156 A. Harker — Graphical Methods in Field Geology. 



of dip and that of slope. Then PQ represents the angle (Z7) at 

 which the surface of the ground cuts the strata, and we have at once 



cos Z7=cos X cos F+sin X sin Fcos Z .... (6). 



This enables us to find the true thickness of a bed from the breadth 

 of its outcrop, for the ratio of the former to the latter is evidently 

 sin TJ. Further, if the great circle PQ meet the horizontal great 

 circle in B, B represents a vei'tical plane through the outcrop ; so 

 0/2 is perpendicular to the direction of outcrop, and since OM'm 

 perpendicular to the direction of strike, Eitf represents the deviation 

 (F) of outcrop from strike. We readily obtain 



tan X sin F=cot PBM=tan 7 sin (V—Z) 

 and so tan F=tan F sin ^~ (tan Fcos^— tanX) . . . (c). 



The problem of the " secondary tilt " is troublesome trigono- 

 metrically, though its graphical solution is sufficiently simple. I 

 give only the results : if strata having an original dip X receive a 

 secondary tilt of amount T in a direction making an angle S with 

 that of the original dip, then the final dip Y, and the angle Z which 

 it makes with the original dip are given by 



cos F=cos X cos T — sin X sin T cos S 

 and 



tan ^=sin S (cos X cos| T — sin X sini T cos S) 



-^ jsin X cos T+cos X sin Tcos S+2 sin X sin- S sin-i Tj . . , (d). 



For trigonometrical calculation the spherical projection is of 

 course the most convenient, but as suggesting graphical construc- 

 tions another projection presents advantages. The planes are 

 represented by the points where they cut, not a sphere, but a 

 horizontal plane at unit distance above the origin. Let the normals 

 to the strata, the ground-surface and the horizontal plane, drawn 

 through 0, meet the plane of projection in P, Q and Z respectively ; 

 then OZ is unity, ZP is tan X and ZQ is tan Y, X being the dip of 

 the strata and Y the slope of the ground, and PZQ = Z, the angle 

 between the directions of dip and slope (Fig. 3). In practice only 

 the plane of projection with the traces on it of the various lines and 

 planes is required {Z P Q in Fig. 4). In Fig. 3 POQ = U, the angle 

 at which the strata are cut by the ground; if we imagine the 

 triangle POQ turned about PQ into the plane of projection, we 

 get Fig. 4, which at once leads to the construction given below 

 (Problem vii.). 



The constructions given for Problems x. xi. xii. and xiii. follow 

 from equally simple considerations. 



By this kind of projection we can represent the dip of any strata 

 both in direction and amount by a line drawn from Z in a diagram 

 on the plane of projection, for the line may be drawn to indicate by 

 its direction the direction of the dip, and its length will be the 

 tangent of the amount of dip. Similarly the slope of the ground 

 can be represented by a line drawn from Z. 



