liv GEODESY. 



Then from Fig. 3 it appears that 



X :^ I sin a, 

 y ^ 2 I sin- \a. 



But from Figs. 2 and 3, 



I ■=^ p^ cot <^, 



la.^ r W = p„ AX cos ^, 

 whence 



a = AA. sin ^. 



Hence, in terms of known quantities there result 



X = p„ cot <ji sin (A\ sin <^), 

 ji' = 2 p„ cot (^ sin^ \ (A\ sin (;6). 



Numerical exaviple. — Suppose ^ = 40° and AX :^ 25° = 90000". 



Then 



log 90000" = 4.9542425, 



log sin 40° = 9.8080675 — 10, 



log 5785o-"88 = 4.7623100 ; 



AX sin, (^ = 16° 04' io."88, 



\ (AX sin </,) = 8° 03' o5."44. 



log. log. 



sin (AX sin ^) 9.4421760 — 10 sin \ (AX sin ^) 9.1454305 — 10 



cot <\> 0.0761865 sin \ (AX sin ^) 9.1454305 — 10 



p„. Table II 7.3212956 cot ^ 0.0761865 



/)„, Table 11 7.3212956 



2 0.3010300 



X 6.8396581 y 5-9893731 



a- = 6 912 865 feet y = 975 828 feet. 



The equations (i) are exact expressions for the co-ordinates. But when 

 AX is small, one may use the first terms in the expansions of sin (AX sin <^) and 

 sin^ ^(AX sin ^) and reach results of a much simpler form. 



Thus, 



sin (AX sin ^) = AX sin </> — J(AX sin <^)'' -|- , . . , 

 sin^ i(A<^ sin </>) = J(AX sin </.)- - 3^g(AX sin «^)* + . . . ; 



whence, to terms of the second order. 



X = p„ AX cos ^ [i — i(AX sin c/))^], 



y = i Pn (AX)- sin 20 [1 - ^'o(AX sin^)^]. 



(2) 



If the terms of the second order in these equations be neglected, the value of 

 X will be too great by an amount somewhat less than J(AX sin 0)^ . x, and the 

 value of y will be too great by an amount somewhat less than x^(AX sin </))^ . y. 

 An idea of the magnitudes of these fractions of x and y may be gained from the 

 following table, which gives the values of J(AX sin c^)- for a few values of the 

 arguments AX and <^. 



