DIFFERENCE BETWEEN SOLID ANGLES AND TRACES. 6i6 



A section of a mineral that has sharp, narrow lines of cleavage or of 

 (lemarkation must therefore lie near to C (figure 6): that is to say, it 

 must be nearly perpendicular to the planes whose traces the lines represent. 

 Sections over 60° from C are uncommon. The range of probable sections 

 is indicated by the dotted circle in figure 6. The angle traced by a solid 

 angle in section "will, accordingly, most likely differ about 5° from the solid 

 angle, and will incline to be a trifle more acute if the solid angle is acute ; 

 yet it is far less likely to be much more acute than much more obtuse. 



The fact that not all sections are equally frequent or likely to be noticed, 

 and that from the breadth of their borders we can form some judgment as 

 to their orientation, renders a mathematical investigation of the probable 

 traced angle practically useless. It makes a pretty theoretical problem 

 only. 



Solution for Faces of one Zone. 



§ 6. The equation (1) of § 4 and figure 6 may be used to find the position 

 of a section if the three traces between which angles are measured are sup- 

 posed to belong to faces of the same zone. We .may combine or superpose 

 two corresponding figures patterned after figure 6. The curves representing 

 the loci of sections in which a given solid angle gives a certain traced angle 

 have also a simple and easily found equation when the plane of projection is 

 one of the sides of the solid angle. In figure 3 let CP^ = «, <,C : S = 0, 

 cot < SC : aS'Pj = cot [3 =^ m, < CP^ : CS = (f ; then, substituting in the 

 well-known trigonometric formula (Chauvenet, Trig. 10), 



cot a sin b =■ cos b cos c -\- sin c cot A, (1) 



we have — 



cot a sin = cos cos (f -\- sin <p m. (2) 



Now, let r = tan k CS= tan i = esc — cot 0. We have also ^. = esc 6 + 

 cot 6, and accordingly esc ='h(r + \ and cot 0=l( — j . Substitute 

 these values of esc and cot in (2), after dividing through by sin 0. Then — 



cot « = H r + - J cos cp + 2 ( - — r j m sin <p ; (3) 



or — 



2 r cot a = r^ cos c- + cos c? -f m sin (p — r^ m sin <p. 



The locus of S is therefore expressed in the following equation : 



2 r cot a m siii -f cos (f 



■ r^ -4- -• = — • • v^) 



' m sin (f — cos tp m sm (p — cos (s 



LV— Bull. Geol. Soc. Am., Vol. 2, 1890. 



