K-PARTITIONS OF N. 143 



Dp=»6.-2642— *6^_r2808— ♦6^_2-5584.*6^.3-392— ♦6,_,- 

 558— *6^.a-2808 ; 

 for, when aj=l, /)=0, &c. 



Substituting this value of D^, and for h its equivalent in 

 terms of a?, A, and c, then adding the sum just found to (eP*) 

 H-CeP*)'* we obtain the expression following: — 



__(l350arJ+5400a;)*2^_i 

 +7926a;-*6,-— 8424a;(*6^_, +*fix-») 



+ ll76a;-»6^^3— 1674^(*6;,_2+*6^«4) 

 — .47556A-*6^+(50544A— 42l20)*6^_l+(10044^ 



— 6696)*6,.2 



__(7056A— 3598)*6,.3+(10044A— 3348)*6;,_4+(50544A 



~8424)*6,., 



+ I80(cfp-i-rfp_64-rfp_i2 4- • • • to A terms)]. 



where x=60A4-6-(A—l)+c; A70;cZ7, 7O; a:— l=60/i+j9; 

 and c^ is the coeflficient of *60j._p in sPx* 



Since ft may have ten different values 0, 1, 2 . . .9, the 

 constant circulator is of 60 terms, and of the form S/J.*60^_p, 

 to which form it may appear necessary to reduce it, if 7?^^ is 

 to be found from gPx, as this is found above from gP,. 



I shall content myself with a few observations by way of 

 shewing that this reduction is at least unnecessary to prove, 

 what we may reasonably expect from our expressions for the 

 5-partitions, the 4-partitions, &c. of a?, that the value of eP* 

 is the integer nearest to the sum of the terms in x. 



In order to see that this is so, it is necessary to be con- 

 vinced that the circulating constant is always numerically less 

 than i.(602.l22)=259200. When *6x=l, p is one of the ten 

 numbers 5, 11, 17 ... 59; and it is easily seen by inspec- 

 tion of the quantities dp in sP^) that no consecutive ft of (/s. 



