236 REV. T. P. KIRKMAN ON THE PARTITIONS 
ting the j-nodal /-reticulations of the 7-gon, which involves 
no tentative process, nor any reference to figures. The 
subject is far too extensive to be here discussed, and the 
process of computation is very tedious, owing to the great 
number of formule due to the various symmetry and ge- 
neration of the results, which, for small values of 7, 7, and 
k, are collected in but small instalments. I shall content 
myself here with giving a complete account of the 7-reti- 
culations of the pentagon; and I think it highly probable 
that any mathematician, who may verify my propositions, 
will fully satisfy himself as to how far I am in possession 
of the whole theory, 
1. The entire number of 7-reticulations of the pen- 
tagon is 7778, of which 413 are symmetrical. 
2, Of these, the 4-nodal 7-reticulations are 62 sym- 
metrical and 1010 unsymmetrical. 
3. The 5-nodal 7-reticulations are 85 symmetrical and 
2000 unsymmetrical. 
4. The 6-nodal 7-reticulations are 99 symmetrical and 
and 2282 unsymmetrical. 
5. The 7-nodal 7-reticulations are 69 symmetrical and 
1340 unsymmetrical. 
6. The remainder have more than 7 nodes, or less 
than 4. 
In this enumeration no figure is counted which is either 
the repetition or the reflected image of any other. 
18. I have to correct two oversights in the sixteenth 
and twenty-second articles of my Paper “ On the triedral 
partitions of the xv-ace,” in the present volume. 
In the sixteenth article, not only doubly irreversibles 
but also the doubly reversibles of Art. 15 are constructed, 
so that these ought to be subtracted before dividing by 
two. 
in the twenty-second article, the triply reversibles of 
Art. 20, as well as triply irreversibles, are constructed ; it 
