ON THE GEOMETRICAL REPRESENTATION OF CAUCHY’S 
THEOREMS OF ROOT-LIMITATION. 
[From the Transactions of the Cambridge Philosophical Society, vol. xn. Part II. (1877), 
pp. 395—413. Read February 16, 1874] 
There is contained in Cauchy's Memoir “ Calcul des Indices des Fonctions,” 
Journ. de l’École Polytech. t. xv. (1837) a general theorem, which, though including 
a well-known theorem in regard to the imaginary roots of a numerical equation, 
seems itself to have been almost lost sight of. In the general theorem (say Cauchy’s 
two-curve theorem) we have in a plane two curves P = 0, Q = 0, and the real inter 
sections of these two curves, or say the “ roots,” are divided into two sets according as 
the Jacobian 
d x P • d y Q d x Q • d y P 
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 Q are the real part 
and the coefficient of i in the imaginary part of a function of x + iy with, in general, 
imaginary coefficients (or, what is the same thing, we have P + iQ= f{x + iy) + i(f> (x + iy), 
where f $ are real functions of îc+ iy) : the roots of necessity are of the same set ; 
and the question is to determine the number of roots within a given circuit. 
In each case the required number is theoretically given by the same rule, viz. 
p 
considering the fraction j=r, it is the excess of the number of times that the fraction 
changes from + to — over the number of times that it changes from — to +, as 
the point (x, y) travels round the circuit, attending only to the changes which take 
place on a passage through a point for which P is =0.