ASTRONOMY. 45 



58. In an ellipse where the eccentricity is 

 small, or where a and b differ but by a small 

 quantity c, this general formula may be reduced 

 to more simplicity, by extracting the root of the 

 denominator, and rejecting the powers of b 

 greater than the first ; we have then 



mD = a(l ?-?+!? sin A*). 



a. This value of m D may be changed into another, more 

 convenient in calculation, by substituting for sin A* 



its value, 1 cos 2 A , from which is obtained 



b. At the Equator, A = 0, and cos 2 A = 1 ; so that 



mD = a (1 -- -)=0 2c. 



c. At the Pole, A = 90, 2 A= 180? ; and since 



cos 180= 1, wD = 0+c. 



The degree of the meridian at the equator, is there- 

 fore to the degree at the pole as a 2 c to 

 -f c. 



H. In the parallel of 45, 2 A ,-= 90, and cos 2 A = ; 



therefore m D = a -. 



The 



