262 Penjield — Solution of Problems in Crystallography 



tion and subtraction, carried out according to the following 



scheme :* 



A, hlcl 1-2 1-3 2-3 



P^ mno _hn — lcni . ho—lm , ho— In 



B^ mno 

 D, xyz 



iny — nx mz — ox nz — oy 



D^ xyz 



C, pqr^ ^ ^q-yp . 

 C^ pqr 2)k—qh^ 

 A. hkl 



xr—zp . yr — zq 

 pl—rh ql—rk 



The result of multiplying one of the upper expressions by 

 that directly beneath gives the desired intercept. The result 



'l€^'^ 



X 



wo 



in some cases is indeterminate, as, for example, schemes 1 — 3 

 and 2 — 3 applied to prismatic zones, where the third index of 

 the forms is zero, or scheme 1 — 2 applied to a zone between 

 some prism and base, where the first and second indices of the 

 forms maintain the same ratio. When a numerical solution is 

 desired the following equation is generally made use of : 



Cot AC — cot AD ..... 



T^r— — -r—Fi — ir-r^ = ratioual intercept. 



Cot AB — cot AD ^ 



Provided AD equals 90°, its cotangent equals zero, and the 

 problem is then much simplified. It may then be easier to 



* Note. — The scheme of cross-multiplication and subtraction here sug- 

 gested is essentially like that given in Dana's Text Book of Mineralogy, page 

 31, Groth's Physikalische Krystallographie, page 584, and other works, but 

 by adopting, as has been done, a somewhat more symmetrical arrangement 

 and procedure, the operation is more easily remembered and carried out. 



