352 THE CRYSTAL FALLS IROI^ -BEARING- DISTRICT. 



If Arrco , and the coordinates are referred to axes in the middle of the 

 rock, these equations reduce to equations (6) and (7). 



By differentiating the right-hand side of equation (10), placing the 

 result equal to zero, and solving for x, the positions of the stations at which 

 H' is a maximum may be determined. This gives: 



2 A 2 A 



Calling the difference of the roots, or the measurable distance between 



2« 

 the maxima, 2d, and siibstituting for h its value -~. — —r 2a being the true 



thickness of the rock, we have: 



.'*'=£^ ■ ■ ■ ■ w 



For rocks of high dip, therefore, the distance between the maximum 

 points is but little gi-eater than it would be were the dip vertical, and it 

 increases inversely as the angle of dip. 



A general algebraic determination of the points at which H' is and 

 V is a maximum is impossible, since it involves the solution of equations of 

 a degree higher than the fifth. HoAvever, by assuming numerical values 



TT/ -y/ 



for A, li, and a (or h) curves expressing the relations between — and — and 



GO GO 



X can be plotted, from which the maximum and zero points can be deter- 

 mined in any desired number of special cases. 



Let us first assume that Arr 3 (or that the rock dips at an angle of about 

 70° 34'), li — 2, and a — %. The ordinates to the curves of fig. 1, PL XLVII, 



TT' V 



give the values of — and — corresponding to different values of x. The 



GO GO 



TT' 



ordinates to — do not represent the deflections S of the horizontal needle 



GO 



from the meridian, but quantities that are connected with those deflections 

 by equation (1). The deflections, however, vary as H' varies, and will have 

 maximum and minimum values at the same points. 



From this figure it appears, first, that the nearer maximum is situated 

 on the dip side of the rock; secondly, that the point of no deflection is not 

 over the middle plane, but is nearer the upper edge; thirdly, that the hori- 

 zontal force of the rock is numerically less at the nearer than at the more 



