48 



SHORTER CONTRIBUTIONS TO GENERAL GEOLOfJY, 11)21. 



Adding these two last equations and factoring, 

 we get 



S,K = .s (sin a tan 5 cos a + sin a) ■ —5-7 . — 7 



sui |3 tan 5 — tan p 



and as 



sm <t) 



1 



COS p (sin /i tan 5 — tan p) | 



By means of a similar though simplef con- 

 struction it may he shown that the formula for 

 "depth to a stratum" is d = s (sin a tan 5 cos 

 c + sin a), in wliicli the term sin a is positive or 

 negative as is determined by the value of a. 

 It therefore appears that tlie expression 



1 



cos p (sin 13 tan 8~ tan p) 



in the equation for the value of (/ is a factor 

 which must be applied where the hole is other 

 than vertical. If the bole or shaft is vertical 

 this factor reduces to luiity. 



In the construction above given, the angles 

 0- and 5 have been drawn in opposite directions, 

 ant! also the angles cr and p in opposite direc- 

 tions. It has been found that four different 

 forinidas can residt by the use of other con- 

 structions, and tiie composite formula covering 

 all cases is as follows: 



.v (sm a tan 6 cos crismo-) 

 1 



(7) 



cos p (sin /3 tan 5 ± tan p) | 



The following rules govern the use of this 

 composite formula : 



Use + sin (T, when a and 6 arc in opposite direc- 

 tions. 



Use — sin <r, when a and 6 tire in the same direc- 

 tion. 



Use + tan p, when u and p are in opi^osite direc- 

 tions. 



Use — tan p, when a and p are in the same ilirec- 

 tion. 



GRAPHIC REPRESENTATION OF THE FORMULA 



As stated ])efore, only that part of formula 

 (7) wMch relates to the measurement of 

 ''depth to a stratum'' will be plotted here. 

 The whole formula could be plotted by the 

 same methods, but in this paper other methods 

 will be given for its solution. 



Consider the formula d = f< (sin a cos a tan 5 

 ± sin a). This may be split into two formulas, 

 like equation (1), and plotted by the same 

 methods. Inserting an auxiliary variable t', 

 we have 



t' 



d 



(8) 



f' = sin a cos o- tan 5 + sin a (9) 



These two equations have been plotted exactly 

 as cfjuation (1) was plotted, and the result is 

 shown on Plate VII. The two charts, Plates 

 VI and VII, are analogous in every respect, 

 and IK) further explanation is required. 



USE OF CHART. 



The depth to a stratum is computed from 

 the chart (PI. VII) exactly as the thickness of 

 strata is computed from Plate VI. All direc- 

 tions are identical. 



GRAPHIC SOLUTION OF RIGHT TRIANGLE. 

 OUTLINE 



In the consideration of the two preceding 

 problems of thickness of strata and distance 



h 



Fn'.URK ti.— Ki;^hI-anglo(l lri;iui;lc -sho\\ini; tin- ifhition^ DlsloixMlistance, 

 horizon 1 1ll di-^tance, dillorciu'.- of elevalion, and vi-ri ical anglr ln'twi-eu 

 I wo si :i1 1(^11 puinis. 



to a stratum, it has been assumed, in the right 

 triangle determined by the two stations Sj 

 and S;, and the plane of reference (see fig. 6) 

 that the slope distance and angle of slope were 

 given. It may be, however, that instead of 

 .s and a any one ()f the five additional combina- 

 tions is given as follows: a and //, a and c, .s and 

 /(, X and ( , or Ji and < . It is desired to derive 

 graphically the values of .s and cr when any of 

 these combinations are given. 



TRIGONOMETRIC FORMULAS. 



It may be seen at a glance that the solution 

 of this |)roblem is essentially a. graphic repre- 

 sentation t)f the sine, cosine, and tangent con- 

 ditions of a riglit triangle. The formulas in- 

 volved are as follows: 



(■ = s sin a (10) 



h = .v cos c (11) 



e = li tan<7 (12) 



