466 WILLIAM HALLOWES MILLER. 



Another very large class of problems in crystallography is based ou 

 the relation of faces iu a zone ; that is of faces which are all parallel to 

 one line called the zone axis, and whose mutual intersections, therefore, 

 are all parallel to each other. If now h kl and p q r are the indices 

 of auy two planes of a zone (not parallel to each other) any other 

 plane in the same zone must fulHl the condition expressed by the 

 simple equation 



UU-\-YV-\-VfW=^0 



where u v and w are the indices of the third plane and u v w have the 



values 



\x=^k r-l q Y = I p-h r w = h q-k p 



Since h k I and p q r are whole numbers, it is evident that u v w must 

 also be whole numbers and these quantities are called the indices of 

 the zone. The three whole numbers which are the indices of a plane 

 when written in succession serve as a very convenient symbol of that 

 plane, and represent to the crystallograjiher all its relations; and in like 

 manner Miller used the indices of a zone enclosed in brackets as the 

 symbol of that zone. Thus 123, 531, 010 are symbols of planes, and 

 [111], [213], [001] symbols of zones. 



An additional theorem enables us to calculate the symbols of a 

 fourth plane in a zone when the angular distances between the four 

 planes and the symbols of three of them are known, but this problem 

 cannot be made intelligible with a few words. 



The few propositions to which we have referred involve all that is 

 essential and peculiar to the system of Professor Miller. These given, 

 and the rest could be at once developed by any scholar who was fa- 

 miliar with the facts of crystallography ; and the circumstance that its 

 essential features cau be so briefly stated is suflicient to show how 

 exceedingly simple the system is. At the same time, it is wonderfully 

 comprehensive, and the student who has mastered it feels that it pre- 

 sents to him in one grand view the entire scheme of crystal forms, and 

 that it greatly helps him to comprehend the scheme as a whole, and 

 not simply as the sum of certain distinct parts. So felt Professor 

 Miller himself; and, while he regarded the six systems of crystals of the 

 German crystallograjiliers as natural divisions of the field, he con- 

 sidered that they were hounded by artilicial lines which have no deeper 

 significance than the boundary lines on a map. How great the unt'old- 

 iug of the science from Ilaiiy to Miller, and yet now we can see tlie 

 great fundamental ideas shining through the obscurity from the first. 

 What we now call the parameters of a crystal were to Ilaiiy the fnnda- 



