64 
MR. GEORGE W. WALKER ON THE MAGNETIC RE-SURVEY OF THE 
but without further preface I may say that the following theoretical cases were 
worked out in detail, in order to illustrate the difficulties that may arise in actual 
cases. I venture to hope that they may have some constructive value. 
The very simplest source of disturbance we can contemplate is an isolated magnetic 
pole beneath the surface, the equal and opposite pole being sufficiently remote to 
produce no effect. The effects of an isolated pole are so simple and obvious that 
discussion would be superfluous. If the observed forces which we have to explain are 
correlated, isolated poles are, by the data, of little help. ^ 
The next simple source is a doublet. Now a doublet need not be confined to a 
small region, for a uniformly magnetised sphere is equivalent, at all external points, 
to a doublet placed at its centre. Moreover, the sphere may be naturally magnetised, 
or magnetised by the earth’s induction. 
We shall consider in detail three cases: (l) a doublet with its axis vertical; 
( 2 ) a doublet witli its axis horizontal; (3) a doublet with axis inclined at tan~^ 3, 
a case which closely represents a sphere magnetised along the direction of the 
resultant earth’s force in the British Isles. 
Case 1. A doublet with its axis vertical. 
Let the south pole be upwards, and let the magnetic moment be /u. Take axes 
through the centre of the doublet, x and y horizontal and z vertical. Then the 
magnetic potential is 
0 = 
where 
Hence if is the depth of the doublet beneath the surface we find that the forces at 
the surface are 
Vertical component V = — 
where 
Badial component 
K = - 
3^t V 
■F(ffiTrp’ 
= (a-'+rr'Vr. 
The curves in fig. 1 show the forces to scale. The abscissae are the values of the 
distances from C, the point on the surface vertically above the doublet at D. The 
unit of distance is i, the depth of the doublet. The ordinates are the values of the 
forces on a scale of 1 cm. = 
The radial force is everywhere towards C, being zero at C, rising to a maximum 
when r = ^ and then diminishing as r increases. 
The vertical force is down near C, being a maximum at C. It becomes zero at 
r = 2*, rises to a positive maximum at r = 2 and then diminishes. 
Note that the value of V at r = 2 is only about s^th of the value at r = 0, while 
the maximum of R is rather less than one half the maximum value of V. 
