(.410 ) 



For (he foriner (he proof foUows by repeated application of" (he 

 law of sines, which gives according to the figure 



u V t a V V t 



.-tin tp" fiin xp' i s'intp shi ^i" sin (f' sin y^ si/i \p" 

 wliile 



X = If'' — </' = »!'" — <f\ ^ = ^Z -h M'' + <f" = V + V' 4- V'"- 

 From these we get 



sin {cp -[- <f') sin {tp 4" <fi") ' -^in {<P + (fi') sin {<p -{- <p") 



si?i if) sin (p' sin {<p -\- <p' -\- <p") sin (p 



The relation between r and o, is most easily obtained l)v deter- 

 mining (p' i'vom the first formula aiul substi(u(ing this value in the 

 second. We thus obtain 



r/ir-l) = Q. 



This shows that a rational r leads to a rational (>, which was to 

 be proved. 



The last |)art of the proof can still be simjtlified, according to a 

 suggestion by Lokknt/, if we assume the Mknkt,.u s' Theorem as known. 



The desired })roof is also given, when from 



— =r r' and — = r' 



V u 



V 



t" 



- = o' and := o' 



V t 



follow. 



The former of these we have considered above; relative to the 

 latter, Mknklals Theorem gives according to the figure 



BB _ O A' BB' 



AD ~~ AA' ' OB' ' 



i. e. 



t" _ n' ^ V — v' 



. ,11 I ' ' ' 



t — t U ■ — tl V 



Hence the ratiojiality of /i''/i and r'/n gives directly the rationality 



of t"/t. 



2. The determination of the })ermissibie order n of an axis of 

 symmetry follows from any one of the fundamental i)rinciples of 

 Crystallogra[)liy, but oidy for the case when n > 5, because each of 

 these principles places five simlhti- crysfallographic elements in relation. 

 We usually so proceed, that the general j)roperty which the principle 

 gives for the cases ;/ > 5 is also demanded for the cases n <^ 5 

 We can however for the latter /iii/ileti number of cases rely upo)i 



