PREFACE TO PART II. xxi 



The above theory assumes that the integration is taken over the whole 

 surface of the Earth, and that the observations are uniformly distributed 

 over the Earth's surface, otherwise the coefficients of the neglected terms 

 on the left-hand side of these equations will not vanish, and each equation 

 may have other terms which are too important to be neglected, and so 

 it will not be so easy to separate the magnetic constants from one another. 



Section VI. contains the Theory of Terrestrial Magnetism for the Earth 

 regarded as a spheroid and gives the theory of the determination of the 

 magnetic constants. Let r, 6', X be the polar coordinates of a point on 

 the spheroidal surface referred to the Earth's centre as origin and axis of 

 figure as initial line ; let 9 be the geographical colatitude (the angle which 

 the normal makes with the axis) and let /A = cos 6 and // = cos 9'. 



The angle of the vertical i// = 6' 9. 



The values of the sines and cosines of these angles for values of 9 

 differing by 1 from to 90 have been computed, the eccentricity c of 

 the elliptic section in the plane of the meridian being derived from Bessel's 

 dimensions of the Earth as given in Encke's tables in the Berliner Jahrbuch, 

 1852. 



The expressions for the magnetic potential and for the magnetic forces 

 X, Y, and Z, in terms of the Gaussian magnetic constants g, h, will be of 

 the same form as those given above for the sphere. 



Where X is the total force towards the north perpendicular to the 

 Earth's radius, 1"" the total force perpendicular to the geographical meridian 



dV 



towards the west, Z the force towards the Earth's centre, where X= ., 



rau 



Y=- 



dV dV 



TT, . r- , and Z = -, (east longitudes being considered positive). 

 r sin 9' d\ dr 



If X' be the horizontal force in the meridian towards the north, 



Y' the horizontal force perpendicular to the meridian towards the 

 west, 



Z' the vertical force on the spheroidal surface of the Earth, 

 then X' = Xcos\jj + Z sin \jj, 



Y'=Y, 

 Z' = X sin i// + Z cos i//. 



