SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF AN EQUATION. 503 
from the fundamental theorem given in Lemma (B.) art. (z.), for the case of Q = Q^a^, 
and will be best understood by showing its operation in a few simple cases. 
Thus let Q = Qi Qa*. 
Then 
Let Q = Qj Q2 Q3. 
Then 
Let Q = Qj Q2 Q3 Q^. 
[Q] = [Q,] X M ~ M X ['Q J . 
[q] = [q,]x[qJx[Q 3] 
- [q;] X ['Q2] X [Q3] - [^^i] X [Q;] X ['Q3] 
+ [Qi] X X ['^^3] • 
^J^hen [Q] = [Q.] x [Q J X [O3] X [Qj 
- [q;] X ['Q J X [Q3] X [QJ - [Q J X [Q'J X ['Q3] X [QJ - X [O J X [o;] X ['Qj f 
+ [Q'J X ['Q;] X ['Q3] X [Q 4] - [o;] X ['Q J X [q;] X ['Q 4] - [o.] X [Q;] X ['o;] X ['Q4] 
-[q;]x['q;]x['q;]x['Q 4], 
and so in general if Q = QiQ2.. .Q,., [Q] maybe expanded under the form of the sum of 
2'”^ products separable into i alternately positive and negative groups containing 
respectively 1, (/—I), (/— 1) •••(i— 1), 1 products. 
Art. (k.). In every one of the above groups forming a product the accents enter in 
pairs and between contiguous factors, it being a condition that if any Q have an 
accent on the right the next Q must have one on the left, and if it have one on the 
left the preceding Q must have an accent on the right, and the number of pairs of 
accents goes on increasing in each group from 0 to i— 1 . This rule serves completely 
to define the development in question 
The sign of equality is employed here to denote the relation between a concrete whole and the aggregate 
of its parts. 
■f The number of distinct factors entering into these products, taken collectively, is evidently iq- 2 (i — 1) 
+ (/— 2), i. e. 4(i— 1). 
+ When each partial type consists of a single element, every doubly accented Q, will vanish, and every 
singly accented CL will become unity ; hence we may derive the rule for the expansion of the cumulant 
[a I aj in terms of a^...ai, which will accordingly consist of 
■ («! . a2...«.) + 2; 
(«i 
..a^) +&C., 
■ ®e + l Clg • ttg+i X Clj. . ttf+i 
the indices e and/", 1 andy, &c. being understood to be all distinct integers (which agrees with the knowm 
rule for the expression of the denominator of a continued fraction in terms of the quotients). The number of 
terms in this expansion, in consequence of the vanishing of the quantities affected with a double accent, reduces 
from 2‘-* down to theith term in the series commencing with 1, 2, 3, &c. defined by the equation m,_i. 
1 / 
a + a/5\'+i 
1 ( 
1 - V5\' 
^ 2 ) 
a/5 ' 
2 / 
the number, therefore, of products in which double accents occur in the general expansion of [wj ujo.,.Wi] is 
9i-i_ 1 (i + ^5y+i^ 1 (i-V5Y+^ 
3 u 
MDCCCLTII. 
