100 THE QUANTITATIVE METHOD IN BIOLOGY 



§82.— ALTERATIONS OF THE CHESS-BOARD SYSTEM 



(continued from § 80). GRADATION IN A CHESS-BOARD 

 SYSTEM X OR IN THE SYSTEMS ;v + i, %+ 2, ETC.— In the 

 examples mentioned in §§ 80-81, gradation (Part VIII.) has been 

 purposely overlooked. In reaUty, however, it is very difficult 

 to find a chess-board system in which gradation does not exist. 

 Here I content myself with a schematic example. 



I start from the regular system x represented in Fig. 13, i. 

 I suppose (Fig. 13, 2) : (i) that x is divided in the direction 

 EW into two segments, x+i, the limit sy-sy being a plan of 

 symmetry ; (2) that each segment a; + 1 is divided into three 

 longitudinal segments x + 2 (a, b, c ; c, b, a) ; (3) that each 

 longitudinal segment ;t + 2 is divided into a series of segments 



x+^ following one 

 another in the direction 

 NS] (4) that each seg- 

 ment x + ^ is adorned 

 with a central spot. 



Taking each prim- 

 ordium separately, I 

 suppose (5) that the 

 breadth of the segments 

 x + 2 {a, b, c ; c, b, a) 

 decreases regularly in 

 both directions (east- 

 ward and westward) 

 from the plan of sym- 

 metry to the lateral 

 borders ; (6) that in 

 each longitudinal seg- 

 ment x + 2 the length of 

 the segments x + ;^ de- 

 the decrease being less 



Fig. 13, I. System x 



creases from the base to the summit, 



rapid in c, c than in b, b, and in b, b less rapid than m a, w, 

 (y) that the gradation of the breadth of the spots is governed 

 by the same law as the gradation of the segments in the direction 

 EW ; (8) that the gradation of the length of the spots (dimen- 

 sion NS) is governed by the same law as the gradation of 

 the length of the segments (of each longitudinal row) in the 

 direction NS (the spots disappear at the extremity A^ of the 

 segments a, a, because their length becomes o). 



In this way we obtain the system represented in Fig. 13, 2. 

 This example gives us an idea of the endless variety of the 

 derivatives of the biaxial system produced by gradation. (In 

 the direction of each axis segregation may be observed. See 

 § 38, p. 45.) However complicated the derivatives may be, 

 it is possible to analyse them by measuring each primordium 



