7-PABTITION8 01 



refer to them, except foi the remark, which inspection of that 

 page will verify, that 



<k+<*«+rf«+<J»+<k+*.=91, 

 the sums of the first and sixth horizontal lines ; and that the 

 sum of the six numbers in every other horizontal line at p. 

 139, beginning with </ 6 , </ I2 , d m d i4i d x , d^ d^ and d Mi is 

 always = — 485. These horizontal lines appear as vertical 

 lines in the fonnula for ^P, just written. 



The constant circulator, or term free from x, is obtained 

 from the above formula by giving to k its 10 successive 

 values, 0, 1, 2 ... 9. As both the sum of the 60 elements 

 of that term and the elements themselves are required, the 

 first for the determination of the terms in x of 7 P X , and the 

 latter for that of the circulating constant in it, we must here 

 obtain these data, before attempting to find 7 P*. 



Adding up the six concluding lines of the formula above 

 written, we obtain 



7^{ 66 564A— 68715*°+180(vf*rKa+ . . . toJterma)); 



where * l> =rf,,+rf Afl + ... to 6 terms, 



*<F=.t v p=$\, and 

 *«* 'is* *i* *24i *m» *«« ««» are each equal — 485. 

 Giving to k all its values, we have the sum 



-i- j 66564.46^8715•10 + 180( 9 *• +8 ^ 7 '"^? +5#,4 ) I 

 720»( 7 ^+4* JO +3<»+2# tt +#* 



==r^ J 299538— 687150+180(13-91— 32485) J, 



272430__ i 

 "" T^O 1 """^^ "^ 1 "^" 4 " ' ' " + ^* 

 the sum of the 60 elements of the circulating constant, whose 



form is ?/J*60 jr _ p - 1 ^ 3 . 



If it could be proved true for 7 P X , as it is for all the lower 

 partitions of x, that the sum of the terms in x expressed the 

 number of the partitions, this sum just found, together with 



