563 
figure of the earth. 
(t— r) cos20 ~(S + 4) cos 4 0j, and the radius of cur- 
vature = a^i+4 + ^~4 — ^ c os2«-(^f + ^a)cos 4« 
Let this = y. It was found that f* a = X 2 — 4 e . A 3 — X 4 , or 
sin* 9 = sin* l — 4 e . sin* / . cos 2 1 ; from which cos 2 5 = 
cos 2 / + e — e cos 4 l : hence 7 = a h + T+TS-T- 
-^-COS 2 l + — T A ) cos 4 • Now since / is the angle 
made by the normal with a fixed line, the increment of the* 
arc, corresponding to the increment <57 of the latitude, is 
ultimately y $ l, and the arc = J y ; integrating therefore 
from l — L to / = L', and making a ( 1 -J- ~ 
the arc included between the latitudes L and L' = 
R ^L' — L j— ~J . (sin 2 L' — sin 2 L) 
+ (Hf — Hr) • ( s ' n 4 L' — sin 4 L ) j • 
(21.) An arc of a parallel in latitude /, comprehending the 
difference of longitude D, is most easily obtained by observ- 
ing that the decrement of its radius upon increasing l is equal 
to sin / x the increment of the arc = sin / x y $ /, and there- 
fore the radius = -ft* sin / ; the integral being corrected 
so as to become o when l = 90°. We must therefore integrate 
R sin l 1 1 — J 
1 3 e 
3 e *\ 
l ^ 
4 / 
cos 2 
l + cos 
This gives the radius of the parallel = R||i + ~ 
cos/ — (f + H — coss/ + (it - cos 5 /} ; and 
therefore the arc corresponding to the difference of longitude 
