26 ON THE GEOMETRICAL REPRESENTATION OF [561 
561] 
In the second case, that is, when the intercalations are contrary, they counteract 
each other in forming the intercalation of the circuit : it is the difference of the 
numbers of letters, or twice the difference of the indices, which is evenly even, and 
the half of this, or the difference of the indices, which is the index of the circuit : 
one intercalation is (+ P — Q), and the other is (— P + Q) : and the circuit will agree 
with that which has the larger index. 
In particular if the circuit consist of a single unclosed trajectory, taken forwards 
and backwards; then the trajectory taken one way is (+P — Q), taken the other way 
it is (— P + Q) ; the number of terms is of course equal, and the circuit is (PQ) 0 . 
16. Consider now two circuits ABC A and AGDA, having a common portion CA, 
or, more accurately, the common portions AC and CA : write down in order the inter 
calations of 
ABC, CA, AC, CD A : 
the two mean terms destroy each other, and we can hence deduce the intercalation 
of the entire circuit ABC DA. 
Suppose first, that ABC and CDA are similar ; then if CA is similar to ABC 
it is also similar to CDA, that is, AC is contrary to CDA : and so if CA is contrary 
to ABC, then AC is similar to CDA. 
To fix the ideas suppose CA similar to ABC, but AC contrary to CDA, then 
ABCA is similar to CA ; but ACDA will be similar or contrary to AC, that is, contrary 
or similar to CA, that is, to ABC A, according as index of A C > or < index of CDA. 
Suppose Ind. AC < Ind. CDA, then ACDA is similar to ABC A. 
Now Ind. ABCDA = Ind. ABC Ind. CDA, 
Ind. ABCA = Ind. ABC + Ind. AC, 
Ind. A CDA = Ind. CDA - Ind. A C, 
and thence 
Ind. ABCDA = Ind. ABCA + Ind. ACDA, 
the whole circuit being in this case similar to each of the component ones. 
But if Ind. .4(7 > Ind. CDA, then ACDA is contrary to ABCA. 
And Ind. ABCDA = Ind. ABC + Ind. CDA, 
Ind. ABCA = Ind. ABC + Ind. CA, 
Ind. ACDA = - Ind. CDA + Ind. AC, 
and thence 
Ind. ABCDA = Ind. ABCA - Ind. ACDA ; 
and the investigation is like hereto if CA is contrary to ABC but AC similar to CDA. 
17. Secondly, if ABC and CDA are contrary, then if CA is similar to ABC it is 
contrary to CDA, that is, AC is similar to CDA ; and so if CA is contrary to ABC 
it is similar to CDA, that is, AC is contrary to CDA. 
Suppo: 
similar to 
to each otl 
Also 
and thence 
and the in 
18. Ii 
according i 
the entire 
Moreo’s 
these agair 
elementary 
true as re: 
any circuit 
19. Ii 
function of 
of a parar 
between gi 
rational an 
question ol 
succession 
of a single 
to the axi: 
y 0 is the 1 
for y in th 
the same 
to each ax 
the rectang 
or QPQ as 
first inter» 
the several 
be), we ha 
the differei 
see how th