384 RESEARCHES ON FUNGI 



fifth set of gills were present, they would number 8x and be inter- 

 polated in as many spaces. The total number of gills on the whole 

 pileus would be the sum of 



x -\~ x + 2# + 4tx -\- Sx 

 or, for n sets of gills, 



x -}- x -\- 2x -\- 4x -{- 8x to n terms, 



the sum of which is equal to 



x X 2"- 1 . 



If a perfectly symmetrical mushroom had 132 long gills and three 

 distinct sets of gills, the total number of its gills would be 



132 + 132 + 264 == 528 



and, if it had four sets of gills, the total number of gills would be 

 132 + 132 + 264 + 528 = = 1,056, 



so that it is evident that every additional set of gills produced at 

 the periphery of the pileus doubles the previous total. In nature, 

 owing to the large number of factors which affect the growth of the 

 cells making up the fruit-body, such perfect symmetry is never 

 attained but only approximations thereto. The gills of each set 

 vary considerably in length about a mean, and these variations 

 become increasingly greater in each set as one proceeds from 

 Che stipe centrifugally. 



If one examines the under side of a pileus, one finds that the gills 

 are distinctly more crowded at the periphery than in the region of 

 the stipe and a little way from it. It so happens in the pileus shown 

 in Fig. 137 that the circumference of the pileus, where the gills 

 end, is just four times the circumference of the stipe. There are 

 639 gills at the margin of the pileus. If the crowding of the gills 

 were the same in the region of the stipe as at the pileus-periphery, 

 then there ought to be 159 gills at the stipe ; but, as a matter of fact, 

 there are only 132. The extra crowding of the gills at the pileus- 

 periphery is due to the fact that the gills in approaching their outer 

 extremities become shallower and shallower (cf. Fig. 134, p. 377), 

 and in accordance therewith thinner and thinner, so that each one 

 occupies less and less space where it is attached to the pileus, much 



