normal to Four Faces of a Crystal in one Zone. 103 



But 



cosPY_sinPDE cosJPZ _ sioPED cos PX ^ sin PFE 



c;^iTZ~sinPDF' cosPX~sinPEF' cos PY sinPFD' 



cosGY _ sin GDE cosGZ _ sin GED cos GX ^ sin GFE 



cos GZ ~ sl^lGDF' cosGX^sinGFE' cosGY sinGFD" 



Therefore 



sin PDE _ oV sin GDE sin PEP _ ?rtW sin GED 

 SlPDF ~ 'M sin GUF' sin PEF ~ o'v! sin GEF ' 



sin PFE _ n!v! sin GFE 

 sin PFD ~ m!v' sin GFD" 



Hence if 



^ o'v' sin GDE , ^ m'w' si n GED 

 *^^^=^'skrGDF' ^""'^-T^s-b^EF' 

 _ n'w' sin GFE 

 *^^'^~;^'sinGFD' 



tani(PDF-PDE) = taniEDFtan( J -^), 



tani(PEF-PED) = taniFEDtan (^-<^)> 



tani(PFD-PFE) = taniDFEtan {^-^j- 



Hence, two of the angles EDF, FED, DFE, and the segments 

 into which they are divided by PD, PE, PF, being known, the 

 position of P is determined. 



Having given the symbols and positions of four poles, and the 

 distances of any two of them from a fifth pole, to find its symbol. 

 Retaining the construction and notation of the precedmg 

 article, let the symbols and positions of D, E, F, G, and the 

 distances PD, PE be given, to find the symbol of P. Since PD, 

 PE are known, the segments into which any of the angles ED F, 

 FED, DFE are divided by PD, PE, PF may be computed. The 

 ratios of u', v', w' may then be found from any two of the three 

 equations 



i/ _ n' sin GDF sin PDE w' _ o^ sin GEF sin PE D 

 U~o' sin GDE sin PDF' v! " m' sin GED sin PEF' 

 u' m' sin GFD sin PFE 

 1?~ n' sin GFE sin PFD' 

 The indices w, v, w are then given in terms of u', v', w' by (13). 



