PROBLEMS 247 



hole mouth, on the said horizontal plane we can get the 

 positions of it and point C, where the hole hits the stratum, 

 from the condition h + gradient height of BC = H, the 

 total height of B above the said plane = height coordinate 

 of B less that of A = B^ — A^, where z is the space (height) 

 coordinate subscript above any given datum like sea level. 

 Let the horizontal distance from 5 to C be x and the 

 gradient of BC will be a function of x. The distance 

 between C', the vertical projection of C on the plane, and 

 the strike line through A is ^iC", then 



xt&ii a -\- AiC t&n 8 = H (19) 



and AiC may be obtained from the right-angled triangle 

 AiC'A2, which is right angled at Ai, thus: 



AiC = C'A2 sin di = {B'Ao - x) sin di 



where di is the angle between the borehole andstratumstrikes. 



Again, from triangle AB'A-i we get B'Ai = AB' —. — — 



sm uif 



where d^ is the angle between the line from the known point 

 A to the horizontal projection of B at B' . That is to say 

 02 is always known from the survey notes. Therefore 



A,C' = (aB' ^-^ - :^ sin d^ = AB' sin d 

 \ sm di } 



2 — X sm Q\ 



Substituting in Eq. (19) above we get 



X tan a + {AB' sin Q2 — x sin 0i) tan 6 = H 

 Whence 



H — AB' sin 62 tan 5 



X ^^^ ; 



tan a — sin di tan 5 



(20) 



and all the quantities on the right hand side are known so 

 getting X, we may easily fix the space coordinates of C by 

 first obtaining h. The signs of the various terms in this 

 expression vary according to the position of the given 

 magnitudes in space, thus: 



