COMPUTATION OF TIIIt'KXKSS OF STRATA AND DISTANCE TO A STRATUM. 



47 



N 



Pass a vertical plane through Sj and S, and 

 revolve this plane about the line joining tlie 

 projections of Sj and S, into the horizontal 

 plane. The station Sj will fall on S,' and the 

 right-angled triangle SiS^S,' will show m true 

 proportions the quantities c, s, /(, and c. Pass 

 a vertical auxiliary phme, perpendicular to the 

 line of strike, tlirough So. Its trace on the 

 horizontal plane is the lino Pj. Lay off SjS," 

 equal to c and draw S/'L', makuig the angle 

 SoL'S," equal to 6, the dip of the stratum at 

 station S,. On revolvmg tlie right-angled tri- 

 angle S,S2"L' 90° about the line SX' and then 

 90° about the vertical through So, the point L' 

 will fall on L. Througli \j draw tlie Ime LM 



Figure 5.— Geometric representation of the disianec toastratum when 

 the dip of the strattim, the position of a point on the stratum relative 

 to the starting point of measurement, and the iiorizontal and vertit-al 

 directions of the line of measiirenienl are given. 



parallel to the strike line. The line LM is the line 

 Lii which the dippuig stratum intersects the hori- 

 zontal plane through S,, tintl the strike line SoT 

 is the line of Lntersection of the dipping plane 

 with the plane tlu-ough SjC and the station S,. 

 Let SjK be the projection of any sloping tun- 

 nel or drill hole which makes an angle p with 

 the horizontal plane and has an azimuth /3 

 with reference to the strike line. Through 

 S,K pass a vertical plane. This plane will cut 

 the strike line S,T in point T ;ind the line LM 

 in point M. If we revolve this plane about 

 S,M into the horizontal plane, the point T, 

 whose distance above the horizontal plane is e, 

 will fall at T', M will be unmoved, and SjK' 

 will be the revolved position of the tunnel or 

 drill hole. The angle p will be shown in true 

 value. Draw the line T'M, cutting SiK' in K'. 



The line T'M is the revolved trace of the verti- 

 cal auxiliary plane through SjM and the dip- 

 ping stratum tlirough S,, the point K' is the 

 revoved position of the point in wliich S,K 

 pierces the inclined stratum, and M' is the 

 revolved position of the point in which the 

 vertical through M cuts the tunnel or drill hole. 

 If the auxiliary plane is revolved back to its 

 original position the projection of K' on tlie 

 horizontal plane will be found at K, and line 

 DK, drawn parallel to the strike line, is the 

 projection, on the horizonttil plane through vSj 

 of the line in which the dip plane is cut by 

 plane Po, the horizontal plane through K. 



The distance vS,K' is the length of the tunnel 

 required and is to be derived in terms of the 

 known slope distance .s' and angles <t, 5, ;ind p. 



TRIGONOMETRIC FORMULA, 



From llu^ figuro: 



S,K' = 



S,K 



cos p 



/( sin a 



(S,T + TM + MK) 



1 



cos p 



TM = 



S,T= 



sj;. 



sin B 



sinjS 



e. 

 "tans' 



S sm a cos a 

 sin/3 



1 s sin cr 



MK = M'Iv'cosp = 



])Ut MM' = SiM tan p 

 /k sin a. cos c 



V sin (3 



.s tan p 

 sin (3 tan 5 



TM 



- + 



sin (3 sin p tan 5 



MM' HM 



— Til *^°^ '' 



(S,T + TM) tanp] 

 s sin 



an 6/ 



an p 



sm /3 tan 



(sin a tan 5 cos (7 + sin a) 



x sin a 



HM= . „, , 



cos p sin (3 tan 5 cos p 



T'H = e - HT = .s sin <j - TM tan pj 



-Aa) 



Ab) 



■ s sin a ■ 



s sin (J 



.s' sin a 

 =i\n 13 tan 5 



tan p 



(sin /3 tan 5 — tan p) 



(0 



sin /3 tan 5 



Then by substitution from equations (a), (li), 



and (c), 



- ,„ s tan p sin a tan 5 cos o- -I- sin <t 

 sinj3 tan 5 ' sin /3 tan 5 -tan p 



From the values of S,T, TM, and MK just 

 found 



S T I TM - ''' ^^^ °^ ^"^ <^ I ^"^ ^''"^ '^ 



MK 



sin (3 sin /3 tan 5 



Q 



~ — ;r— (sin a tan 5 cos a + sin a) 



sill 13 tan 5 



s tan p sin a tan 5 cos a + sin a- 



sin /3 tan 5 



sin /3 tan 5 — tan p 



