231 



way, none crossing another, so that one at least shall pass through 

 each of the s points, we shall construct the asymmetric ROF in ques- 

 tion among the results a certain number of times, namely, in every 

 position of ROF in which erasure of the d diagonals and of the s 

 diaces will reproduce the primary R'O/; and if R'0/be polar, there 

 may be many such positions of ROF for the same position of R'O/. 

 And it is evident that we shall construct equally every plane reticu- 

 lation ROF of every symmetry which can reduce by the same process 

 of effacement to the same primary R'O/. 



Nothing is easier than to determine the number of asymmetric 

 constructions thus obtained of ROF [2 T M>], if the number of all pos- 

 sible ways of drawing the d diagonals can be found. 



We have to employ in turn every possible partition of the * points, 

 of which one is 



.+a m (m = u), 



There is a given number of ways of depositing a l points on any 

 one of the u submargins, 2 points on any other, &c 



All that is difficult is to determine, when a disposition of the 

 a \ + a z+ ' ' + a m points is made, in how many ways d diagonals can 

 be drawn in the 



(T+w=)r-gon, 



so that one at least shall pass through each of the s points. Let this 

 number be 



It is given always by the equations following : Let t = 1 ; then 



rd ai a 3a3 ...=rda 9a3 ...-(r-l)d at a 3 ...'-(r 

 Let a l > 1 ; then 



r (ai -i) 0303... 



where 



H(r-l) U(d+ 1) n(r-rf-3) U(d) 



is what I have called in my memoir " On the /^-partitions of the 

 R-gon and the R-ace" (Phil. Trans. 1857), the (c?+ l)-di visions of 

 the r-gon. 



This number rd ai a 2 a 3 ... being thus given for every partition of s, 



