290 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tendency toward the production of corresponding crystal faces, the 

 degree of probability for the production of a crystal face in any zone 

 depends directly on the number of times the primary crystallizing 

 forces for that zone have been resolved in order to produce a sec- 

 ondary, tertiary, quaternary, etc., force perpendicular to the given 

 crystal face. If we represent two primary directions of particle- 

 attraction by and °° and assume this for the first normal series (N ) 

 the subsequent results of division and redivision expressed in descend- 

 ing normal series will be : 



No 0, co 

 Ni 0, 1, oo 

 N 2 0, 



J-M U> 5> 3. 5> 2' 5' 3> 4» *-, 3, 2> 3' ^> 2' °> ^> 



The principles of this method of developing normal series may be 

 then stated as : 



1° The probability and therewith the frequency of a number is 

 the greater the nearer the number stands in development to the origin, 

 i. e., the lower the order of the series in which it first occurs. 



2° For each derived number and the corresponding form there is a 

 certain intensity. The probability and frequency of a form is the 

 greater the greater its intensity. 



Any series representing forms lying in a zone and derived from the 

 crystallographic indices of those forms may be reduced to the series 

 form 0. ... co by the addition of a constant to each number of the 

 series, by multiplication by a constant, by inversion or by remodeling 



by the formula in which v is anv intermediate member in 



Vo — v 



a series, Vi the first member and v 2 the last member of the series. A 

 zone or zone section when developed by these rules and compared 

 with some member of the normal series N n may be identical with its 

 normal type, in which case it is probable that this zone or zone section 

 is of normal development and has suffered little or no disturbance. 

 If, on the other hand, such a zone contains terms which do not accord 

 with the normal series such terms invite criticism and should be criti- 

 cally considered from all other points of view, such as zonal intersec- 

 tions, size and definiteness of the faces observed by the author describ- 

 ing them, etc. Such terms are designated "extra" forms and are 

 enclosed in parentheses. Where terms of the normal series N n are 



