472 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
Now since 
m.M—1 noo 2.n—1.n—2. 
(a+ iy)*=a" + nia 'y ——5 2" scaling ih i 
m.N—1 no 9 
(a—ty)*=2" — nia "y——5— 2" y+ . 
we see that the above equation is equivalent to (a + iy)" + (a —ty)"=0 
If we put x=7 cos 6 and y=7 sin 6, this becomes 
r{(cos 0+7 sin 0)”+(cos @—7 sin #)"}=0, 
or 27" cos nO=0. Hence for the infinite branches we have cos n6=0, or 
pa 3 5 2n—-1 
17) Fe taps Spel Prey UPA A IS sate ego li} 
a7 30 On 2n—1 
and O=F, 5 on? On 5 Gib S Se T. 
This shows that there are » real asymptotes, the directions of which are 
arranged at equal intervals through the first two quadrants. Also since the 
equation 
oe Wn alley Soon ails n—1. M—2.N—3 4.4 =, 
Ca ag Des gy >a is 
is satisfied by x=y cot5_,y cot 5 27, &e., the first member of it must be iden- 
om 2n 
tical with the prods 
(2 y cot 3 )(x—y y cot 3 creda _(@ x—y cot” 1 
Having thus determined the direction of the asymptotes, we have next to find 
where they cut the a-axis. Take the asymptote for which 0=(27—1)z :2n; 
then, observing that the equation to P may be put in the form 
(0+ iy) + (ei) tr (wy +(e)" 
+pj{(atiyy? +(e) *}+ .... =0, 
and that «—y cot 2” = Tis a factor of the first term, we have 
“a—y cot ats T 
—1 2n bee 
“—-y Cob at eee pay Lena S(a + ty)" + (a —ty)""} 
