lxiV INTRODUCTION. 



LENGTH OF A DEGREE OF THE MERIDIAN AND OF ANY PARALLEL. 



The dimensions of the earth used in computing lengths of the meridian 

 and of parallels of latitude are those of Clarke's spheroid of 1866. 1 This 

 spheroid undoubtedly represents very closely the true size and shape of the 

 earth, and is the one to which nearly all geodetic work in the United States 

 is now referred. 



The values of the constants are as follows: 



a, semi-major axis = 20926062 feet; log a = 7.3206875. 



b, semi-minor axis = 20855121 feet; log b = 7.3192127. 



2 1-2 



e 2 = - — =— = 0.00676866 ; log e-= 7.8305030 - 10. 



a 1 



With these values for the figure of the earth, the formula for comput- 

 ing any portion of a quadrant of the meridian is 



Meridional distance in feet = [5.5618284] A</> (in degrees), 



— [5.0269880] cos 2 4> sin A<£, 

 + [2.0528] cos 4 cf> sin 2 A<f>, 

 in which 20 = fa + fa, A0 = fa — fa; fa, fa =end latitudes of arc. 



For the length of 1 degree, the formula becomes: 



I degree of the meridian, in feet = 364609.9 — 1 857.1 cos 2 4> + 3-94 cos 4 0« 

 The length of the parallel is given by the equation 



1 degree of the parallel at latitude <j>, in feet = 



36553848 cos 4> - 310.17 cos 3 <j> + 0.39 cos 5 fa 



Table 85. -Length of one degree of the meridian at different latitudes. 



This gives for every degree of latitude the length of one degree of the 

 meridian in statute miles to three decimals, in meters to one decimal, and 

 in geographic miles to three decimals — the geographic mile being here de- 

 fined to be one minute of arc on the equator. The values in meters are com- 

 puted from the relation: 1 meter = 39.3700 inches. The tabular values rep- 

 resent the length of an arc of one degree, the middle of which is situated 

 at the corresponding latitude. For example, the length of an arc of one 

 degree of the meridian, whose end latitudes are 29 30' and 30 30', is 

 68.879 statute miles. 



Table 86. Length of one degree of the parallel at different latitudes. 

 This table is similar to Table 85. 



1 Comparisons of Standards of Length, made at the Ordnance Survey Office, South- 

 ampton, England, by Capt. A. R. Clarke, R. E., 1866. 



