44 



SHORTER GONTRIBUTIOXS TO GENERAL GEOLOGY, 1!)21. 



ayst(>m of coordinates. The latter procedure 

 is here shown, the sohition o;iven hy Lipka ^ 

 behig I'ollowetl very closely. 



In fi<;ure 4, let a and f be plotted on two 

 parallel straight ILnes, as shown; and let 5 be 

 represented by some hypothetical curvilinear 

 luie. Let the zero point of each of the two 

 parallel scales be connected by a base line, 

 whose length is /,■,• and let the two outer scales 

 be so placed that this base line lies perpendicu- 

 lar to both. In this way a system of rectan- 

 gular Cartesian coordinates will be assured. 

 Take for an origm of such a coordinate system 

 the intersection of I- A\ith the t' scale. Draw 

 any nomographic index line coniiectmg the a 

 antl /' scales and cutting the 5 scale. From 

 the intci'section of the index line with 8. draw 

 a line |)arallel t" /.' to meet th(> a scale and 

 aniithiT parallel to the a scale to meet Jc. 

 From 111;' intersection of the mdex line with 



Then in order to satisfy the equation, it is 

 necessary that 





and 7.-3:7 =m,./V"') 



Solving tlie first equation, we find that 

 _ I'm, fs (?r) 



(5) 



jVnd solving the second equation and sub 

 stituting in it the value of Cj frome quation (5) 

 We find that 



■/n. Ill- I . I 11^ I 



(6) 



m^+in, /.,()/') 



The values z and .', an^ the rectangular co- 

 ordinates of any point on the curvilinear scale, 

 representing a definite value of 5, measured 

 from the intersection of J: and the f scale as 

 an origin. The locus of the curvilinear scale 

 S can then bo determined, for a seriis of as- 

 signed values of 5 will give the coordinates of 

 a series of points which may be joined 

 togi'ther into a smooth curve. 



To j)lot such a curve, however, a fixed 

 value was assigned to the variable a. There- 

 fore for every assigned value of a a new curve 

 will result. In the preparation of the chart 

 a series of such curves may be computed 

 for a regular series of values of a. If only 

 a single curve were charted it would be cali- 

 brated in terms of 5, in a way sunilar to the 

 parallel st raight-line scales. But with a se- 



FiGUEE 4.— Diagram to illustrate thi' unlliod uf <l-trrimning the locus otlhi' rieS of SUcll CUl'VCS the points OU each CUrVC 

 curvilinear scale in an alinement cliart cMusLsliug of two parallel straight- that represent like ValueS of 5 are joined 

 line scales and a curvilinear scale. i /• • * -i- • 



tosrether, forming auxniary intersecting 



the /' scale, draw a line j>arallel to /■. In the 

 diagram, 



y-z : z-x : : A'-c, : z^ 



Zij/ - 22, = l-z - h: - 2-1 + 2,X 

 A-.K - 2i.r -I- 2i // = A'3 



(k-zj x-z,ii = l-z 



, Zx kz 



This equation is evidently in a. form similar 

 to the one to be plotted — that is, 



j\{u)-f,(r).f,(ir)^i\{w) 



Ther<'f(in', assigning scale moduli of ///, and m., 

 respectively to/', ((/) and/JD, we may say that 



X = ///.,/, ( (/ ) and y=- )'i-;f-. ( '.') 



5 Lipka, .Toseph, op. cit., pp. liti.-1'i 



curves that may be regarded as loci of tlefinite 

 values of 5. The original cuiwes may then be 

 regarded as loci of definite values of a, and we 

 shall have a series of intersecting curves repre- 

 senting the relationship between the variables 

 5 and a. 



PREPAKATION OF CHART. 



The complete formula for the thickness of a 

 stratum was charted by the methotls here 

 described. (See PI. VI.)" The details of the 

 process have to do mainly with the selection 

 of suitable scale moduli and the selection of 

 such values for the variables .<;, a, 8, and a that 

 the resulting scales will have an adecjuate and 

 balanced calibration. 



Little need be said of the ])rep!iration of 

 equation (2)— that is, t = t's. From the pres- 

 ence of the ± sign in the general fornmla it 



