25 
Quarterly Meeting, January 26th, 1858. 
Dr. J. P. Joule, Vice-President, in the Chair. 
James Whitehead, M.D., Henry Roscoe, Ph.D., F.C.S., 
Mr. Davis Benson, Mr. Joseph Baxendell, F.R.A.S., Mr. 
C. P. Stewart, and Mr. Joseph Casartelli, were elected 
ordinary members of the Society. 
The Rev. H. H. Jones communicated an account of the 
fall of rain at Rugby as observed by Mr. Frederick Fuller, 
from which it appeared that 21.814 inches fell in 1855, 20.548 
in 1856, and 25.209 in 1857. 
A Paper was read by the Rev. T. P. Kirkman, A.M., 
F.R.S., entitled “General Solution of the Problem of the 
Polyedra.” Given any Q-partition and jo'-partition of 2e, 
such that 
(A) A-pB-p . . +Q=a+5+ . . 4-J3— 2(Q'— l-py— l)=2e 
to determine the number of p'-acral Q'-edra, having for faces 
an A-gon, a B-gon, &c., and for summits an a-ace, a J-ace, 
&c., e being the number of edges.* 
Let Q' columns, headed by ABC ... in any order, form 
p' rows headed by abc ... in any order, and let M„ be the 
point common to the column under M and the row headed by 
n. If we write M„=l, or M„=0, 
according as the M-gon has or has not the w-ace for one ot 
its summits, i.e., as the w-ace has or has not the M-gon for 
one of the n faces about it, we shall read M units under M, 
and no more, showing the summits of the M-gon, and n units 
in a row with «, showing the faces about the n-ace. And we 
have the equations, 
A=iAa-pAft-p . . . -pAp = A^-pAj-p . . . -pAp, 
B^Ba-pBft-p . . . +Bp = B„-pBj-p . . . +B^, 
Q=Q„+Q,-P . . . -pa^ = Q^+QI+ . . . +Q;; 
(Aa) 
a=Aa-pBa-p . . . +Qa — Aa-pBo-p . . . +QL 
i^zAj-pBft-p . . . -pQfi = Aj-pBj-p . . . +0.4, 
=* r 
p = A ,+ B ,+ ...+% = a;-pb;-p . . . -pq;; 
* Vide a paper “ On Polyedra,” Manchester Memoirs, vol. 12. i 
Proceedinqs No. 9. 
