GIVE THE IMAGINARY ROOTS OF AN ALGEBRAIC EQUATION. 473 
a—y cot — en 
+ pe (c+1y)” +(“e@—w)” (x+ty)"? + (a —iy)"t 
EAT det ‘iyi: —Oetike a b4ches. eter tspanon) 
é : on — 
If in the term multiplied by p, we substitute y cot Es a for x, and neglect 
the following terms, which vanish when « and y are infinite, we shall obtain 
the equation to the asymptote. On making this substitution, the fraction, 
2r—1 
“—y cot 5 
Tv 
0 
Gtyy tea? takes the form 7: 
Differentiating both numerator and denominator with respect to x, the value of 
the fraction is the same as that of 
1 
n(e+iy)""*+n(a—ty)"* 
and the term multiplied by p, therefore becomes # sand the equation to the 
asymptote is 
x“x—y cot n+ f=0. 
The intercept of the asymptote on the «z-axis is therefore —p,:n, or is the 
same for all the asymptotes, which therefore all intersect the x-axis in the 
same point. This result might have been at once obtained by observing that, 
if we increase the roots of the original equation f(z)=0 by the same quantity 
pr: n, or transfer the origin of co-ordinates for that distance to the left, the 
term p,x"' disappears from the equation. 
We have next to determine the asymptotes of Q. Expanding the equation 
(Q), we get 
na" * + (n—1)pya"? + (1 — 2) psa” P+ FDad 
arare (nm —1)(n—2)a"~* + (x —1)(n—2)(n—3) pa" 4+ 2... 
+0Nn (n—1)(n—2)(n—B)(n— Ha. bo . . . =, 
n.nm—1.n—2.n—3.n—4 n- 
y+ ‘ a” a 
or ae i Na 
5! 
+ ps} (n— Tar? B12 Santee Spite 
3! 
APe Mase ra. ake ph oO, 
