GIVE THE IMAGINARY ROOTS OF AN ALGEBRAIC EQUATION. 475 
Hence we have approximately 
Po Sin @ cos (n—2)a 
ny cos (n—1)a 
xZ=y cot a— 
Now since the roots of f(x) =0 are all real and p,=0, we know from the theory 
F : ks or — ; 
of equations that p, is negative. And since a= ts and is therefore <z, 
sin a is positive. Again 
cos (x —2)a=cos es T=COS (v-3-7 + a)n= (—1)"*" sin (27 — 1)= 
IVOn- : 
cos (n—1)a = cos a 7 = COS (r-3-2 + ana (- 1)" sin(27—1) ne 
Now r ranges from 1 to n, and (2”—1) = from = to an) ors —= ; and sin 
2n 2n 2n 2n 
 Or—1 
2n 
and sin (27 — 1)= — is positive when (27;— 1) = —<m, or 2r—1<n and r —— 
7 is therefore always positive. Lastly, (27 — 1) = — ranges from * tO Qr—=, 
The final result is that x—ycota is positive when (y being positive) the 
asymptote lies in the first quadrant, and negative when it lies in the second 
quadrant ; or in both cases the infinite branch lies below the asymptote. 
We have thus proved that the curve P has m branches, which cut the x-axis 
at right angles ; and that it has ~ infinite branches, lying below the 7 asymptotes 
which are arranged at equal intervals through the first two quadrants. 
_ For the infinite branches of Q, we have similarly, putting B=rz : n, 
sin 6 , sin (n—2)B 
pts SLID approximately ; 
a=y cot B—RAS* 
and it may be proved, in the same way as for P, that (y being positive) the infinite 
branches lie below the asymptotes. We thus see that Q has »—1 branches, 
that cut the x-axis at right angles in points which lie singly between two cor- 
responding points of P; also that it has »—1 infinite branches lying below the 
asymptotes which bisect the angles between the asymptotes of P. Again, since 
J («)=0 has no imaginary roots, the curves P and Q cannot intersect ; and it 
follows that from each point of intersection, either of P or Q, with the x-axis, a 
branch proceeds up to the corresponding asymptote, without cutting any other 
branch of either P or Q. This may be proved, as regards P and Q separately, 
directly from the equations. 
Suppose the roots of f(x) =0, arranged in descending order of magnitude, to 
We aoe) other 
VOL. XXX. PART II. 4D 
