46 



SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1921. 



Trace tho two lines representing given 

 values of angle of slope and angle of dip to an 

 intersection in the 5-a gridwork of curves. 

 With a straight edge, or a transparent straight- 

 line index, connect tlie point on the a scale 

 with the o-a intersectitwi, mid the conti:iua- 

 tion of this luie will give an intersection on the 

 t' scale. Then connect the mtersection on 

 the t' scale with the point on tlic s scale which 

 represents a given value of slope distance, and 

 the continuation of this line gives an intersec- 

 tion on the / scale, whicli when read shows tlie 

 thickness of the strata. It will l)e noticed 

 that both the t' and I scales are divided into 

 upper and lower parts, just as the a scale is. 

 Also there are two .v scales. When the first 

 operation gives an intersection on the upper 

 t' scale the second operation is performed 

 Iike\vise on tiie upper s and t scales; and con- 

 versely when tiie first operation gives an inter- 

 section on the lower f' scale the second opera- 

 tion is jierfoi'med t>n the l()wer x and / scales. 



The s and t scales are calibrated 100, 200, 

 300, etc., instead of 1, 2, 3, etc., because tlie 

 answers will usually be of that magnitude. 

 If desired, however, these calibrations may 

 be regarded as 1, 2, 3, or 10, 20, 30, or 1,000, 

 2,000, 3,000, according to the use to which the 

 chart is to be put, just as the ordmary slide- 

 rule calil)rations are used. 



Another use to which the chart may be put, 

 in addition to nndijig the thickness of strata, 

 is the solution of ecjuation (1) for any unluiown 

 quantity, if the other four are known. Thus, 

 a, a, s, ajid 1 may be known, and it is desired 

 to find 5. A line connecting the t and s scales 

 will intersect the /' scale. If this intersection is 

 connected with the given point on the a scale, 

 the resulting luxe will intersect the given <t line 

 at a point which wlien read will show the 

 recjuired value of 5. 



niSTANf E TO A STRATUM. 



OUTLINE. 



It is recpiired to find the length of a tunnel, 

 shaft, or <lrill hoh^ from some scdected point to 

 some definite point tm a stratum, when the 

 following data are given: 



1. The horizontal and vertical location of the 

 starting point of tlu' tunnel, siuift, or drill hole. 



2. The horizontal and vertical location of a 

 second point, wiiich may lie annvhere on the 

 surface of the stratum that is to be intersected. 



3. The azimutli angle between the strike of 

 the rocks and the line connecting these two 

 stations. 



4. The azimuth angle between the strike of 

 the rocks and the direction of the tunnel, 

 shaft, or drill hole. 



5. The angle of dip of the I'ocks. 



(i. The angh^ of dip of the tunnel, shaft, or 

 drill hole. 



In connection with Nos. 1 and 2, which may 

 be considered the beginning antl end ])oints of 

 a traverse, any two of the following measure- 

 ments will suffice: (a) Angle of slope between 

 the two stations, (5) difference in elevation 

 between the two stations, (c) slope distance 

 between the two stations, {d) horizontal dis- 

 tance between the two stations. Therefore six 

 sets of data are given, and these, together with 

 the answer (the tunnel distance), will neces- 

 sarily produce a trigonometric equation of 

 seven variables. 



This is the most general form of the problem 

 of distance to a stratum. Tlie proldiMn usually 

 considered by geologists, particularly in oil 

 geology, and referred to as "depth to a stra- 

 timi," is a special case of the more general 

 problem, wherein the line joining the two 

 points is vertical. In such a case the pitch is 

 90° and the line joining the two points has no 

 horizontal azimuth angle. In other words, 

 two variables are eliminated. The formula for 

 the general problem will be developed, but for 

 this paper only the formula for the special 

 case — that is, depth to a stratum — will be 

 charted. 



GEOMETRIC CONSTRUCTION. 



Let Si (fig. 5) be the startmg pouit of the 

 tunnel, shaft, or drill hole, and let S, be a point 

 which is on the surface of the stratum that is 

 to be intersected but is not in the horizontal 

 plane through vS,. Let S, and S, be repre- 

 sented by theu' projections on the horizontal 

 plane tlu-ough Sj and let S.T be the strike of 

 the stratum at S,. The line SjC, parallel to 

 .S3T, is also the strike Ime, and AB is any 

 reference line tlu-ough Sj in the horizontal 

 plane. Also let Ti be the horizontal distance 

 from S, to S,, s the slope distance, c the differ- 

 ence of elevation, a the vertical angle at S, 

 between the horizontal ]>lane and the station 

 point S,, and a the azimuth of the line joining 

 vS, and S, with reference to the strike line. 



