of Longitude from the Latitudes and Azimuths of two Statioyis. 25 



■r« sin a sin /5' •, sin a sin /3' , j [. 



But, -^ — = r> and -^ — > = ; therefore, 



' sm jK COS X ' sin ft.' cos A. 



Sin (u. — m) — e^ cos A. sin ?« A'. -: — -r, , 



^~ ' sm ^ 



Sin (m' — uJ) = e- cos x' sin ?»' A . . „. . 



\ ~ ' sin /5' 



If we examine the equations (x), at p. 243 of this Journal for 

 October last, it will readily appear that 



Cos A sin m A' = cos h' sin m' A . 

 In reality this equation is not rigorously exact in the case of 

 two distant stations, since, in obtaining the equations x, it has 

 been supposed that a = R in the value of cos (p. If we go 

 back to the principle of the investigation, we shall find this 

 rigorous formula, viz. 



Cos X sin w A' cos tp = cos x' sin ?«' A cos <p', 

 f and <p' being the angles of depression of the chord joining 

 the stations below the respective horizons. We therefore ob- 

 tain this exact equation, viz. 



Sin {(J'—m) cos (p = sin {m'— /x') cos 9' ; 



which shows that the two sines multiplied by cos f and cos <p' 

 fail of being exactly equal, because the common chord is not 

 equally depressed below the two horizons. In order to judge 

 of this point with precision, I have investigated the depression 

 <p at the station of which the latitude is A, as follows, 



Sin ^ = -^ (1 - -J sin^ A + e^ x^) ; 



and, as we have a similar expression for sin (p', we obtain these 

 values, 



^ / ■y'^ ay" sin'2 X „ y- x''- \i 



Cos4>=: (1- ^ + e'^--''i^r 

 Cos?) - (^1-^^ +e -^^ ^ 2a^ y 



From these expressions it appears that cos <{> and cos <$>' ap- 

 proach so near to equality that we may safely reckon sin (jx — ?«) 

 and sin (;n'— /x'), which are always small quantities, equal to 

 one another, in any position of the two stations on the surface 

 of the globe. Thus we have generally, 



Sin [}t. — m) = sin (?«'— jx'), 



m + ?«' = ju, + ju.'. 



It is to be observed that the azimuths of which we are speaking, 



have nothing to do with any geodetical line on the surface of 



the spheroid, or with any spheroidical triangle. The azimuth 



New Series. Vol. 5. No. 25. Jan. 1820. E at 



