cauchy’s theorems of root-limitation. 
25 
3F [561 
561] cauchy’s theorems of root-limitation. 25 
lly the letters P and Q 
calation may contain an 
h the other letter, and 
ain an odd number of 
¡ontaining one more of 
QP, or else PQP or 
uence were QPPQ, this 
or else — P and + Q. Hence in the case of a circuit the intercalation is either 
(+ P — Q), say this is a positive circuit, or else (— P + Q), say this is a negative circuit. 
There is of course the neutral circuit (PQ\ for which the intercalation vanishes. 
11. Consider a circuit not containing within it any root; as a simple example let 
the circuit lie wholly in one colour, or wholly in two adjacent colours, say sable and 
gules: in the former case the sequence, and therefore also the intercalation, vanishes: 
in the latter case the sequence is + Q — Q, and therefore the intercalation vanishes: 
md end with the same 
¡ince any letter thereof 
ferently. A little con- 
i evenly even, or, what 
tius imagine the circuit 
viz. P we pass from 
to get back into sable 
:ure to sable), but then 
the number of letters 
le of colours P argent 
the intercalation then 
viz. in either case the intercalation is (PQ\. 
12. Consider next a circuit containing within it one right-handed root; for instance 
let the circuit lie wholly in the four regions adjacent to this root, cutting the two 
curves each twice; the sequence and therefore also the intercalation is + P—Q + P — Q; 
viz. this is a positive circuit (+P — Q) 1} where the subscript number is the half-index, 
or half of the number of P’s or of Q’s. Similarly if a circuit contains within it one 
left-handed root, for instance if the circuit lies wholly in the four regions adjacent 
to this root, cutting the two curves each twice, the sequence and therefore also the 
intercalation is —P + Q — P + Q, viz. this is a negative circuit (— P + Q\: and the 
consideration of a few more particular cases leads easily to the general and fundamental 
theorem : 
er. 
13. A circuit is positive (+P — Q)s or negative (—P+Q)s according as it contains 
of letters in the inter- 
■ an integer + A But 
half-index is therefore 
within it more right-handed or more left-handed roots; and in either case the half-index 
8 is equal to the excess of the number of one over that of the other set of roots. If 
the circuit is neutral (PQ) 0 , then there are within it as many left-handed as right- 
handed roots. 
*- and Q-sequence we 
bserve that a passage 
ur (azure to sable or 
colour (sable to azure 
e blue curve may be 
o argent), this is + Q, 
argent to azure), this 
in any manner, but 
ting the passage into 
3 or Q), denoting the 
independently of the 
' — as the case may 
14. The proof depends on a composition of circuits, but for this some preliminary 
considerations are necessary. 
Imagine two unclosed trajectories forming a circuit, and write down in order the 
intercalation of each. The whole number of letters must be even: viz. the numbers 
for the two intercalations respectively must be both even or both odd. I say that if 
the terminal letter of the first intercalation and the initial letter of the second inter 
calation are different, then also the initial letter of the first intercalation and the 
terminal letter of the second intercalation will be different: if the same, then the 
same. In fact, the intercalations may be each PQ or each QP, or one PQ and the 
other QP: or each PQP, or each QPQ, or one PQP and the other QPQ. Supposing 
the letters in question are different, then the intercalations may be termed similar; 
but if the same, then the intercalations may be termed contrary. 
any even number of 
- + — + &c., viz. the 
sign. Hence the signs 
on remain alternate : 
n only + P and - Q 
15. In the first case, that is when the intercalations are similar, the two together 
form the intercalation of the circuit; the sum of their numbers of letters (that is, 
twice the sum of their indices) will be evenly even, and the half of this, or the sum of 
the indices, will be the index of the circuit; each intercalation will be (+ P — Q) or 
else each will be (— P + Q); and the circuit will be (+ P — Q) or (— P + Q) accordingly, 
c. ix. 4