and 
EXPRESSED IN TERMS OF THE ROOTS OF (j>X, fx. 
437 
(E..e,...e.)x(H|.H5,...h,)x 
■E,E„...E,, 
X 
[E-E,. ...E, 1 
[Hii H|, . 
X 
Ill . 
L >'+1 
..Hi 
n _ 
JEj E . . . E„ 
Vv+] 
Hg Hg ...Hg 
the sura of this pair of terras will be of the form 
r- 
[E, 
X 
[E. 1 
l\ 
E, 
_H|iH|,. 
-HiJ 
Hi Hi ...H| 
L ^+1 + 2 ^n__ 
Ej— Eg 
"El 
E„ E„._...E„ 
_ ^v + 2 
rE2 
X 
[E: 1 
E. 
_H|iH^,. 
1 
H? H| ...Hi 
E 2 — El 
■E 2 
E„ E, ...E, 
^V +1 ^V + 2 ^ 7 ) 
p 
where Q, it raay be observed, does not contain Hi — H 2 , so that ^ remains finite 
when H,=H 2 . 
The above pair of terms together make up a sura of the form 
P 1 <p(Ei, E2)4/E2 -<p(E 2, Ei)vl;Ei 
Q'Ei-E 2 - 4/E1X4/E2 
which (as the numerator of the third factor vanishes when Ei=E 2 ) remains finite on 
that supposition. Hence the whole sum of terms in (22.) which is made up of such 
pairs of terms, and of other terms in which Ej — E 2 does not enter, remains finite 
when El— E 2 = 0 , and therefore generally when D=0, and similarly when Hi — H 2 = 0 , 
and therefore also when A=0; hence the expression for ^ in (22.) is an integral 
function of the E and H series of quantities, as was to be proved. 
Art. (17-)- Let us now proceed to determine the dimensions of the coefficient of x‘, 
the highest power of x in this value of when supposed to be expressed under the 
form of an integral function (as it has been proved to be capable of being expressed) 
of ^1 ^2 . . . ; ?Jl JJ2 • • • 5 
This coefficient is the sum of fractions the numerators of each of which consist of 
two factors, which are respectively of vXv and of (m — v)x{n — r) dimensions in 
respect of the two sets of roots taken conjointly, and the denominators of two factors 
respectively of v .{m — v) and vx {n—v) dimensions in respect of the same. 
Consequently, the exponent of the total dimensions of the coefficient in question 
=v X — v){n — v) — v{m — v) — {v.{ri — v)) 
=i{pi—v—v) X {n—v — v) 
= (m— 
