214 
ON WARINGS FORMULA. 
[925 
+ &C., 
coefficient of b m ~ 6 we have all the combinations of c, d, e, .(or say all the 
non-unitary combinations) of the weight 9, and where the numerical coefficient of 
b m ~ 6 C G d d e e ... (c + d + e + ... = 6), 
is 
= (-> 
,o+e+g+.. 
m.m — (9 — 8 + 1). m — (9 — 8 + 2) ... m — (9 — 1) 
lie. lid. lie ... 
Thus for the term b m ~ s c 2 e 1 , 9 = 8 ; c, d, e = 2, 4, 1 respectively (the other exponents 
each vanishing), and the coefficient is 
. .m.m — 6.m — 7 . a n 
(-) 3 TIT! ’ 
as above ; and so in other cases. 
For the MacMahon form 
1 + bx + + ... =(1 — ax)(l - fix)..., 
or say 
y n ^yn- 1 + JLyn 2 + ...=(y-a)(y-fi) 
c d 
we must for b, c, d, write b, -—w, ..—k , ... respectively : we thus have 
JL . Z ±.¿1.0 
(~) m • s m = 6* 
1.2 
+ m 1.2.3 
b m ~ 2 
b m ~ 3 
m 
1.2.3.4 
or say 
+ \m . m — 3 
+ &c., 
(_)m n ( m - l) S m = II (m — 1) 
— II m 
+ Ilm 
— Ilm 
+ Ilm 
1.2 J ) 
1.2 
d 
1.2.3 
e 
1.2.3 
.4 
m — 3 / 
b m ~ 4 
bm—2 
fyn-s 
fom—4