176 



verfcical columu with a familj or eacli oue of all tlie families on the liorizoutal 

 line on wMcli the former family A lies, and at the same time you make 

 another set of cards, on eacli of wliich is writteu iu the reciprocal order the 

 same combination as that just referred to above ; for example, a set of such 

 cai'ds for Salicacese-Tamaricacege, .... etc, and another set of cards for Tamari- 

 caceee - Salicacefe, .... etc. Take these two sets of cards together, and arrauge 

 them in any order you please, say in the alphabetical order*. If you fiud 

 that all of yoOT cards are each iu a pair, then you will find that your system 

 satisfies the condition proposed. If you find, on the other hand, any oue card 

 not in a pair, this shows you that the system does not satisfy the condition, 

 so far as the two families mentioned ou the card are concerned, and that, in 

 either of tlie two horizontal Hnes leading to one of the two families in the 

 middle column, one or the other of the two families is missiug. lu such a 

 case, you examine which of the famihes is missiug, and in which of the Hnes 

 the omission occurs, and place the missiug family on the Kne showing the 

 omission. At the same time, you make two new cards, one with a combiua- 

 tion of the two families, and the other with the same combiuatiou iu the 

 reciprocal order. Then, with either of the two new cards, you double the 

 origiual card, which has uutil theu beeu siugle, aud with the other uew card, 

 you double the other card with the same combination iu the reciprocal order, 

 which card in such a case you will surely find uumistakablly single somewhere 

 in your collectiou. If you do the same thiug' — adding famiHes to yom* system, 

 and cards to your collectiou — with all the single cards, theu you wiU perfect 

 your system, so far as the families iu the latter are concerned. The above 

 method which has beeu 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 seeu, we have simply coutemplated 

 each relation of each two famihes separately. But, if we thiuk of exhibitiug 

 the above relatious not separately, but joiutly, or relatious of each group of 

 three or more families, the system must become a very compHcated one; and 



* The best method is to give, before constracting the system, a mimber to each of the 

 f amilies of the f ramework — 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 paj^er, as this idea has first occurred 

 to me, ■when I was reading the proofs. 



