474 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
or (x + 2)" —(e— ty)" +p, (w+ ty)" —(2@— wy) 
+p, (a+ wy ?—(x—iyy "+ .... =0. 
The directions of the asymptotes are given by (x +7%y)"—(x—iy)"=0, and put- 
ting «=r cos 0, y=r sin 0, this leads to 7” sin n0=0, 
whence nMO=0, a7, 27, ... n(M—l er, 
eer) as 20 3a n—1 
and ss Rate 
VW vi) Td W 
The first of these values corresponds to the x-axis, which is not an asymptote, 
but is part of the locus of the equation sin (vz,) f(x)=0; and the other values 
show us that the »—1 asymptotes of Q are all real, and that their directions 
bisect the angles between the directions of the asymptotes of P. It may be 
proved, as in the case of P, that all the asymptotes pass through the same 
point on the x-axis at a distance —p,:n from the origin, so that the m asymp- 
totes of P and the »—1 asymptotes of @ all meet in a point, and midway 
between each adjacent two of the former lies one of the latter. 
We will next examine whether the infinite branches lie above or below the 
asymptotes. For this purpose we resume the equation (A), (see p. 473), and 
expand w in a series proceeding by negative powers of y. It will, however, 
simplify the process very much if we transfer the origin to the point of inter- 
section of the asymptotes, or, what comes to the same a put p,=0, which 
may be done without any loss of generality. Putting ee ii 9, 7=a, equation 
(A) becomes 
a=y cot a—_~_4 Pane poi(atiyy?+(e—-wy r+... 
(e+ iy)" +(@—iy)" 
Our first approximation to the value of a is x=y cot a, and we shall get a 
second approximation by putting this value of 2 in the term involving p,. 
Making this substitution, we get 
ps 
= arp {(cos a+7 sin a)"~’+ (cos a—7 sin a)"~*} 
(a+ ty)" +(a—w)"” 
| Sat 
= = 3, C08 (n—2)a. 
Also we have seen that, when we make the same substitution, 
w—y ch _ becomes = : einaeit 
(2+iy)"+(e—w)" ” = nab iyy + n(a—iyyr 2ny"—! cos (n—1)a 
