AND IMAGINARY BOOTS OF EQUATIONS. 
661 
at infinity to the first and fourth branch. A cubic asymptote touches the intermediate 
branches in the point at infinity corresponding to <p=0. For we have 
and writing v for — jj, 
®=?(r£d=^ 3- p + *’ - * 1 ’- ■ ■ •)> 
, 3*/ . 1 , , \ a 3*/o 3 . 13 . 27 _ \ 
ef B + ••••). ® t = 7( 3 -2?+-8 ^ -Fs^ ••••)■ 
Hence we may determine 
A, B, C, I) so that- A#+Bw+C#+D— | shall ==W 1 +|&m/+ . — 
and I find 
A=\, 
3 T 
B=- 
C=- 
D =-r 
6 72 
Thus the cubic asymptote will have for its equation 
which is a divergent cubic parabola with a conjugate point, viz. the point for which 
v= — -? £+ -v-\--=0, or '/}=-•> £= — — • 
8 5 6 9 8 ^ 128 
(95) It is obvious from the preceding article, that we may expand f in terms of v by 
the descending series 
£= A#+Bv+C#+D + ® + . . . . 
But we may also obtain an ascending series for £ in terms of v which will exhibit the 
nature of the curve of the cusp-node at which point <p= oo. Let <p=- 3 then 
l 
<P 3 (<P + 1) 
= 1 — i v-\-co 2 — jy 3 ....), 
Hence 
V* —OJ 4 (4:-{-4&) — 3<y 2 -j- 2 6t) 3 . . . ). 
#=^ 4 ( 4 N /2* + 5,y2a, 2 -f, v /2a, 3 ... ), 
v 3 =c% 8* 2 + 12a 3 ... ), 
V^=u\ >/2a 3 . . . 
), 
&C.==&C. 
4 t 2 
