22 
ON THE GEOMETRICAL REPRESENTATION OF 
[561 561] 
In the case where the circuit is a polygon, and most easily when it is a rect- 2. Coi 
angle the sides of which are parallel to the two axes respectively, the excess in point 
question can be actually determined by means of an application of Sturm’s theorem an( j azure • 
successively to each side of the polygon, or rectangle. 
In the present memoir I reproduce the whole theory, presenting it under a com 
pletely geometrical form, viz. I establish between the two sets of roots the distinction if to have 
of right- and left-handed: and (availing myself of a notion due to Prof. Sylvester*) handedly, 
I give a geometrical form to the theoretic rule, making it depend on the “ inter- or, what 
calation ” of the intersections of the two curves with the circuit: I also complete the colours is 
Sturmian process in regard to the sides of the rectangle: the memoir contains further 
researches in regard to the curves in the case of the particular theorem, or say as 
to the rhizic curves P = 0, Q = 0. 100 ^ 1S 
The General Theory. Articles Nos. 1 to 19. 
1. Consider in a plane two curves P = 0, Q — 0 (P and Q each a rational and 
integral function of x, y), which to fix the ideas I call the red curve and the blue 
curve respectively *f*: the curve P = 0 divides the plane into two sets of regions, say 
a positive set for each of which P is positive, and a negative set for each of which 
P is negative: it is of course immaterial which set is positive and which negative, 
since writing — P for P the two sets would be interchanged: but taking P to be 
given, the two sets are distinguished as above. And we may imagine the negative 
regions to be coloured red, the positive ones being left uncoloured, or say they are 
white. Similarly the curve Q = 0 divides the plane into two sets of regions, the 
negative regions being coloured blue, and the positive ones being left uncoloured, or 
say they are white. Taking account of the twofold division, and considering the 
coincidence of red and blue as producing black, there will be four sets of regions, 
which for convenience may be spoken of as sable, gules, argent, azure: viz. in the figures 
we have 
P Q 
— — sable, shown by cross lines, 
— + gules, „ „ vertical lines, 
+ 4- argent, left white, 
+ — azure, shown by horizontal lines, 
sable and argent (— — and + -f) being thus positive colours, and gules and azure 
(- + and + —) negative colours. See figures [pp. 32, 38] towards end of Memoir. 
* See his memoir, “A theory of the Syzygetic relations, &c.” Phil. Tram., 1853. The Sturmian process 
is by Sturm and Cauchy applied to two independent functions <px, fx of a variable x; but the notion of 
an intercalation as applied to the order of succession of the roots of the equations </> (x) = 0, f (x) = 0 is due 
to Sylvester, and it was he who showed that what the Sturmian process determined was in fact the inter 
calation of these roots: but, not being concerned with circuits, he was not led to consider the intercalation 
of a circuit. 
t It is assumed throughout that the two curves have no points (or at least no real points) of multiple 
intersection; i.e. they nowhere touch each other, and neither curve passes through a multiple point of the 
other curve. 
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