N WILCOX : SPERMATOGENESIS. 23 



geneous, or, if they are not homogeneous, that there is an exact halving 

 of the component particles of the elements of the Vierergruppe. But 

 Brauer considers the four elements of a group identical because they all 

 arise, by two divisions, from one. 



Again, if this whole process be only to secure a reduction of the mass 

 of chromatin, the doubling of the chromatic elements, and the long, 

 laborious process of mitosis would be unexplained and unjustified, as 

 Weismann has pointed out ; for a halving of the mass could be brought 

 about by amitotic division. According to Weismann, the formula for 



Brauer's Vierergruppe would be (Hacker, '93) ■< > . We start 



with one element, a ; this undergoes two longitudinal splittings, and 

 then two separations by the two maturation divisions, and we then have 

 just what we started with. The series formulated would be 



a a a a 



a a a a 



a _a 



a a 



But Henking, Weismann, Hacker, Riickert, vom Rath, and others 

 allow only one longitudinal splitting, and their formula for the Vierer- 

 gruppen, as stated by Hacker ('93), and accepted by vom Rath ('93), 



a a 



This evidently permits only one reduction division in the 

 b h 



Weismannian sense. Vom Rath and Weismann are therefore inconsistent 



when they hold to a longitudinal splitting in the spirem condition, and 



yet consider both maturation divisions as reductions. If the Vierer- 



gruppen have the formula i t j, r j there are but two sorts of ids, a and 



h, and it is simply impossible to get more than one reduction division. 



K from the nucleus arises by division two nuclei, and , 



h h ^ h h 



this is by Weismann's own definition an equation division, and only 

 when these two cells become by division the four ultimate products of 

 maturation a, b, a, b, can we speak of a reduction. 



Hacker at first considered both divisions as reductions ('92), but 

 later ('93) he rightly came to the conclusion that the longitudinal 

 splitting in the spirem stage was a preparation for one division, and 

 that the final separation of the sister elements thus produced consti- 

 tutes an equation division, — a "modified equation division," he calls 

 it, because the splitting, which ordinarily occurs at the equator of the 

 spindle is here precociously introduced in the spirem condition. 



