Bowdifch on the Ohlatencss of the Earth. 



47 



IS not of revolution. The term of r thus neglected, renders* tbe 

 expressions of the length of an arch of the meridian, and that of 

 the perpendicular to the meridian, also the azimuth angle » 



gi\ 



en by La Place in Page i25, vol. 3, erroneous. The corrected 

 values are found in the following manner. 



i 



Putting the expressions (3) and (5) equal to each other we get 



sin. ^(,^ {1 + h. COS. S (f + ^)] + h. cos. (2 p -f Q. (6) 



in which the last term was neglected by La Place, and if we put 



this term equal to u^ we shall have 



u 



f 



sin. ■^\ [i + h. cos. S (9+e)}+w^^, 



u 



// 



h, cos. 2(f 4 C). 



(7) 



This value of u''' gives [-A-) = 0, so that the expression of the 

 length of an arch of the meridian s given in Yol. 2, Page 115^ uo- 



der the form 



s 



t-h at. 



i»/+{; 









fd^uf 



+ &C. 



(S) 



will be increased by the term at u'^; observing that % represents 



the difference of the latitudes of the two extreme points of the 



arcs 5, and u' is the value of m' when s = 0. 



Vol. 2^ La Place has deduced the following expressiou of s^ 



namely, 



In Page i S5, 



s 



i 





{1+A.cos, 2(f + ^}}. [1 +3. cos. S^|/— 3e.sin.2 4/} (9) 



To which we must add the term a 

 which means its correct value will be 



h, COS. S (f + C) by 



s 



{i + /i.cos.3(9+^)}.[l+3cos.24.-8i.sin.24/|-fctt,^.cos.2f^+e), 



r 



If the earth is s^upposeJ not to be a spheroid of revolution^ and 

 an arch be measured upon its surface so that the direction of its 

 first part is parallel to the celestial meridian, and its last part 

 forms the angle »• with the celestial meridian, corresponding to 



t 



■^ 



_iivr 



^V 



