K-PARTITIONS OF N. 137 



Again, if x be even, 



-.25A+5Bh-5D-E=-4 and 

 iix be odd,— 25A+5B+ E=0, i.e., 



' 20-122; 



and. 5B=25A-E==|^?r^ 



B= ^:?i, 



2012* • 



We have now deduced that 



When a:^5=c, this must be equal to the single circulator 



for GP^p.^)' has no terms at all. Therefore 

 — 9c2— 72c+(90c+l35).*2,_i + 20.122-F 



= (-180c+180)*2,.2, 



F=20^2 (9c2+72c— 90c— 90c*2c.4r-135.*2,_i+ 180.*2,.2) 

 =2^(-l44.*5,_i_108.*5,_3-108.*5,_4+0.*5,). 



It remains to find the sum of the constant circulators in 



4P,-i, 4Px-6 • • • • 4Pc-i, to (e+1) terms; x=be-\-c; cyO, 



The constant in 4?^^ being Sa^.*12x_r, that in 4Par«i is 



Sa,..*i2j:_i.^, that in 4PX-6 is Sar.*12j._6_n that in 4Pr-i-5«» 



is Sa^.*12;p_i»^.^. When 5m is a multiple of 12, Op will 



appear with the same circulator which it affects in 4Px_i ; for 



*12jf_i_i2n-p=*12^_i_p. As m cannot be less than 12, Op will 



affect twelve different circulators *12j_0 in the first twelve 



functions 4PX.1, 4Px-5» &c. ; and again the same 12 in the 



T 



