114 
This question of reticulations is closely connected with the 
theory of the polyedra, and is not a whit less complex and 
difficult. 
I am in possession of a complete solution of this problem 
of reticulations, of which, however, I shall for a season defer 
the publication. My object in this Paper is chiefly to place 
on record some of my numerical results. And I think it 
highly probable that any mathematician, who may take the 
trouble to verify them, will completely satisfy himself as to 
how far I am in possession of the entire theory of these 
reticulations. 
It is worth while to remark, that investigations of this 
nature have recently acquired a new and important interest, from 
the fact, that the Imperial Institute of France has chosen the 
subject of the polyedra for its grand Mathematical Prize (the 
gold medal of 3,000 francs) for the year 1861. The proposal 
runs thus — “ Perfectionner en quelque point important la 
Theorie des polyedres.” The memoirs are to be sent in 
before July 1, 1861, written either in Latin or in French. 
Among the propositions that I have to communicate, are 
the following : 
1. The number of 7-reticulations of the pentagon is 7774, 
of which 413 are symmetrical. 
2. Of these, the 4-nodal 7-partitions are 62 symmetrical, 
and 1006 unsymmetrical. 
3. The 5-nodal 7-partitions are 85 symmetrical, and 2000 
unsymmetrical. 
4. The 6-nodal 7-partitions are 99 symmetrical, and 2282 
unsymmetrical. 
5. The 7-nodal 7-partitions are 69 symmetrical, and 1340 
unsymmetrical. 
The remainder have more than seven nodes, or less than 
four. 
In this enumeration no figure is counted which is either the 
repetition or the reflected image of any other. 
