GEODESY. Xlv 



In order to express the analytical relations between the different latitudes let 



D </> = the astronomical latitude, 



\\i = the geocentric latitude, 

 6 = the reduced latitude. 



Then, referring to equations (i) and (2) under 

 section 4 above, and to Fig. i, it appears that 



dy 



tan \\) 



y 



1 



X 



tan e = ^. 

 i>x 



Hence 



tan if/ = —2- tan cf) = (i — e^) tan (f>, 



tan 6 



(i - e-y tan <f, = (i - e")-^ tan if/. 



<f> — if/ = m sin 2 (f) — m- sin 4 <^ -j- . 

 (fi — = n sin 2 (p — h n^ sin 4 c^ -|- 



For the adopted spheroid 



and 



log (i — e-) = 9.9970504, 



(f> — if/ (in seconds) = 7oo."44 sin 2 4> — i."ig sin 4 </>, 

 — 6 (in seconds) = 35o."22 sin 2 (f> — o."3o sin 4 (f>. 



6. Radii of Curvature. 



p„, = radius of curvature of meridian section of spheroid at any point whose 



latitude is 4> =.P0, Fig. i, 

 p„ = radius of curvature of normal section perpendicular to the meridian at 



the same point = J^Q, Fig. i, 

 Pa = radius of curvature of normal section making angle a with the meridian 



at same point. 



p,„ = ^ (i — ^2) (i — e^ sin^cfy)-'', 

 p^ = a (i —e^ sin^ <f))-\ 



I 



Pa 



COS" a 



+ 



sin^ a 



Pn 



= - (i -| ^^^ cos"^ ^ cos^ a) {\ — e" sin'^ <^)*. 



log (i — e^ sin'^ ^)--- = -(- log (i + «) 



— /A ;z cos 2</) 

 -\- \ \x.n- cos 4(^ 



— \ \x. n^ cos 6^ 



+ : . . . 



/x = modulus of common logarithms and ;/ is same as in section 3. For the 

 adopted spheroid — 



