A. Harker — Graphical 3IefJiods in Field Geology. 155 



direction from the true dip, and for connecting the dip, thickness 

 and depth of beds : these tables are given in the appendix to the 

 Survey Memoir on the Geology of the South Staffordshire Coal-field, 

 and reproduced in Jukes's "Manual of Geology." Graphical methods 

 have been used for finding the true dip from two apparent dips : 

 a method partly graphical but requiring a table of cotangents is 

 given in Phillips's " Treatise on Geology " (p. 298, 5th ed.), and also 

 by the Rev. E. Hill (Geol. Mag. 1876, p. 334) ; a purely graphical 

 method by Mr. W. H. Dalton (Geol. Mag. 1873, p. 332) ; and an 

 approximate method by Mr. Penning (Geol. Mag. 1876, p. 236), 

 reproduced in his " Field Geology." As Prof Green has pointed 

 out {ih. p. 377), the last-named method is equivalent to taking the 

 angle for its tangent, and so applicable only to small angles of dip.^ 

 Mr. Dalton's solution {Joe. cit. p. 334) of another question, to find the 

 effect on strata already inclined of a second tilt in a new direction, 

 is only an approximation, and cannot be applied if the dips are 

 considerable. It is erroneously assumed that the inclination of the 

 strata in a direction at right angles to that of the second tilt is 

 unaltered by the tilting. 



I propose to show that graphical methods are capable of wider and 

 simpler application than they have yet received, and may be made 

 really useful in field-work. 



Various Modes of Treatment. 



Questions relating to the intersection of planes, etc., may be treated 

 in various ways, all equally simple. Firstly, we may draw figures 

 to represent the planes themselves. For instance, let ABC, Fig. 1, 

 represent the position of certain strata, ABD a. horizontal plane, CD 

 being vertical ; then AB is the line of strike and A D, perpendicular 

 to it, the direction of true dip; the angle CAD (= X say) is the 

 amount of dip and CBD (= T) the apparent dip in a section 

 making with the direction of true dip an angle ADB (=: Z). Then 

 we have directly (Fig. 1) 



AD= CD cot X, BD=CD cot T, AD = BD cos Z. 



Therefore cot X = cot T cos Z, ") , ^ 



or tan Y =: tan X cos Z, j" • • • • V / 



which are the formula from which Mr. Jukes's tables are calculated. 

 Again, we may conveniently consider instead of the planes them- 

 selves the normals to them from a fixed origin 0, and represent 

 them by the points in which the normals meet a sphere of unit 

 radius. For instance, in Fig. 2, let Z represent the horizontal plane, 

 P the plane of the strata, Q that of the surface of the ground ; then 

 ZP represents the dip of the strata (= X), ZQ the slope of the 

 ground {=¥), MN or P^Q the angle (Z) between the direction 



^ To indicate the degree of the approximation, suppose the two ohserved dips 

 to make an angle of 60° with one another ; then if the amounts of the dips be 15° and 

 20° respectively, the error in determining the direction of true dip by Mr. Penniug's 

 method is less than 1° ; if the dips be 30° and 40°, it is about 4° ; if 45° and 60°, it is 

 11°; and if 60° and 80°, the error amounts to 29° ! 



