12 DAY, Variation in a Carboniferous Brachiopod. 



represent the actual numbers of shells occurring in the 

 corresponding squares. 



While the general agreement of the actual density of 

 individuals in each square with the theoretical expectation 

 is fairly evident, it is better seen in the condensed form of 

 the graph represented in Fig. 6 where the squares rep- 

 resent intervals of '15, in the indices (corresponding with 

 the thickened squares in Fig. 5). Any systematic deviation 

 of the actual numbers from the theoretical expectation 

 will indicate a correlation between breadth and depth, 

 The existence of such a deviation can most readily be 

 seen by summing the numbers in quadrants as has been 

 done in Fig. 7. (the theoretical expectations have here 

 been reduced to chances per thousand, as also in Fig. 6, 

 for more ready comparison with the actual numbers). 



The only kind of correlation which can be reasonably 



expected is one in which the depth ( jz) increases either 



directly or inversely as the breadth (-^J. If such a corre- 

 lation were complete the position of every individual on 

 the graph would fall on a line passing through the modal 

 point x. 



In the case of direct correlation this line must be 

 contained in the quadrants a, a ; with inverse correlation 

 it will fall in the quadrants /;, b '. If such a correlation 

 exists, but is incomplete it will be shown by a concentra- 

 tion of the individuals towards such a line, resulting in a 

 concentration in the pair of quadrants in which the line 

 falls. The degree of this concentration is clearly a 

 measure of the correlation, and may be expressed as 



{a + a')-(b + b') 



{a + a') + (b + V) 

 In this case we have 



(308-5 + 287)- (144+ 245-5) = 2o6 = . 2I 

 (308-5 + 287) + (144 + 245-5) 9 8 5 



