442 THE THEORY OF TERRESTRIAL MAGNETISM. 



T , x 



- 



Y= 



ttence 



dV 

 Y d\ 



On the sphere of unit radius r=l, and 



dX . a tlX ^V^ dV 



ja = smVj = (l p) j J+M-J , 

 dB dp ' dp* dp. 



dY_ <IY__ CTT p. dV 



dd = dp~d\dp + l-p t d\' 



Hence 



wj. ydX_, z\h[dV d*V dV d*V 2p 



d6 dO Ldp dXdp dX t//x 2 1 /u, 2 dX dp J ' 



dY ydX _ dVd'V dV _ 

 dK d\ rfu, t/A. 2 dX ' 



and 



As an example of the application of the above theory, we will find 

 the approximate place of the crossing of two branches of the line of 

 equal declination in the neighbourhood of the point = 80, A. = 260. 

 Take this point as the origin, and take x and y as the longitude and 

 latitude respectively of some near point referred to this origin, taking 

 10 of longitude and 5 of latitude as the units of x and y respectively. 

 Then if x and y be the coordinates of the point of crossing of the two 

 branches, we have 



dY dX / dY dX\ Y 



^~ = = X - +X - - 



Y ,_. 



* 



-- _ 



dxdy dx* dy dx dy 



Y dY dX I dY dX\ f v (PY v 



and A -= i j = = A -5 i -=- + A -i } 



ay ay \ ay ay/o 



, f v d*Y v d>X dXdY dYdX\ 



T I -A- , 7 I ~j j~ + j 5 ; j- X. 



dxdy dxdy dx dy dx dy / 



