GIVE THE IMAGINARY ROOTS OF AN ALGEBRAIC EQUATION. 471 
x that satisfies the equation f(z) =0, then /(z) contains the factor 2—h/ once, 
and only once; and when x becomes equal to # and y vanishes, the limit of 
y : (e—h) is finite, and the limit of y : (v—A) is infinite. This is also an imme- 
diate result of the symmetry of each curve with regard to the x-axis. 
If f(x)=0 has two equal roots, /’(7)=0 has one root equal to them. In 
this case P has a double point, through which passes a branch of Q cutting the 
v-axis at right angles, as shown in fig. 15, where the dotted curve belongs to Q. 
If f(2)=0 has three equal roots, two of the roots of /’(v)=0 and one of the 
roots of f’(z)=0, are equal to them. In this case P has a triple point, which 
is also a double point on Q, as indicated in fig. 16, where, as before, the dotted 
branches belong to Q and the others to P; and the middle branch of P cuts 
the z-axis at right angles. In general, if m roots of f(z)=0 are equal, P has 
a multiple point of the order m on the #-axis, and this is also a multiple point 
of the order m—1 on Q. If m is odd, one branch of P cuts the z-axis at right 
angles, and the others are inclined to it, two and two, at equal angles; and one 
branch of @ lies between each adjacent two of P. Ifm is even, one branch of 
() cuts the z-axis at right angles, and the other branches of Q and all the 
branches of P, are symmetrically arranged as just described. 
We will henceforward suppose that the roots of f(«)=0 are all real and un- 
equal, in which case, as already shown, there are x branches of P and n—1 
branches of Q, all of which cut the z-axis at right angles. We have next to 
consider the nature of the infinite branches, and for this purpose we must 
develop the equations (P) and (Q). The former gives us 
ee Die At SD ig a ew Dy 
—F)n(n—A)w "+(n—1)(n—2)pya?>+. 2. 2 +2py 0 
+f jn(n— \(m=2)(n—B)ae*+ 2 a... eee ee ea 
Or, re-arranging the terms according to their dimensions, 
dia Un py, Wot LN 2 yy He 
af at ry es heli ty 
get US a EO DE nae a ) 
+psja = ray ne aT q “yt. j 
A= hn =3 
_ ea aaa ty Sos i OS 
The directions of the infinite branches, 7 in number, are given by the terms 
of highest dimensions ; and equating these to zero, we have 
a w-n—1 n.n—1.n—2.n—3 ,-4.4 
i gee cee 
ay? + — rim Gor Pt 4 ag A EO. 
