330 
is positive or negative; say these are the Jacobian-positive and 
the Jacobian-negative roots, and the question is to determine 
for the roots within a given contour or circuit, the difference of 
the numbers of the roots belonging to the two sets respectively. 
In the particular theorem (say Cauchy’s rhizic theorem) 
P and @ are the real part and the coefficient of 2 in the 
imaginary part of a function of #+2y with in general imaginary 
coefficients (or what is the same thing, we have 
P+iQ=f(a+ty) + ip (w+ ty), 
where f, ¢ are real functions of a +7iy): the roots of necessity 
are of the same class: and the question is to determine the 
number of roots within a given circuit. 
Tn each case the required number is theoretically given by 
: Lay mi Vaan 
the same rule, viz. considering the fraction — it is the excess 
Q 
of the number of times that the fraction changes from + to — 
over the number of times that it changes from —to+, as the 
point (a, y) travels round the circuit, attending only to the 
changes which take place on a passage through a point for 
which P is=0. 
In the case where the circuit is a polygon, and most easily 
when it is a rectangle, the sides of which are parallel to the 
two axes respectively, the excess in question can be actually 
determined by means of an application of Sturm’s theorem 
successively to each side of the polygon, or rectangle. 
In the present memoir I reproduce the whole theory, pre- 
senting it under a completely geometrical form, viz. I establish 
between the two sets of roots the distinction of right- and 
left-handed: and (availing myself of a notion due to Prof. 
Sylvester) I give a geometrical form to the theoretic rule, making 
it depend on the “intercalation” of the intersections of the 
two curves with the circuit: I also complete the Sturmian 
process in regard to the sides of the rectangle: the memoir 
26—2 
