( ilJ ) 



experience, and apply (lie [)riiicipl(' only lor llic formci" imHinllrd 

 mimber of cases. 



Since t.iie principle of llie ralional indices pcrnnis sni'laces of llic 

 ci'Vtstal polyhedron lo be li-anslated parallel lo llieniselves, lliei-efore 

 foi' ils applicalion axes of syminelry of llic lirsl and second kind 

 lia\e exactly the same \alne. A diilerence lies only in that for axes 

 of the second kind // ninsl necessarily he an exen nnnil)ei'. 



We start xvith a constrnction npon a sphei'c of nnit radins, lhi-oii,i;h 

 the center of which we lay all the directions that come into con- 

 sideration. (Fig'. '1). Let .1 be trace of the //-fold axis, l\, l\ . . . /^, 

 the traces of the normals of 5 related snrfaces (1), (2), . . . (5) of 

 the polyhedron, sncli that if' = 2.t///. The J^ks are theji designated as 

 the poles of these snrfaces. 



Fig. 2- 



Let K^, K^, K^, be the traces of the lines of iidersection of the 

 snrfaces (2,3), (3, I), (J, 2), so that 



Let AV K.-^, /{,, form the system of axes, and (4) and (5) the pair 

 of snrfaces for the application of the |)rinci])le of rational indices. It 

 is now^ a qnestion of determining the intercepts which the snrfaces 

 (4) nnd (5) make npon the edges K;. 



If w^e give the snrfaces snch positions that they are tangent to the 

 sphere in their poles, then the sections (fki are identical with the 

 reciprocals of the cos {Ki l\) wdiere /■=:i:l,2,3, A = 4, 5. Conse- 

 qnejitly the xalnes of these cosines are to be determined. 



If we write y;, instead of J A',-, (/^ for .17^/, and X'-/ for the X 7^/, . I A'/ 

 then from the LKiAPh in the figure we get 



eo.^ (A' Fi) z=z. cox (f cos yi -f sin (f si» yj cos y}> \ ... (4) 



