933] 
333 
933. 
TABLES OF PUBE BECIPBOCANTS TO THE WEIGHT 8. 
[From the American Journal of Mathematics, t. xv. (1893), pp. 75—77.] 
In the tabulation of Pure Reciprocants it is convenient to write a = 1; we thus 
have for all the reciprocants of a given weight a single column of literal terms which 
(as in the Semin variant Tables) I arrange in alphabetical order AO, and the several 
reciprocants have then each of them its own column of numerical coefficients: the 
form of the table is thus similar to that of the seminvariant table, the only difference 
being that for reciprocants the final terms are not in genei’al power-enders: as in the 
seminvariant table, the columns of the table are arranged inter se with their final 
terms in AO. As remarked in my paper, “Corrected Seminvariant Tables for the 
Weights 11 and 12,” Amer. Math. Journ., t. xiv. (1892), pp. 195—200, [926], it is 
not in every case the top term of a column which should be regarded as the initial 
term; but to the extent 8, to which the reciprocant tables are here carried, this remark 
has no application. 
I recall that the notation is the modified one employed by Halphen, and by 
Sylvester* in his 12th and subsequent lectures, viz. a, b, c, d,... denote 
1 <fy x &y x &y 
2 dx 2 ’ ^ dx 3 ’ 2 4 dx 4 ’ 120 da f ’ *" 
respectively. As already noticed, a is put = 1, but it is to be in the several terms 
restored in the proper powers so as to obtain for the reciprocant a homogeneous 
expression of a degree equal to the original degree of the final term; thus d — 3be + 2b 3 
is to be read as standing for a?d — 3abc 4- 2 b 3 . 
The ultimate verification of the expression for a pure reciprocant consists (as is 
known) in its annihilation by the operator 
V = 2a 2 db + oabd c + (6ac + 3b' 2 ) da + (7ad + 7be) d e + (8ae + 8bd + 4c 2 ) 0/ + &c., 
or, say 
V = 20 6 +5 bde + (6c + 3Z> 2 ) d d + (7d + 7bc)d e + (8e + 8bd + 4c 2 ) 0/ + &c.; 
[* American Journal of Mathematics, t. ix. (1887), p. 7.]