SIS APPENDIX CALCULATION OF THE 



REGULAR OCTAHEDRON. 



But from the structure assigned to the octahedron, 

 it is evident that the axis a o represents double the 

 number of molecules that are represented by an edge 

 a by and we may therefore in relation to the numbers 

 of molecules represented, consider 



a n 2 g a. 

 Hence g a : a i : : 2 g a a m i am. 



and g a-\-a i : a i :: 2 g a : a m. 



From this ratio we find 



a m = gg* ai 

 g a + a i. 



And by a similar proceeding we shall find 



m = 



h a -f af. 



Therefore 8g ' - *** "/ 



g a-\-ai h a -\- af. 



And 2 g a . a i : g a -f- a i ' :: 2 h a . af ; h a -\- af 



I + ^Z 

 ha 



1 : J. + _L::1:L + _I 

 cri g / A a 



Therefore 1 +-! = 1 + JL 

 at g a af h a 



1 1,1 1 



and, =: + 



a z af ha ga 



whence _ ~ _ 4- --- . 

 ^ r q p * 



* This formula was mentioned to me in conversation by Mr, Levy, 

 some months before it occurred to me as the result of the investigation 

 given in the text. But it was mentioned without any allusion to the 

 means l3y which it had been obtained; and these, I since learn from 

 Mr.L., were different from those employed above. 



