62.1 ox KyLILATICK AL TKl ANGLES. 



The direction cosines /, /', ' of tliese principal radii ari 



i A ^( r I ' 





Now if the centre of this (/ ni n) equilateral face is f /y referred 

 to the i^rincipal axes, we have, from (iii) above, 



-V .,(X-2 <r + 3 /-^ ,^f + t!^ (F ,3^ + 3 X-2 «^)^ = (/-^ n^ - /-^ /P)=^ 



.-. (a- - li-'r k- = -J (,r + 3 ;5^)^ + ^ OP + 3 «^f . 



1 6 /'— 4- ^^ 

 and (its radius)- = AT 1^ JI^J ''= _l^ tLl 



V73- ci-J 

 and the lieiglit of the regular tetrahedron being R \/2, we have 

 for the opposite corner .r = Ola- + \ £ + X' ;/ ± R v/2 ./. etc. 



The condition — + ^ -f- ' =: 1 gives 



l=«^V(/2„:>) + i:::v ^' + „:i v >^ ' + 2R^v ^' ±2 R v/ 2 j « + i v ^^ + ,, 2 ^ ^ | 

 <i- II- a- ' (I- II- ' 



.■./•^'= '', + '''+ 2R^^| V ' -V ^ |h-2Rs/2S ' 



<r /)- ' II- (J- » ', * 



.-., P. P being V '''^ v^y, 

 ir ir 



± 2 R v'2 («+P£ + P ,,)="» ("J-' +1/"^' _ 1 j +... _2R^'» vi_ V y* 



= ''-^'-,^"^tr?^ + ...-...=H^*.lv ^ -2rv ^ - V ^V 

 «- («- — /7-)- ' »-' \ 11- a'-J ) 



..O^Vi+l- i,= ± ^^ \n^ ' -2v \[ = -^^i'.sav. 



2^/2' - a- ii-S -2^2 • 



And. if 7 y J is the cenlrc of the tetrahedron. 

 7 = ri//-7 + \; + \ ,;-4- ^ ./, 



= i{\-V<i'l)^,,{ . . )± -^. (C./^'+l) 



^ ^ z 



= iX . + '/X' -f- 



