118 REPORT— 1898. 



and from the values of Z we derive 



r(n + l)c.„-»i,/3„ J r (H;r)2 dfx = JH;;' z^ d,x 



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. 



Let us now take into account the spheroidal figure of the Earth. Let 

 r, 6', X be the polar co-ordinates of a point on the spheroidal surface 

 referred to the Earth's centre as origin and axis of figure as initial line ; 

 let be the geographical colatitude (the- angle which the normal makes 

 with the axis), and let /j = cos d and /t' = cos 0', 



The angle of the vertical ij/ = 6' — 6. 



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

 difiering by 1° from 0° to 90° have been computed, the eccentricity e 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^, /t™ will be 

 of the same form as those given above (see p. 4). Where X is the total 

 force towards the north perpendicular to the Earth's radius, Y the total 

 force perpendicular to the geographical meridian towards the west, Z 

 the force towards the Earth's centre, or 



dV ^ 1 dY . „ dY 



-— — , Y= , -■ • , , and Z= — --- 



rdti' rsmO' d\ dr 



(east longitudes being considered positive). 



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 downward force on the spheroidal surface of the Earth, 



then X'=X cos i//--f Z sin )/f 



Y'=Y 

 Z' =— X sin ^ + Z cos i|/. 



We may conveniently denote the values of the coefficients of 

 g"^ cos mX and /i™ sin in\ in the potential function and in the forces 

 X', Y', and Z' by the symbols V™, X';", Y'™, and Z'™ respectively. 



If r be the radius vector, ;w,=cos 6 and /x'=cos 6', 



then V'^= -i- H'™, and V ™„=r" H'Jj;, 



H'^ being the same function of n' that H™ is of //. 



The expressions for the magnetic forces on the spheroidal surface of 

 the Earth will be as follows : — 



Taking o„ and /3„ to represent magnetic constants depending on in- 



