ON MATHEMATICAL TABLES. 



321 



But taking a, h, e, d to stand for 0, 1, 2, 3 respectively, the actual partitions 

 of the required form are exhibited by the literal terms of the table (these 

 being obtained, each column from the preceding one, by the method of deri- 

 vations, or say by the rule of the last and last but one), and the numbers 

 of the bottom line are simply the number of terms in the several columns 

 respectively. 



(ct b c 7c\ ^ 

 — o' l' 2 ' ) '^°^ different 



values of n and m (where the number of letters is =m-{-l), would be ex- 

 tremely interesting and valuable. The tables for a given value of m and for 

 different values of n are, it is clear, the proper foundation of the theory of 

 the binary quantic fa, 6, c . . . >t ^ x, l)"', which corresponds to such value 

 of TO. Prof. Cayley regrets that he has not in his covariant tables given in 

 every case the complete series of literal terms (viz. the literal terms which 

 have zero coefficients are, for the most part, though not always, omitted in 

 the expressions of the several covariants). 



33. But the question at present is as to the nuinbey of terms in a column, that 

 is, as to the number of the partitions of a given form : the analytical theory 

 has been investigated by Euler and others. The expression for the number 

 of partitions is usually obtained as = coefficient of .^•" in the development, in 

 ascending powers of x, of a given rational function of x : for instance, if there 

 is no limitation as to the number of the parts, but if the joarts are 1, 2, 3, m 

 ^viz. a part may have any value not exceeding m), each part being repeatable 

 an iudefinite number of times, then 



N'umber of partitions of n= coefficient .^'" in r— ; :——. ^^ — -z -. 



(1 — .^•) (1 — X-) (1 — .r') . . (1 — x'") 



md we can, by actual development, obtain for any given values of m, n the 

 lumber of partitions. 



These have been tabulated )h=1, 2, . . . 20, and m=oQ (viz. there is in 

 ,his case no limit as to the largest part), and « = 1 to 59, 



Euler, Op. Arith. CoU. t. i. pp. 97-101 (given in the paper " De Partitione 

 !^amerorum," pp. 73-101, 1750) ; heading is " Tabula indicans quot variis 

 nodis numerus n e numeris 1, 2, 3, 4 . . . m per additionem exhibi potest, 

 eu exhibens valores formulae n'-'"K" The successive lines are, in fact, the 

 •oefficients of the several powers x°, x^ . . .r" in the expansions of the functions 



i;'l-.i-.l- 



-x' 



l-.r.l- 



l-x'' 



34. The generating function for any given value of m is, it is clear, = 



nto that for the next preceding value of m, and it thus appears how each 

 iue of the table is calculated from that which precedes it. The auxiliary 

 lumbers are printed; thus a specimen is 

 1875. Y 



