■ËJÜ&MMMH 
147] 
OF THE ROOTS OF AN EQUATION. 
421 
the partition 21 3 gives the new coefficients 3, 6, 12 of the partitions 31 3 , 2 2 1 2 , 21 4 , 
and the coefficient 10 of the partition l 5 gives the new coefficients 10, 60 of the 
partitions 21 4 and l 6 , and finally, the last line is obtained by addition. The process 
in fact amounts to the multiplication separately of each term of cd — 
1 (2 2 1) + 3 (21 3 ) + 10 (l 5 ) 
by b = (1). It would perhaps have been proper to employ an analogous rule for the 
calculation of the combinations c q d r ... not containing b, but instead of doing so I 
availed myself of the existing Tables (b). But the comparison of the last line of each 
Table (a) (which as corresponding to a combination b p was always calculated in 
dependently of the Tables (b)'j with such last line as calculated from the corresponding 
Table (b), seems to afford a complete verification of both the Tables; and my process 
has in fact enabled me to detect several numerical errors in the Tables (b), as given 
in the English translation of the work above referred to. It is not desirable, as 
regards facility of calculation and independently of the want of verification, to calculate 
either set of Tables wholly from the other; the rules for the independent calculation 
of the Tables (b) are fully and clearly explained in the work referred to, and I have 
nothing to add upon this subject. 
The relation of symmetry, alluded to in the introductory paragraph of the present 
memoir, exists in each Table of either set, and is as follows: viz. the number in the 
Table corresponding to any two partitions in the outside column and the outside line 
respectively, is equal to the number corresponding to the same two partitions in the 
outside line and the outside column respectively. Or, calling the two partitions P, Q, 
and writing for shortness, combination (P) for the combination represented by the 
partition P, and for greater clearness, symmetric function (P) (instead of merely (P)) 
to denote the symmetric function represented by the partition P, we have the following 
two theorems, viz. 
Theorem. The coefficient in combination (P) of symmetric function (Q) is equal 
to the coefficient in combination (Q) of symmetric function (P); 
and conversely, 
Theorem. The coefficient in symmetric function (P) of combination (Q) is equal 
to the coefficient in symmetric function (Q) of combination (P). 
M. Borchardt’s formula, before referred to, is given in the ‘Monatsbericht’ of the 
Berlin Academy (March 5, 1885)\ and may be thus stated; viz. considering the case of 
n roots, write 
(1, b, c, ... x) n = (1 - aæ) (1 - ßx)...(1 - kx) =fx, 
then 
1 And in Grelle, t. liii. p. 195.—Note added 4th Dec. 1857, A. C.