14 
THE SYMMETRIC FUNCTION TABLES OF THE FIFTEENTHIC. 
where m is greater than x x , while the fourth states that a function of the 
roots of an equation of the nth degree has the same coefficient of the term 
involving (X^ . . . X r-1 ) as the same function of the roots of an equation 
of the (r— l)th degr'ee. His completely reducible form is 
© 
t 
where P is the partition conjugate to Q, the formula giving Cayley’s result (9). 
Roe’s normal forms are those for which his fundamental and reduction 
formulas do not provide a reduction. The formula is 
+1 
(T"“lh . . . n /n 
yn©(ra—l)^ . . . 
r=n 
. V 
r= 1 
(1 + M r ) X 
0 + + 1 Ri . . . (r — 1 )X- 1 +,?— l(f-j- 1)^+1 . t _ \ 
1 (m— l)!“i . . . (m —r — l)^-i(m — — r + !)/“>•+1 . . . O ot /_ 
Thus he reduces his normal coefficients to the sums of coefficients of tables 
of lower weight. For example 
R) 2 134° 
, 40 3 , 
/ 0 2 13\ /0 2 3 N 
R 4 40V (1 + 0) \30h 
(1+0)1 
' 0 T 
HO 2 . 
(—3+1=4 
that is, 
coefficient of a 3 a l in 2a? = — (1 +0) X coefficient of a 3 in 2a? — (1+0) 
X coefficient of cq in 2a x = 3+1=4. 
Roe gives also a formula for calculating the constituents in each of the 
first four lines (or columns). Those for the first two follow: 
First line 
w(?l 1 + ^-2+ • • • + X n —1) 
! ) 
xfx 2 !. 
aj 
( 20 ) 
where ra = X 1 + X 2 + • • • + X n . 
Second line 
(po^i 
n 
(w-l)10. . .0, 
1 J/ (w — 1) (/C+X 2 + • • • 2)! 
1-1 ^ -l)ra,!...JL! 
+ +n — 1 ) 
w(x i +X 2 + 
Xi! X 2 
V 
( 21 ) 
where h = 2 when m = 2 and h = 1 when m > 2 . 
Roe’s formulas were used by the writer in the calculation of the 
fifteenthic here appended. The work has been verified by the use of the 
law for the sum of the coefficients in a row or column, (14), and by Roe’s 
formulas, ( 20 ) and ( 21 ). 
