REPRESENTING MEASURED ARCS OF THE MERIDIAN. 39.'? 



be compared is as follows. If the two semi-axes of an elliptic 



spheroid of revolution be designated by a and h, and if 



a — h 



7 = n. 



a + b ' 



then the length of the arc of the meridian between the equator 



and the latitude <$ is the integral 



s = a{\- e')J - 



V {l—e^ sin <f>-) ' 

 or, developed, 



s = a{\— nf (1 + w) N { cf — a sin 2 4) + i a' sin 4 (f> 

 — ^«" sin 6 4. + . . .} 

 w^herein 



N = 1 + (I) V- + (|a_^) V + . ., 



,, 3 3.5 3„ 3.5.7 3.5, 



^'^ = y^ + 274 • 2*^+27476 • 274^^^ + •- 



„ , 3 . 5 „ 3.5.7 3 , 



N a = n^ 4- ;; . — «^ + . . ., 



2.4 2.4.6 2 ' 



^« =27476^+2747^78- 2'^' + -- 

 and so forth. 

 If we desire to make this expression dependent on the length 

 of the mean degree of the meridian [g) instead of on the greater 

 semiaxis, we must put ^ = 1 80°, -whence we obtain 



180^ = a{\- nf (1 + n) N tt, 

 and thus 



s = ^ {'^ —a sin 2 4) + I «' sin 4 4; — 4 a" sin 6 ^ + . .}. 



TT 



Hence follows the expression for the distance between the pa- 

 rallels corresponding to the latitudes <p and 4', 



^ — s = ^ { 4' — 4 — 2 « sin (<!>' — 4)) cos (4' + 4) 



+ I a' sin 2 (4' — 4) cos 2 (4' + 4). 

 If for brevity we write I for the amplitude 4' — 4, and 2 L 

 for the sum of the latitudes 4' and 4, and understand by «; the 



number = 206 2 64"* 8, and if we express / in seconds, 



we thence obtain 



3600 , , , , 



is' — s)=l — 2wa. sin / cos 2 L + w «' sin 2 / cos 4 L 



9 



— f CO a" sin 3 / cos 6 L + . . . 



VOL. II. PART VII. 2 D 



