GEODESIC INTESTIGATIONS. 



159 



cot L, 



R, (l-e2) sin I' + R„ e^. sin I" 



Similarly, 



cot L„ = 



E, cos I' 

 B,, {l-e"-) sin I" + B, e^ sin V 



(39) 



(40) 



M,,. cos I" 

 _ ' It is evident that L, and L,, are functions of the latitudes of the 

 stations <S" and <S"' ; and (no matter what may be the relative positions of the 

 stations) L^ L^ ■will remain constant magnitudes, provided the latitudes are 

 constant in magnitude. When V and I" remain constant in magnitude, then 



obviously — — and remain constant in magnitude. 



^ sin B„ sin B, ^ 



The expressions for cot L^, cot i^,, in terms of the latitudes only, as given 



in equations (39) and (40) are of importance. They enable us to find the 



rigorous values of \ 2', \ 2", &c., when the values of the latitudes V , I", are 



known, and that either the difference of longitude or one of the azimuths 



is also known. 



®The accuracy of the following most important set of formulae can be 

 relied upon. The simple investigations, by means of which they have 

 been evolved, are omitted in order to economize space in the "Transactions." 

 Expressions for the angles of depression of the chord of the geodesic arc. 



R.. 



cos 



12' 



= 



k 



COS 



12" 



= 



R, 

 h 



COS I" 



sin A! 

 COS V 



sin A!' 



J 



V 



2' 



R/R,, (cos /' COS I" cos w -f ^ sin V sin I") 



E, • 1c 



52 



(41) 



I- (42) 



sin i 2" = 



R, 

 tan 12= -^ 



R,R„ (cos V cos I" cos CD -|" ■^3" sin I' sin I") — a? 



tan 1 2" 



tan i 2' 



sin A' 

 cos I" 

 sin A" 



R,_ 



R^ cos I' 

 sin A' 



R„ • Tc 



1 cot w sin A' + sin V cos A' ' "^ 



sin 0) cos V 



1 cot £0 sin A!' -}- sin I" cos A!' 



sin (w ■ cos Z" 



\ (43) 



J 



tan i 2" = 



tan i 2' = 



cos V sin CD 

 sin ^" 



(7,2 a^ \ -^ 

 cos Z cos I cos CD -1 sin V sin Z" — | 



. ( cos V cos I" cos CD + — — sin V sin Z" — . — "^ — \ 



cos Z' sin CD 



cos Z' jg^ sin A' cos ^" -}- R^^ sin ^" cos ^' 

 sin A!' R^ sin Z" -\- R^^ sin Z' 



, 1^ 2" cos Z'' i?/ sin A! cos ^" -f R^, sin ^" cos ^' 



sin A' R, sin Z" -j- i2„ sin Z' 



And from these we immediately obtain the following relations : — 



cos 12' R„ cos I" sin A" 



R, cos Z' sin ^' 

 ^„ 



cos V sin ^' 



K44) 



(45) 



cos 



1 



2" 



sin 



1 



2' 



sin 



1 



2" 



tan 



1 



2' 



tan i 2 ' 



cos I" sin A" 



(46) 



