xlviii 



GEODESY. 



It appears, therefore, that the first three terms of (3) will give A J/ with an 

 accuracy considerably surpassing that of the constant Aq. In the use of (3) it will 

 generally be most convenient to express A^ in degrees, and in this case Aq must 

 be divided by the number of degrees in the radius, viz. : 57.2957795 [i. 7581226]. 

 Applying this value and writing the logarithms of Aq, A^, etc., in rectangular 

 brackets in place of Aq, Ax, etc., (3) becomes 



Meridional distance Ayl/ in feet. 



AJ/= [5.5618284] A^ (in degrees) 

 — [5.0269880] cos 2^ sin A<^ 

 -f- [2.0528] cos 4^ sin 2A(^ 



(4) 



2^ = P2 + ^1. 



A^ = ^, — 01, 



01. 02 =^ ^"d latitudes of arc. 



Formula (4) will sufBce for the calculation of any portion or the whole of a 

 quadrant. The length of a quadrant is the value of the first term of (4) when 

 ^ =^ 45° and A^ = 90°, since all of the remaining terms vanish. 



Nu7nerical examples. 



Then 



Suppose 



<^i = 0° and (fio = 45^ 



2<^ = 45°' 

 A<^ = 45°. 



cons't 

 45 



log. 



5.5618284 

 1-6532125 



ist term -|- 16 407 443 feet ist term 7.2150409 



cos 2^ 9.8494850 



sin A<j!> 9.8494850 



cons't 5.0269880 



10 

 10 



2d term — 53205.7 feet 2d term 4.7259580 



The third term of the series vanishes by reason of the factor cos 4 </> = cos 90° 

 = o. The sum of the first two terms, or length of a meridional arc from the 

 equator to the parallel of 45°, is 16 354 237 feet. 



2°. Suppose 

 Then 



<;J)i = 45° and ^2 = 90°. 



2<^= 135°' 

 A</>= 45°- 



The numerical values of the terms will be the same as in the previous example, 

 but the sign of the second term will hep/us. Hence the length of the meridional 

 arc between the parallel of 45° and the adjacent pole is 16 460 649 feet. The 

 sum of these two computed distances, or the length of a quadrant, is 32 814886 

 feet. 



