and used in his system for the indices of a plane the values _h_:_k_:_l_, which are also in the ratio of whole numbers, and usually of simpler whole numbers than _m_:_n_:_p_. This seems a small difference; for _h_ _k_ _l_ in the last proportion are obviously the reciprocals of _m_ _n_ _p_ in the first; but the difference, small as it is, causes a wonderful simplification of the formulae which express the relations between the parts of a crystal. From the last proportion we derive at once (1/_h_).(_a_/_A_) = (1/_k_).(_b_/_B_) = (1/_l_).(_c_/_C_), which is the form in which Miller stated his fundamental law. [G] For example, the native crystals of sulphur have _a_:_b_:_c_ = 1:2.340:1.233. Crystals of gypsum have _a_:_b_:_c_ = 1:0.413:0.691. Crystals of tin-stone have _a_:_b_:_c_ = 1:1:0.6724. And crystals of common salt have _a_:_b_:_c_ = 1:1:1. If _P_ represents the "pole" of a face whose "indices" are _h_ _k_ _l_, that is, represents the point where the radius drawn normal to the face meets the surface of the sphere circumscribed around the crystal (the sphere of projection, as it is called), and if _X_, _Y_, _Z_ represent the points where the axes of the crystal meet the same spherical surface,[H] then it is evident that _X Y_, _X Z_, and _Y Z_ are the arcs of great circles, which measure the inclination of the axes to each other, and that _P X_, _P Y_, and _P Z_ are arcs of other great circles, which measure the inclination of the plane (_h_ _k_ _l_) on planes normal to the respective axes; and, also, that these several arcs form the sides of spherical triangles thus drawn on the sphere of projection. Now, it is very easily shown that (_a_/_h_).cos _P X_ = (_b_/_k_).cos _P Y_ = (_c_/_l_).cos _P Z_; and by means of this theorem we are able to reduce a great many problems of crystallography to the solution of spherical triangles. [H] The origin of the axes is always taken as the center of the sphere of projection. Another very large class of problems in crystallography is based on the relation of faces in a zone; that is, of faces which are all parallel to one line called the zone axis, and whose mutual intersections, therefore, are all parallel to each other. If, now, _h_ _k_ _l_ and _p_ _q_ _r_ are the indices of any two planes of a zone (not parallel to each other), any other plane in the same zone must fulfill the condition expressed by the simple equation u._u_ + v._v_ + w._w_ = _o_, where _u_ _v_ and _w_ are the indices of the third plane, and u v w have the values u = _k_._r_ - _l_._q_ v = _l_._p_ - _h_._r_ w = _h_._q_ - _k_._p_.