Colonel A. R. Clarke on the Figure of the Earth, 89 



The lengths of one degree in and perpendicular to the meri- 

 dian, viz. 8, 8', are 



8 = 364609-12 - 1866*72 cos 20 + 3*98 cos 40, 



8' = 365854-72 - 624-40 cos 20 + 0'80 cos 40. 



Also the following : — 



° p sin 1" 

 = 7-994477820 + -002223606 cos 20- -000001897 cos 40, 



lo Vsinl" 



= 7-992994150 + -000741202 cos 20- -000000632 cos 40. 



Having seen that the surface of India cannot be represented 

 by a spheroid of revolution, it is necessary now to inquire 

 what ellipsoid best represents all the observations as the figure 

 of the earth. On this hypothesis, the equator being no longer 

 a circle, the ellipticity of a meridian is not a constant, but is a 

 function of the longitude — say I, from Greenwich. We have 

 consequently to replace our previous v by v + ic cos 21 + z sin 21; 

 and the longitude V of the greater semiaxis of the equator will 

 be given by the equation w sin 2V —z cos 2Z / =0. But this 

 substitution cannot be made in the longitude-equations — they 

 no longer hold good, having been formed on the distinct sup- 

 position of the earth being a surface of revolution, and they 

 must now be put aside. If the earth should be found to be 

 really ellipsoidal, this circumstance will involve a considerable 

 increase of the labours of the geodetic computer. The " meri- 

 dian " on an ellipsoid is somewhat vague. If it be taken as the 

 locus of points of constant longitude w, its equation in combi- 

 nation with that of the ellipsoid is 



b 2 xsmco — a 2 ycosco = (1) 



But it may also be defined as a line on the ellipsoid whose 

 direction is always north and south. Suppose that a point 



X 1J z 



on the surface of the ellipsoid -^ + p + — 2 = 1 moves always 



towards a given fixed point x'y'z', and let it be required to 

 determine the nature of the curve traced by the moving point. 

 Two consecutive points on the curve having coordinates cc, y, z, 

 sc + dx, y + dy, z + dz give the condition 



^- 2 dx+-l 2 dy+~dz = (2) 



The equation of a plane passing through x, y, z and a/, y f , z' is 

 A(*'-^) + B0/-y) + C(c'-*) = 0. 



