Colonel A. R. Clarke on the Figure of the Earth. 91 



SN, and SP. Let the semiaxes of the equator be expressed 

 bv the relations 



a 2 = k 2 (l + i); b 2 =k 2 (l-i), 



where i is a very small quantity whose square is to be neg- 

 lected. Then the coordinates x, y, z of any point, as S, are 

 proportional to 



c 2 

 (1 + 1) cos (o : (1 — i) sin co : ptan </>. 



Substitute these in (6), (7), (8), and we get finally the fol- 

 lowing results for the angles between the lines in question : — 



MSP=£ — isin<f>sin2<»(l+ 2 . ., , C t 72 3-7 ), 



2 r \ c 2 sm 2 <p + k 2 cos 2 <£/ 



xrq p _ ^ • c 2 sin (/> sin 2co 



~ 2 c 2 sin 2 </> + k 2 cos 2 <£ 



MSN=isin0sin2e». 



In the figure of the earth, as determined in the paper in the 

 ' Memoirs of the Royal Astronomical Society ' for 1860, there 

 is a difference of a mile between the greatest and least radii of 

 the equator. Although this seems but a small departure from 

 the form of a circle, yet i — 52 // '3o (in parts of radius unity), 

 and the angles expressed above become somewhat large quan- 

 tities. Supposing S to be on a meridian midway between the 

 greatest and least radii of the equator, the angle between the 

 " meridian " and the " north line" is 52 //, 33 sin cf> ; and the 

 defect of MSP from a right angle is about double this quan- 

 tity. So large an angle as this should be detected by firstrate 

 geodetic observations, though it would require a somewhat 

 long measurement of meridian and parallel. It is to be re- 

 membered that, SM, SN being directed towards the north, and 

 SP towards the minor axis of the equator, SM lies between 

 SP and SK 



And in an ellipsoidal earth the direction of the principal 

 sections of the surface (that is, of maximum and minimum 

 curvature) are no longer coincident with meridians, north lines, 

 or parallels. Supposing that S is not in a very high latitude, 

 one of the lines of curvature, as SR through S, will lie some- 

 where in the direction of SP, and the second line of curvature 

 will be perpendicular to SR. It may be shown that the angle 



IT C 



RSN= 7T — i sin 2o) sin 6 sec 2 6 -rr, ,> 



an expression which does not hold in high latitudes; for in 

 the vicinity of the umbilics, the lines of curvature are approxi- 

 mately confocal conies having the umbilics as foci. The defect 



