34 Proceedings of the Roijul Irish Academ//. 



If we arrange these figures, we can determine certain constants which are 

 of great importance. The prohaUe error, for example, will give us the measure 

 of unreliability of the counting and abstraction of the samples. To this 

 eifect the numbers found in the various counts have been arranged in classes, 

 where five is the number of units in the class-range and 138 has been taken 

 as a nlean, though correctly 136'4. 



Now, the probable error is given by the formula : — 



+ 0'6745 X standard deviation = + 0'6745 x o-. 

 The standard deviation a is given by the formula: — 



1:2 (X' 



sum of [(deviation of class from origin)^] x frequency of class 

 - (distance from origin to mean)^ 



"^' 



number of variates ; 



This is /^ - (1-6)2 = /l84-5 _ (1-6)= 



.-. (T = 13-49. 



The probable error = ± 0-6745 x 13-49 

 = ± 9-1029 



Thus we have a figure showing to what extent the number of Ceratium in 

 one of our counts is probably correct. 



The probable error in the withdrawal of organisms by the pipette and 

 the quantitative enumeration by counting is small, but at the same time is 

 greater than it ought to be. We may take from the above, however, that for 

 this particular case a difference of about 30 above or below the mean in a 

 count would be required before we could assume an actual difference in the 

 constitution of the catch. We are not aware that a series of experimental 

 counts with a plankton organism has been worked out before in this way. 



From a paper in " Biometrika," vol. v, p. 351, it would appear that in such 



counts as ours the o- should = /^, where vi is the mean number per unit 



volume and M is the number of unit volumes counted. This in our case is 



ir-, but v^l36 is rather different from \/'l82, and shows there was some 



heterogeneity in our samples. 



The serious errors in quantitative work are in the process of catching by 

 means of net or other apparatus. It is obvious that unless the organisms are 

 more or less uniformly distributed in the water horizontally, the sample 



J 



