32 
ARCHDEACON PRATT ON THE EFFECT OF LOCAL 
Let X be the amplitude of any one of these arcs^ expressed in seconds of a degree ; 
jM- the latitude of its middle point ; 
a the number of feet in the arc ; 
a the semi-axis major of the ellipse to which the arc belongs ; 
e the ellipticity of the arc. 
Then, by the Integral Calculus, 
a f /I , 3 sin A ^ \ . 1 o / . - sin A \1 
_=x|1-£(^2 + 2 cosXcos 
neglecting the cube and higher powers of s ; 
af, . /I . 3 sin A ^ \ 
,„/l, 3 sinA sin A xl 
+ ^ ( 2 + 2 “IT ) “l6^ + ^^ ~A~ ^ f- 
The coefficient of in this may be written thus : 
1 r 3 , 15 sin A , ^ sin A ^ ^ /sin A\^/ 5 A \ , „ 1 
4(4+4 — cos X +6 —cos 2^+9 (—) (l-g i^)cos'‘ 2 f.|. 
Since tan?i is always greater than X, the coefficient of cos^ 2 [jt, is always positive ; and 
therefore the greatest value which the coefficient of can attain is when cos 2/^0=! ; 
in which case it equals 
1 /3 , 15 sin A , ^ sin A\ , 9/sinA\^/, 5 A \ 
iU+T — )+4(— ) 
9 /sin A\ ^ , 
or 3 — g X^, neglecting higher powers of X, which is alv/ays small. As the ellipticity is 
a very small fraction, and this coefficient is not a large number, the square of s may 
be neglected without any perceptible error. Hence 
^—a] ^+H2 + 
3 sin A 
COS 
1 , 3 sin A 
“a” 
E = COS 2/a, 
X = -( 1 +£.E). 
or if we make 
then 
Let X1X2X3 ciiCi2«3 EiEaEg be the values of X, a, E for the several arcs. 
Hence 
^.^^=H-£(E-E3), 
or 
e= 
j 
Ao a 
2 “1 
Put 
i=A 
u 
Ei-E^ 
A, 
5-1 
E1-E2 
then 
