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vertical column with a family or each one of all the families on the horizontal 
line on which the former family A lies, and at the same time you make 
another set of cards, on each of which is written in the reciprocal order the 
same combination as that just referred to above; for example, a set of such 
cards for Salicacese-Tamaricacer,.... etc., and another set of cards for Tamari- 
cacex-Salicacem,.... etc. Take these two sets of cards together, and arrange 
them in any order you please, say in the alphabetical order*. If you find 
that all of your cards are each ina pair, then you will find that your system 
satisfies the condition proposed. If you find, on the other hand, any one card 
not in a pair, this shows you that the system does not satisfy the condition, 
so far as the two families mentioned on the card are concerned, and that, in 
either of the two horizontal lines leading to one of the two families in the 
middle column, one or the other of the two families is missing. In such a 
case, you examine which of the families is missing, and in which of the lines 
the omission occurs, and place the missing family on the line showing the 
omission. At the same time, you make two new cards, one with a combina- 
tion of the two families, and the other with the same combination in the 
reciprocal order. Then, with either of the two new cards, you double the 
original card, which has until then been single, and with the other new card, 
you double the other card with the same combination in the reciprocal order, 
which card in such a case you will surely find unmistakablly single somewhere 
in your collection. If you do the same thing— adding families to your system, 
and cards to your collection—with all the single cards, then you will perfect 
your system, so far as the families in the latter are concerned. The above 
method which has been stated as to the families in the system will also hold 
good as to the series in the same. 
In the present system, as we have seen, we have simply contemplated 
each relation of each two families separately. But, if we think of exhibiting 
the above relations not separately, but jointly, or relations of each group of 
three or more families, the system must become a very complicated one; and 
* The best method is to give, before constructing the system, a number to each of the 
families of the framework —a number corresponding to each of their respective orders —-, and 
then in the present case arrange the cards in the numerical order instead of the alphabetical 
order. This method is but partly followed in the present paper, as this idea has first occurred 
to me, when I was reading the proofs. ; 
