K-PARTITIONS OF N. 133 



The circulating constant can never be so great as J* 12, for, 

 to examine the most suspicious-looking case, when c=2 and e 

 even, it is only =2 — 6= — 4. 



It appears, then, that the tri-partitions of x are in number 

 always equal to the integer nearest to -jVa?'. When c=3, and 

 e is odd, x=.^h ; in this case the number aP, is exactly -jV^* 

 The expression is easily put into the form following : — 



3P^A|x8__0-*6;r— *6x-i—4.*6,-2+3/6,_3— 4.*6x-4 



Using equation (D) to find (4?^) the sura of the ordinary 

 algebraic terms in the equivalent of 4Px, we have 



(4Px) = Aa-a-h 3x2+ Cx-f D 

 — (4Px-4)=--Aa;3+12Aa^— 48Aa:4-64A 

 —Bxa+SBx— 16B 

 — Cx-h 4C 



whose sum must be the algebraic portion of aPx-i? or 



[3P,_i]=A(a^-l)^=TV(a^-2x+l). 

 This gives by equating co-efficients 



A=T*T, B=tV, C= — t^t. 



Wherefore 



If jr=4e+c, c/o^ (4PC-4) ^as no terms, and 

 (4P.)=[,Pc-i], or 

 cH-a^— c c^— 2c+l 



144 ^^~ 12 ' 

 ^_c'-9 c^+23c— 12 

 — "- 144 ' 



=''4^,_3_ .M,-,^ •4,.,J_ MxO 

 144''" 144''' 144 "*■ 



(4?,)=— j a»+3*^_x-*4,_,-3-»4,.,-6-Mx_^ J. 



