THE OCEAN ON THE PLUMB-LINE IN INDIA. 795 
mass of the earth, and these three disturbing forces W, M, and M, which I have been 
calculating Let be the angular velocity of the earth round its axis, « the latitude of 
any point of the surface r its distance from the earth’s centre, « the semi-axis major of 
the mean mendran. Then a,h.r and is the centrifugal force parallel to ai and «, z 
emg the eaiths axis, and ^»,Vcos'‘f is the corresponding part of the above equation, 
e V be the potential for the earth’s mass, supposed a perfect spheroid of equilibrium 
diffenng httle from a sphere; E the earth’s mass. Then V differs fi-om ® only by a 
small variable quantity depending upon the ellipticity : let it equal 7(1 + U). Substi- 
tutmg these and the three distuibing forces, the equation of the surface now becomes 
const =-(1 +U)+ cos’ 
between the several limits, as already explained, or 
const =®(l+U)+ia;Vcos^^+L.^; 
const =-“(l + U+^*^)+L. 
E‘ 
But 
1 = 7 ( 1— 2 sin" ^)=^(l — s+s cos" 
K the equation to the sui-face, s being the ellipticity, when there is no disturbing force. 
Hence the equation in the present case is 
1=;(1 — 2sin"^)+L.|^; 
f=«(l — £sin"^)+L, as 
Let ^ be the angle through which the normal to the surface is thrown backwards. 
JNow the tangent of the angle between r and the normal 
— 1 • on . 1 
, 1 dL 
and ds=rdi=ad6 being an element of the arc of the surface, the Elevation of the 
^rface of the sea in passing northwards =^<ls=L, between the limits X=0 and x=n. 
ence by addmg together the values deduced in pars. 15, 17, 19 to obtain L, we have 
Llevation of sea-level at Karachi above that of Cape Comorin 
=:448-25-f 15-88 + 50-44=514-57 feet. 
This will alter all heights which depend upon Karachi, but is not of sufEcient import- 
ance to anect the horizontal measures of the Survey. 
21. This calculation shows, perhaps, the greatest extent to which the sea-level along 
5m2 
