MAGNETIC OBSERVATIONS. 319 



the magnetic force of the rock may be. It is also evident that in general 

 Hr has its minimum value when H'zz — H sin /?. When the rock strikes 

 north and south or /? zz 0, H^ is a minimum when H' rz: 0. 



4. HORIZONTAL AND VERTICAL COMPONENTS WHEN THE MAGNETIC ROCK 



DIPS VERTICALLY. 



If we assume that the magnetic rock has a uniform strike in any direc- 

 tion, a vertical dip and a surface width or thickness equal to 2a, it is easy to 

 show that the horizontal and vertical components of the rock force are given 

 by the following equations, where x is the horizontal distance of the station 

 of observation from the middle plane of the formation, h is the depth of 

 surface covering, assumed to be uniforin, and co is a constant. 



K-log^l±^^±^ (6) 



yi=2\t&ii-' -^4^-^11-'-^] (7) 



CO I h h ) ^ ^ 



XT/ 



In equation (6) — z= when « rr ; therefore a point of no deflec- 



GO 



tion of the horizontal needle is found vertically over the middle point of the 

 magnetic rock. It is also evident that at corresponding stations on opposite 

 sides of the middle point, the horizontal components are equal, but act in 

 opposite directions. 



To obtain the points of maximum or minimum values of the horizontal 

 component, we differentiate the right-hand side of equation (6) with respect 

 to X, and place the result equal to zero. This gives 



x=^\/ h^ + a^ (8) 



which determines two points, at equal distances from on opposite sides of 

 the rock, at which the horizontal component has maximum values. Writing 

 for X the measurable distance d, and squaring, we have 



cP=h^+a'' (9) 



The thickness of the magnetic formation is therefore always less than 

 the distance between the points of maximum horizontal deflection, except 

 when /izzO, or the rock is uncovered, in which case the thickness and sepa- 

 ration of the maxima are the same. 



