286 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
[932 
The MacMcthon Form of Equation. Art. Nos. 32 to 34. 
32. The equation connecting the coefficients and the roots is here taken to be 
t & c d „ , -.a -i 
1 + y so + y~2 X + i 2 3 ^ + ’'’ = 1 — ax • 1 - pec .1 —yx .... 
As to this it may be remarked that, if we had started with a form of the nth 
order with binomial coefficients, 
n 1 n. n — 1 0 n. n — 1 . n — 2 7 Q .. i / c , . 
1 + zr bx H z—=—■ c% 2 H -—-x—x dx° + ... = 1 — ax. 1 — px. 1 — yx ... (n factors), 
JL ± . A JL . A . o 
then writing herein ^ for x, and also not, n/3, ny, ..., for a, /3, y, ... and putting 
ultimately n = oc, we have the form in question. 
We pass from the ordinary form to the MacMahon form, by writing for 
b, c, d, e, ..., T , 
d 
1’ 1.2’ 1.2.3’ 1.2.3.4 
• - or sa >’ b ■ I- t k' 120' 720'" 
All the results obtained for the ordinary form will, after making therein this change, 
apply to the new form. We thus find 
©<r = a db + (cr — 1) 2bd c + (cr — 2) 3cd d + ... + 1 <rpd q , 
© ff = a'd b + (a — 1) 2bd c + (a — 2) 3ed,i + ... + (a — cr + 1) crpd q , 
©<r — ©a' = (cr' — cr) A, 
where 
A — d]) + 2 bd c •+■ 3c0(j -l .. 
. + crpd, 
or say 
= db 4- 2 bd c + 3cdd + • • 
Also 
F = bd a + cd b + dd c + .. 
.. + rd, 
or say 
= bd a + cd b + dd c + . - 
• * •> 
Q = cd b + 2 dd c + • ■ 
.. 4- ard, 
or say 
= cdj) 4- 2 dd c + .. 
The change a, ß, y, ... into na, nß, ny, ... would change Sd a , Sad a , Sa 2 d a into 
n-'Sda, Soid a , nSa 2 d„ respectively (n = oo): but this change is, in fact, compensated for 
by the introduction into the formulse of the binomial coefficients as above; it is 
— Sa, Saß, ... not —nSa, n 2 Saß, ... which are equal to b, |c, ...; and the conclusion 
is that we have to retain without alteration the symbols Sd a , Sad a , Sa 2 d a : thus in 
the new form as in the old one, we have © 4 $ot 4 = — Sd a . Sa* = — 4$a 3 , see the example 
ante No. 23. 
33. In the new form, a non-unitariant is annihilated by the operator 
A, = d b + 2bd c + 3 cdd + • • • >