^^ 



which we may write in the forms — 



S, = e^ < — cos I" sin (L' — 8,) + sin L' cos (L' — 8,) > 



sin 8,, = e^ \ cos r sin (X" + 8,,) — sin L" cos (L" -f 8,,) > 



And if we expand these and regard cos 8, = cos 8,, = 1 

 (which we can do since 8, or 8,, is always less than 20'^) we 

 easily find — 



. - e^ ' (cos L' — cos I") sin L' 



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



sm 



