GEODESIC INVESTIGATION'S. 161 



That tlie formula is erroneous is easily seen : for, independent of tlie over- 

 sight committed in assuming sin 2 A' = sin 2 A", it is obvious that when 

 I' = I", we must have A ^ 0, no matter what may be the common value of 

 I' and I". This is contrary to what the formula intimates in such case. In 

 fact, the expression is not even a rough approximate to the angle A. And it 

 may be further observed that the formulae given in that work, as pertaining to 

 a true spheroidal triangle, are arrived at by making use of the erroneous 

 expression for A, so that they too are erroneous. 



Expressions for the chord k of the geodesic arc — 



7c~ = (E, cos Vy- -f {R„ cos l"f — 2 i?, E„ cos I' cos I" cos a» -f /AV"! 



. {R, sin V — R„ sin l"Y )■ (51) 



, D COS I" sin 00 y, cos I' sin oi 



K = K,, . . = U . . „ I 



sm A cos i2 sin A cos J 2 



Expressions for the radius p^ of mean curvature of the arc — 



_i^ cosHi2--i-S") 1 



^ -sinHi^' + iS") I 



sin a> {R^R,! cos V cos l"\ if 



sin 2 * \ sin A! sin A' / J 

 closely approximate only. 



Expressions for the length of the geodesic arc between the stations — ■ 



5 = \-k (i2' -f iS'O 



(^r - i2") -l 



sin i (i2' 4- 4-2") , 



" ^ ^^ ' \ (53) 



R,R,, cos I" cosZ'" 



sin 2 V sin A' shiA" /s™ '^ J 

 closely approximate only. 



g^° The complete expressions for the angles of depression of the chord, 

 the angle between the normals, the angle between the normal-chordal planes, 

 and the length of the chord, cannot fail to be of great jDvactical use in ques- 

 tions of Greodesy. It would be an easy matter to give much simpler formulae 

 of an approximate kind, such as are usually supplied in treatises on Practical 

 Geodesy, and in accounts of trigorwmetrical siirvei/s ; but it may be proper to 

 observe that the application of approximate formulee should be a matter of 

 necessity ; for when extensively used in geodesical surveying (as they are 

 certain to be by persons who wish to avoid extensive calculations) they give a 

 low inaccurate character to the work, and leave it entirely impossible to 

 deduce anything reliable therefrom as to the figure of the earth. 



Problem 1. 



Griven the latitudes V, I", of two stations S', 8", and their difference of 

 longitude co ; to find the azimuths A' , A" , at the stations ; the circular measure 

 2, the chord k, and length s of the geodesic arc between the stations, &c. 



We can find the arcs PD^^, RD,, or L^,, L^, from — 



R,, . sin I" 



coti/=(l — e-) tan Z' + e- . ^' ' — y 



R^ 

 R,. 



R, 



cot L^,= {i — e-) tanl -fe- • =p- 



