rOMPUTATION OF THICKNESS OF STRATA AND DISTANCE TO A STRATUM. 



49 



Each of those equations is hi such form that it 

 may be written /i («) =/,(i;) • /sU'), and as 

 shown before such equations are best plotted 

 by means of the Z type of alinement chart. 



PREPARATION OF CHART. 



The method of plottmg such equations has 

 ah'eady been described, in connection with the 



C B 



Figure 7. — Piagrara illustrating the combination of three alinement 



eharts for the solution of right-angled triangles. 



The chart is now complete except for the 

 solution of equation (12), which expresses the 

 tangent condition. For this, the scales e and 

 h, already plotted, are used in conjunction with 

 a scale of tan a which is plotted on the line Bl). 

 An mdex of two Imes intersecting at right 

 angles is used, such as EG and FH (figure 8). 

 The method consists in passing the line EG 

 through the given points on the e and h scales 

 and then sliding the index along EG until the 

 hue FH passes through 0. In this position the 

 angle DCF=(7, and it is recjuired to compute a 

 calibration of tan o- on the line 1)B so that a 

 reading indicated by the line IIF on DB will 

 give the recjuired value oi a. 



In figure 8 



DO:DS:sin (135° -<r): sin <t 



DS 



sm 



DC 



DC sin (7 _ DC ^i n a 

 sin ( 135°- a) ~ sin (45° -fa) 



sin 45° 

 DS=DB • 



sm <r 



sin <r -f cos 0- 



sin <r 

 sin a + cos <T " ' 



(13) 



plotting of equation (2). Two straight-line 

 parallel scales, oppositely directed, are used to 



plotf,(?/) and either of tile other two variables. Therefore the tan a calibration, expressed in 

 ui this instance /3(w). Both of these are iiat- I the diagram by successive values of DS, was 

 ural scales. A straight line join- 

 ing the zero ends of these two 

 scales cari'ies the scale of the 

 third variable, f2iv), and the 

 calibrations are calculated by 

 means of equation (4) , as already 

 explained. 



In order to plot m one diagram 

 all three of these equations, it is 

 necessary to use one or more of 

 the same scales in different solu- 

 tions. Equations (10) and (11) 

 have in common the variable .s, 

 wherefore it has seemed best to 

 make a common scale of s, in the 

 solution of these two formulas. 

 In figure 7 the values of e are 

 plotted on CD, the values of sin 

 a on AB, and the values of .s on 

 the diagonal joinmg the zero ends 

 of these two scales. Two other 

 lines, however, AD and CB, may be drawn to 

 complete the square, and on these the values 

 respectively of cos a and h maybe plotted. In 

 this way a chart composed of five lines is pro- 

 vided for the plotting of e, h, .s, sin <j, and 

 cos cr. 



Figure 8.- 



~G 



Diagram illustrating the use of an alinement chart for the solution of the tangent 

 condition in right-angled triangles. 



computed by multiplying the length of the 



diagonal DB by the expression , 



The chart representing the complete solu- 

 tion of a right-angle triangle is shown in 

 Plate VIII. It is of universal use in the 



