GEODESY. 



xlvii 



The relations (2) will answer most practical purposes when A(^ does not exceed 

 5°. A comparison with the precise formula (3) below shows in fact that the error 



of (2) is very nearly 



\ r A</)2 cos 2</) . A J/, 



which vanishes for 4> = 45°, and which for A<^ = 5° is at most ttvAtftj ^^^> or 

 about II feet. 



Numerical example. Suppose — 



<^, = 37°29'48."i7, 

 <^i = 35° 48' 29."89. 

 Then 



«^ = K<^2 + <^i) = 36° 39' o9."o3, 



A(/) = </>2 — (^1 = 1° 41' i8."28, 



= 6o78."28. 



From the first of (2) 



cons't. log 4.6855749 — 10 



Table 10, log p;„ 7.3193112 



log A</) 3.78 37807 



AJ/= 614705 feet, log AJ/ 5.7886668 



The values of A J/ for intervals of 10", 20" . . . 60", and for 10', 20' . . . 60' are 

 given in Table 17 for each degree of latitude from 0° to 90°. 



For precise computation of long meridional arcs the following formula is ade- 

 quate : — 



AiT/= ^0 ^^ — A\ cos 2^ sin A<^ 

 -j- A-i cos 4^ sin 2A<;^ 



— Az cos 6(/) sin 3A<^ (3) 



-f- A^ cos 8^ sin ^\<^ 



In this, AJ/j <^, and A<^ have the same meanings as above, and ^oj ^i> • • • are 

 functions of a and ^ or of «: and ;/. 

 Thus, in terms of a and n, 



^0 = « (i + n)-^ (i + ^ «2 _|_ ^ ,,4 _|. _ )^ 

 A^ = 3a (i + n)-^ {n — \n^ — ...\ 

 A.= ^ a{x-\- n)-^ {it" — !«*-...), 

 A. = il ^ (i + n)-^ («8 _ . . . )^ 



Introducing the adopted values of a and n, these constants become — 



log. 

 ^0 = 20 890 606 feet, 7-3i995io> 

 Ai= 106 41 1 feet, 5.0269880, 

 A-2= 113 feet, 2.0528, 



A3= 0.15 feet, 9.174 — 10. 



