COMPUTATION OF THICKNESS OF STRATA AND DISTANCE TO A STRATUM. 



45 



results that botli a positive and a negative 

 scale for both t' and t are recjuired. A scale 

 modulus of 10 was adopted for the original 

 drawing, calculation of the / scale being thus 

 eliminated. The t' scale, though uncalibrated 

 in the finished chart, was calibrated for pur- 

 poses of projection in the actual work, and the 

 calibration may be shown to be merely a 

 natural sine scale. The scale modulus of 10 

 likewise eliminated calculation in the prepara- 

 tion of this scale as well as in the preparation 

 of the a scale. The numbers 1, 2, .3, etc., might 

 have been used on the t scale in keeping with 

 slide-rule practice, instead of 100, 200, 300, 

 etc. But as this chart is to be used solely to 

 compute the thickness of geologic strata, it has 

 seemed best to the writer to calibrate the scale 

 in terms of the probable range of answers that 

 wiU be obtained. The s scale has accordingly 

 been numbered to accord with this conven- 

 tion. 



Ill the plotting of equation (3) — that is, t' = 

 sm a sin 5 cos c ± cos 5 sin cr — one point in par- 

 ticular requires explanation. For the posi- 

 tive form of the equation, only positive values 

 of f will result, but for the negative form of the 

 equation both positive and negative values of 

 t' will be obtained. Hence two nets of 5 and a 

 curves would bo required, one with positive 

 and one with negative values; but only a shigle 

 a scale, the positive one, would be necessary. 

 To avoid drafting these two nets of 5 and a 

 scales, both positive and negative a scales were 

 drawn, and only one 5 and a network. In 

 using the chart, therefore, the positive values 

 of a are used for a solution of the normal or 

 positive form of the general equation, and the 

 negative values of a in solving the negative 

 foi'm of the general equation. This procedure 

 is indicated on the chart. 



As stated, the curves of 5 and a will ordi- 

 narily be calculated by some system of coordi- 

 nates and joined together into smooth curves. 

 In the case of this particular equation (f = sin 

 a sm 3 cos <7±cos 6 sin <t) , however, the com- 

 pensatmg form of the fmictions of 3 and a — 

 that is, sm 3 cos <r against cos 3 sin cr — results in 

 a series of curves which are most easily pre- 

 pared by a projective method. It is unneces- 

 sary to go into an analysis of the method, but 

 a statement of the method used is given. It is 

 stated above that a prelimmary suie calibra- 

 tion was used on the t' scale. The positive 



32333°— 2L^ ! 



and negative end points (90° positions) of the; 

 a scale being used as points of projection, two 

 series of radiating lines were drawn to the 

 points of sine calibration on the t' scale. The 

 intersection of these two sets of radiating lines 

 gave the loci of the requu'ed curves. Each of 

 these curves Ls tangent to the base of the isosceles 

 triangle that bounds the network, and each 

 emerges to intersect both sides of this triangle. 

 Each curve serves a double purpose, therefore — 

 as a o- curve and as a curve of the complemen- 

 tary value of 3. As it is hard to trace several 

 curves past a rather flat zone of tangency, the 

 curves are doubly named, in order to avoid that 

 necessity. Every curve cuts every other curve, 

 and hence an intersecting point for values of 3 

 and a can always be found. Only one equivo- 

 cal condition will be noticed, and that is where 

 complementary values of 3 and c are given as 

 field data. Under this condition, the same 

 curve represents both values, and the point of 

 tangency of the curve with the base of the 

 isosceles triangle must be regarded as the jioint 

 of intersection of a 5 curve and a cr curve — that 

 is, one limb of the curve will be regarded as a 

 3 curve, and th(> other limb as a o- curve. The 

 <r or angle of slope calibration was carried up to 

 90°, and this is open to criticism by field geolo- 

 gists, for hillsides of greater slope than 30° are 

 rare. But the chart is also intended for meas- 

 uring geologic sections in mmes as well as in 

 the open, and for this purpose the complete 

 range from 0° to 90° for a is recjuh'ed. 



A small index chart showmg five hypothetical 

 points joined by a compound nomographic 

 mdex line has been addetl as a guide to anyone 

 using the chart. 



USE OF CHART. 



The use of the chart (PI. VI) m obtaining 

 the tliickness of geologic strata is simple. 

 At the left side of the chart is the a scale, on 

 which are plotted the azimuth angles between 

 the strike of the rocks and the line of traverse. 

 This calibration comprises Vjoth a positive 

 and a negative scale, the positive one starting 

 at the middle of the line and extending up- 

 ward and the negative one starting at the 

 middle and extending downward. Use the 

 upper scale where the angle of slope and angle 

 of dip are in opposite directions and the lower 

 scale where the angle of slope and angle of dip 

 are la the same direction. 



