338 
Merism and Sex in " Spinax Niger''' 
hypothesis of intercalation. Lastly there is to be considered the possibility that 
the variations in the number of the meristic series and of its various portions are 
due to random addition to or subtraction of segments from a normal number. This 
is a point which, though of the greatest interest to students of variation, is yet 
extremely difficult for the biologist to test. In the following paragraphs Prof. 
Pearson has very kindly undertaken this task. It is sufficient here to emphasize 
his conclusion that the observed correlations are not in agreement with the 
correlations calculated upon the hypothesis under consideration. In other words 
random interpolation or excision of segments will not explain the variations 
observed, and we are consequently forced to adopt the principle of homoeosis as 
the only conception hitherto offered which affords an explanation of all the facts. 
\_0n the Random Increase and Decrease of Segments and on the Correlations 
between the three Vertebral Regions of Spinax niger. 
By Karl Pearson. 
I must first state that I do not fully grasp either the hypothesis of excalation or 
homoeosis, or the manner in which biometric analysis can be used as a criterion 
between them. But we can, I think, ask how far the existing correlations are in 
keeping with : 
(i) the proportional insertion of segments into a series of three mean groups ; 
(ii) the random insertion of segments into the same groups. 
Let X = number of segments up to anterior spine, which we will call the anterior 
series. 
y = number between anterior and posterior spines, which we will call the median 
series. 
z = number beyond posterior spine, which we will call the posterior series. 
t = total segments = x + y + z. 
Let us suppose all these quantities measured in whole vertebrae as units. 
Let the mean numbers be x, y, z, t. Suppose variations ^, t], r to occur in 
these numbers, so that 
T = ^ + V+^. 
Then if t were always distributed in any definite proportions whatever between 
the three groups, say : 
where + +p3 = 1, we should have 
<^x=PiO't, = p.2a-T , cr^ • jh'^T ) 
or: crr = o-rc + (ry + a^, 
'>'yz = t'zx = rxy = r^t = Tyt = r^t = 1. 
