42 



SHORTER CONTRIBTTTIONS TO GENERAL GEOLOGY, 1921. 



represented by meaixs of a ii<>ino<;raph or aliiie- 

 ment chart, which in reality is a system of plot- 

 tinjif by means of parallel coordinates. Three 

 excellent treatments of this method of graphic 

 analysis have been \\Titten, by D'Ocagne,* 

 Lipka,- and Peddle,'' and the reader is referred 

 to their publications for aji understanding of 

 the theory of the alinement chart. 



In plotting the a})ove formula Palmer * used 

 their thre(i-variable nomograms, thus necessa- 

 rily s<)lving the formula by several independent 

 operations. Thus sin 5 and cos u were mul- 

 tiplied in one operation, and the ])rodiu't mul- 

 tiplied by sLn « in a second operation. Cos 6 

 and sin o- were multiplied in a tliird operation. 

 The products of the second and third opera- 

 ti(ms were then added num(>rically by a foiu'th 

 operation, and this sum niulti]>lied by .s" in a 

 fiftii operation, to solve foi' t. Tlu'ee charts 

 were required for these operations, one to mul- 

 tiply sines by cosines, a second to multiply 

 numbers by sines, and a tliird to multiply num- 

 bers by numbers. Moreover, as the nomo- 

 graphic solution with parallel scales, which is 

 the i>ne employed, is essentially a method of 

 addition and sul>traction, and as all the above- 

 mentioned operations that were jierformed 

 graphically involve multiplication, all the cal- 

 il)rated scales were necessarily l(»ga,rithmic 

 scales. A serious drawback exists in tiie use of 

 logaritlunic scales, because the accuracy of the 

 reading is greater at one end of the scale than 

 at tlie other, and tliis weakjiess is specially pro- 

 nounced in logarithmic scales of the trigo- 

 nometric fimctions. By the method here used, 

 the solution of the (ujuation 



f = s (sin « sin 5 cos a±cos 8 sin a) 



is effected l:)y a compound operation, ui which 

 a single chart and natural instead of logarith- 

 mic functions are employed. 



The above equation, containing five varia- 

 riables, can not be plotted directly by any 

 method in two dimensions knoAvn to the wi'iter, 

 but by separating it into two parts and equat- 

 ing each of these to some auxiliary variable, the 

 equation may be readily charted. Thus the 



1 D'ocagni', Mauriiv, Trail6 dc iiomographie, Paris, Gaulhier-Villars, 

 1SS19. 



2 Lipka, Joseph, Craphital and mt-L-lianical computation, New York, 

 1918. 



3 Peddle, J. li., The i-unstruol ion o( uraphieal iharls, 2d ed., New York, 

 1919. 



* Palmer, H. S., Nomographie solutions of certain stratigraphic meas- 

 urements: Ecou. Geology, vol. U, 1916. 



e(| nation may be 

 follows : 



written in two parts as 



t'-^ 



(2) 



/' = sin a sin 5 cos a ± cos 6 sin a (3) 



where /' is the introduced auxiliary variable. 

 Equation (2) is a problem in division or, when 

 written t = t' s, a problem in multiplication and 

 therefore can not be plotted with natural scales 

 if an alinement chart with parallel scales is used. 

 By employing a nomographic Z chart, how- 

 ever, natural scales may be employed in midti- 

 plication and division, and this is the method 

 which has been used. 



Equation (3), however, is well adapted to 

 graphic representation by an alinement chart 

 with parallel scales, as the primary operation 

 to be performed is addition or subtraction, as in- 

 dicated by the symljol ± . This equation, how- 

 ever, presents a difficulty in that it expresses 

 a relationship lietween four variables — that is, 

 I' , (V, 5, and c. If one of these variables could 

 be regarded as a constant, the equation would 

 be reduced to a three-variable type. The 

 oljvious solution consists in assigning to one 

 variable a series of fixed values and com- 

 puting the resulting curves for each particular 

 value. Two variables will be plotted on two 

 parallel scales, and a third variable, whose 

 position is partly determined by the fi.xed value 

 assigned to tlie fourth variable, will be plotted 

 between the two parallel scales. For each 

 fixed value assigned to the fourth variable a 

 different curve of the third variable will be de- 

 veloped, and the composite result will be a series 

 of curves expressing the third variable in terms 

 of the fourth. These curves may be joined to- 

 gether by a set of auxiliary curves, drawn 

 through points of equal value of the third va- 

 riixble, and a gridwork of intersecting curves 

 will be formed which will express grapliically 

 the true relationship between the tliird and 

 fourth variables. 



In tlie practical application of tliis method 

 the variable t' is assigned to one of the outer 

 parallel scales, in the plotting of both equations 

 (2) and (3). The same scale modulus is used, 

 and as both solutions involve only natural 

 functions, the scale of t' for each solution is the 

 same, and a common support for the scale t' 

 may be used. Three parallel supports are 

 therefore used to plot the variables a, t' , and t. 



