7-PABTITION8 OP X. 141 



The reason of all this may be found in the fact, that the 

 elements d p of the circulating constant in ft P« all satisfy, 

 except only twelve of them, 



while all of them satisfy 



*** ^-±M0. 



We proceed to deduce from the expression for 6 Px, that for 

 7P,, the 7 -partitions of x. 



1 have suffered no little perplexity in the endeavour to 

 obtain 7 P* from the consideration of the equation 



k*z — *P»— * = *— iP*—i » 

 which gave me readily my previous results. But all difficulty 

 disappears if we use the two equations 

 7 P. — 7 P^_ 7 = 6 P^ 1 , and 



7*« — 7*»— u^P^-i+e**-*"!" • • "f"6l*-*» 



which follow easily from the fundamental theorem 



*P»=*-iP»-i+*-iP»-*-i+*-iP*-2*-i+ • • ., or, 

 writing *=7(*- 1)+*, (c70, Z8), 



7P-=«P*-i+.P*-«+.P^i 4 + • • • -feP^i- (A.) 

 Let (jPf) stand for the sum of the algebraic, or non- 

 circulating terms in the right member of equation (A), and 

 GPjr-?) for the sum of such terms in 



7 P^ 7 =,P^+ fl P_ 16 + . . . +^i. (A') 

 It is plain that, if [«P*_J represent the non-circulating part 

 of gPjr-i, we shall have 



(:P.)-(7P-7)=[^^]- (B) 



Let then 



( 7 P.)=A^+B^+Ck 4 +D*'+E*>+ Fx+Qx ; 

 for evidently no power above the 6 th can disappear by the 

 above subtraction. Then (B) is 



+E (x«_(x-7)') +P(^-(*-7)) 

 = (7^0)' j 6 ('- 1 ) a -r-l35(*~l)«+760(a;-l)»-5046(^l) j 



