COMPUTATION OF THICKNESS OF STRATA AND DISTANCE TO A STRATUM. 



43 



The. variables 8 and a are expressed in a grid- 

 work of curves lying between a and /, and the 

 variable s is plotted upon a diagonal line con- 

 necting opposite ends of the t' and t scales. As 

 no numerical value of t' is required, the support 

 of the t' scale is not calibrated. Thus in the 

 operation of the chart a point upon the a scale 

 representing some value of a is connected by a 

 straight line with a point which represents given 

 values of 8 and a in the gridwork of curves and 

 produced to meet the uncalibrated scale t'. 

 The intersected point is then connected by an- 

 other straight line ■\\ath a point on the diagonal 

 line representing some value of .s and projected 

 to the t scale, the readmg on which shows di- 

 rectlv the thickness of the stratum, vein, or 

 formation. 



MATHEMATICAL ANALYSIS. 



EQU.\TIO.^J (2). 



The equation t = t'.9 may be wTitten as 



/i {») =f, (v) -jKiv) 



where t = u, t' =v, and s = w. In figure 3, let t' 

 and t be plotted upon two parallel straight-line 

 scales, oppositely directed. The diagonal line 

 joining the zero ends of these two scales -wall be 

 the locus of the scale z and will be considered 

 to have a length of I:. Draw any nomographic 

 index line joining the t' and t scales and inter- 

 secting the 2 scah\ In the diagram, 



>j : X : : k-z : z ^ 



The above eciuation is evidently in the form 

 /i (") =/: (i'') -/sM'. Therefore, assigning scale 

 moduli of m, and m, respectively to/i(u) and 

 /s iv) , we may say that 



X = m,/, (w) and y = mj^ (v) 



As t' and therefore t and .s must be plotted as 

 natural functions, in order to be coordinate 

 with the chart of equation (3), the Z type of 

 alinement chart is used. The method of analy- 

 sis is that used by Lipka.^ Hence the equation 

 becomes 



Therefore 



m, (k ■ 



./; (w) 



or 



■'i/i(") = ^tI:-, • '"2/2 (w) 



/.(«)=;;r^fe^ -/.W 



' /«! (k — z) 



^ Lipka, Joseph, Graphical and mfchanical t-omputation, pp. 05-66, 

 New York, 1918. 



and from the solution of this equation it is 

 found that 



^^^m^£W (,) 



From equation (4) , by substituting the speciiied 

 moduli and ualues oif^(w), a series of values of 

 z can be computed, which will represent the 

 calibration of the diagonal scale, or scale of .s. 



EQUATION (3). 



Consider the positive form of ecjuation (3) 

 that is to be plotted: 



t' = sin a sin 8 cos c + cos 5 sm a 

 or 



t' — sin a (sin 8 cos u) = cos 6 sin a 



If some definite value is assigned to cr, so that 

 cos 0- and sm a become temporarily constants, 

 the equation may he written 



A{ii)-f,(v).Mw)=f,(io) 



where t' = tt, a = v, and S = n\ In this form we 

 have an equation of tliree variables, one of 



Figure 3. — Pirgram to illustrate the method of calibrating the dia};onaI 

 scale of a Z chart. 



which (w) occurs on both sides of the equation 

 as two different functions — that is, fsiw) and 

 fiiw). Such an ecjuation, when plotted as an 

 alinement chart, will requh'o two parallel 

 straight-line scales and one cm'vilinear scale. 

 The two parallel straight-line scales, represent- 

 uig the functions /i(«) andf^iv), may be drawn 

 and calibrated in the ordinary mamier used 

 ui building the simpler type of alinement 

 chart, but the curvilmear scale representing 

 the two fmictions of the variable w must either 

 be projected graphically or computed by some 



