EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-OON AND R-ACE. 271 
And (Problem w) 
P( 10 , 7),=2,2.|4p‘(10, 6 ).+^KSniO, 6).J, 
the quantities here omitted from the formula being 
Ilf“^'^'(10, 6)„=0, because 10 is not divisible by 4A (Problem e). 
(10, 6)„=0, for the same reason (Problems f, g, h). 
j^2Aajdi(10^ 6}„=0, as it has been proved in the preceding article. 
Ilf“^*(10, 6)„=0, because 10 is not divisible by 4A (Problem b), 
=Rf'^' = 0, for the same reason (Problems k, 1). 
Now the equations of Problem u for P*(10, 6 )„ are those of Problem 1 for Ilc( 1^5 6 )„, 
and we have proved in the preceding article that /i=l and n=4 are the only values 
affording a solution ; which gives 
2(AA = 2(A2)=14. 
\ 2A/ 
Wherefore (Problem w) (7>0), 
P(10, 6).=f|l4-2.(5^I»*'»(10, 6).+|.E“(10, 6).)}. 
The formula for I"(r, A)„ shows that P(10, 6 ) 4 = 0 , for — is not integer, and we 
have just proved that R^*( 10 , 6)„=0 under every form except R^*“^''*( 10 , 6 )„, which 
remains to be determined. 
In Problem d we see that r — n — 2/i is divisible by 4 /i; wherefore ?i=4 and /i = l. 
Therefore, by the formula for R^, 
R^^‘^'(10, 6}4=2„,R'"-’”"(5, 2)=R’«“(5, 2)=1, 
as is easily verified. Consequently 
P(10, 6).=f{14-fE“»"(10, 6),}=il^=3, 
and P(10, 7)=P(10, 7),=tP(10, 6).+|,E”''*(10, 6),=6 + l = 7, 
whence, finally (Problem x), 
1(10, 7)=Jo{D(10, 7)-^R(10, 7)-^P(10, 7)}=^o(1430-10. 14-10. 7) = 61. 
And this enumeration of the 8 -partitions of the 10-gon agrees with that at p. 410 of 
my memoir mentioned in the first article. 
It is evident that this last example would have caused far less trouble, if we -had had 
our register of partitions filled up for inspection up to the 7 -partitions of the 10 -gon. 
There is little difficulty, as I have verified to some extent, in framing, by the aid of 
the results here given, algebraical expressions, containing circulating functions for R*(r, A) 
and P(r, A), in terms of r and A only, and thence by addition, complete expressions of the 
(l 4 -A:)-partitions of the r-gon. But the subject has been pursued far enough for one 
communication. 
