586 
AECHDEACON PEATT ON THE INDIAN AEC OE IMEEIDIAN. 
Also tan ^=(1 — 2g)tan Z, 
ssm2/f, ^'=X— 22sinXcos2w; 
1— cos(^— ^')=1— cosX— 2£sin^Xcos2m=2 sin®^x|l— 2£(1+ cos cos 2m| ; 
sin^^x|l— 2£(1+ cosX) cos2m— £(sin^Z+ sin^/')| 
=4^z^ sin^“x|l— g{l+(2+ cos X) cos2m}| ; 
sm^X=^|l+|g{l+(2+ cosX)cos2TO}j; 
V 4a®- 
| = sin-'|^+||l + (2+ cosX)cos2w| 
= sin“^^+||l+(24- cos X) cos 2mjtan 
Hence s=«^— ^g^X— |«g sin X cos 2m 
= a(2 — g) sin~ -f-«g| 1 + (2 + cos X) cos 2m Itan^ — | ag sin X cos 2m 
2a 
2 2' 
= (a-i-6) sin"*^4-(c^— cosX) cos2m|tan^X. 
Taking the variation with respect to the axes, 
^s=(§( 2+§5) sin"* 
c a + Z> cSa 
cosX) cos2mjtan^. 
Since the terms are small, we may use approximate values ; 
Bs=(^a+^5)^X— 2 tan^X.^«+(^a— cosX) cos 2m|tan^X 
^X— tan^X j +(^a— ^5)5tan^X(l— cos2X) cos 2m. 
Applying this to the case in hand, we have X=ll° 27' 33" and 2m=47° 34' 
These lead to 
gs=-0-0003350(^a+^^i)+0-0006747(^a-^^) 
= 0-0003397^a-0-0010097^5. 
8. Secondly. The distance between the arcs at the mid-latitude. 
The equation to the local ellipse is 
{x-uf [y-^f 
a® 
Neglecting small quantities of the second order, 
x^-\-^'^=zcd-]-2ux-{-2(3y—2s(a‘—x^), 
f^=a^-\-2aa cos d-{-2a^ sin 0—2ah sin^ 0, 
r =a -\-cc cos 0-\-(3 sin 0 — as sin^ 0 . 
