Colonel A, R. Clarke on the Figure oj the Earth. 87 



curve in the shape of its intrinsic equation : and the absolute 

 direction of the curve with reference to the polar axis is given 

 by observing that, at its southern extremity, the actual direc- 

 tion of the surface of the sea makes an angle of 1 ,/# 61 with the 

 curve. Xow. the observed latitude of the southern point being 

 8° 12' 10" -4A. the direction of the normal to our curve (E') 

 at the same point makes the angle 8° 12' 12 //, 05 with the plane 

 of the equator, which determines the curve as to its absolute 

 direction. So also, on referring the Indian meridian to the 

 ellipse (E). determined above as representing the mean figure 

 of the earth, this ellipse at the southern point of the Indian arc 

 has its normal inclined to the equator at an angle equal to 

 observed latitude — 3"-14 ; or <S C 12' 7 "-30. TVe can now trace 

 the difference of the forms of the curves (E, E') by making 

 them coincide at the southern point of the arc. The selection 

 for this purpose of the southern point is quite arbitrary ; any 

 other station would have done equally well. Multiply the 

 expression for p. the radius of curvature, by — sin (p do and 

 then by cos cf> d(p, and integrate : thus we get the following- 

 values of the coordinates of the curve (E') in the meridian 

 plane, parallel and perpendicular to the equator : — 



a/=(A'-B') cos + i(B r -C0 co5 3 

 ^(A'+EOsin^+i^ + COsin^ 



when H and K are disposable constants. The corresponding- 

 coordinates of the ellipse (E) may also be written in the form 



,r=(A-B)coi<j> + i(B-C)coi3ct> + lC cos5c/0 (J , 



Z /=(A + B)sin(/) + ^(B + C)sin3(/) + iC / sin5(/).J 



The values of H and K are now to be determined by put- 

 ting c/) = 8° 12' 12 //# 05 in the expressions for :■■:■' and y' . and 

 (f>=8° 12' l' f -?>0 in those for x and y; then putting x = j j , 



y—y'- 



The normal distance between the curves (E ; E') in latitude 

 <f) is £==(*/— a?) cos cb + Q/— y) sin (/> : this expresses the dis- 

 tance by which a point in (E') is further from the centre of 

 the earth than the corresponding point of (E). Put A! — A= E, 

 B / -B = F, C'-C = G; then 



£=E-§ F cos 2^-^ G cos 4</> + H cos + K sin 0. 



The following Table shows, according to this formula , the 

 departure of the curve best representing the Indian meridian 

 from that best representing the earth as a whole. I add also 

 similar quantities for the Russian and Anglo-French arcs ; the 

 only difference is that, in the case of these last arcs, the local 



</> + iC'cos5c/> + H.") E ,. 



