2g2 MR. F. BALFOUR BROWNE ON AQUATIC COLEOPTERA. 
from g to 9.02 species per collection. If, on the other hand, 
we have only 9 collections containing 9 species and make one 
more collection and find 10 species, the average is raised by 
the additional species from 9 to 9.1, giving thus a difference 
of .08 in the two results. If, on the other hand, instead of only 
finding 10 species in the final collections we had found 20, 
the average would have been raised in the first case to 9.11 
and in the second to 10. 1, a very considerable difference. 
I therefore made enquiries as to whether the law of pro- 
babilities could so adjust such differences as to give me results 
of equivalent value for the varying number of collections ; 
but beyond learning that some application of the calculus 
would lessen the discrepancies, I have not heard of any 
principle in mathematics which could assist me. I have, 
therefoi'e, had to adopt an entirely different method since 
the number of collections per month varies so greatly and 
this method consists in taking a pair of months and including 
only the collections made at collecting places visited in 
both months, that is, leaving out of consideration all the 
collections made in one month if the same collecting places 
were not re-visited in the other month. Thus, for instance, 
I visit collecting places A, B, C, D, E, in March, and in April 
I only visit A, C, and E. Collections made at B and D will, 
therefore, have to be left out in working out my results. By 
this method I can compare March with April and see whether 
the number of species has on the average increased in April 
at the collecting places which were visited in March ; similarly 
I can compare April with May, May with June, and so on, 
and I can from these results draw a curve showing the increase 
or decrease in species per collection from March to April. 
April to May, May to June, and so on. The result of such 
a comparison is shown in the dotted curves in Chart I. 
Now it is obvious that in this method I may be using exactly 
the same collecting places in comparing March with April as 
I use in comparing April with May, in which case, of course, 
the increase or decrease from April to May will, when combined 
with the increase or decrease from March to April, give the 
true relationship between March and May. As a fact, how- 
