561] 
CAUCHY S THEOREMS OF ROOT-LIMITATION. 
2 7 
4—2 
Suppose CA similar to ABC, and AC similar to CD A ; then ABC A is also 
similar to ABC, and AC DA similar to CD A ; viz. ABC, CA and ABC A are similar 
to each other, and contrary to AC, CD A, ACDA which are also similar to each other. 
Also 
Ind. ABCDA = Ind. ABC ~ Ind. CD A, 
Ind. ABC A = Ind. ABC + Ind. CA, 
Ind. ACDA = Ind. CD A + Ind. AC, 
according as the component circuits are similar or contrary, and in the latter case 
the entire circuit is similar to that which has the largest index. 
Moreover, any circuit whatever can be broken up into two smaller circuits, and 
these again continually into smaller circuits until we arrive at the before-mentioned 
elementary circuits, and the theorem as to the number of roots within a circuit is 
true as regards these elementary circuits; wherefore the theorem is true as regards 
any circuit whatever. 
19. In the case where a trajectory is a finite right line, y is a given linear 
function of x, or the coordinates x, y can if we please be expressed as linear functions 
of a parameter u, so that as the describing point passes along the line, u varies 
between given limits, say from u— 0 to u= 1. The functions P, Q thus become given 
rational and integral functions of a single variable u (or it may be x or y), and the 
question of the P- and Q _se( l uence and intercalation relates merely to the order of 
succession of the roots of the equations P = 0, Q = 0, where P and Q denote functions 
of a single variable as above. To fix the ideas, let the trajectory be a line parallel 
to the axis of x; and in this case taking x as the parameter, and supposing that 
y 0 is the given value of y, P and Q are the functions of x obtained by writing y 0 
for y in the original expressions of these functions. Of course the theory will be precisely 
the same for a line parallel to the axis of y: and by combining two lines parallel 
to each axis we have the case of a rectangular circuit. We require, for each side of 
the rectangle considered according to its proper currency, the intercalation PQ, QP, PQP 
or QPQ as the case may be, and also the sign + or — of the initial letter of the 
first intercalation; for then writing down the intercalations in order, with the signs for 
the several letters, + and — alternately (the first sign being + or — as the case may 
be), we have or deduce the intercalation of the circuit, and thus obtain the value of 
the difference of the numbers of the included right- and left-handed roots. We thus 
see how the whole theory depends on the case where the trajectory is a right line.