AND IMAGINARY ROOTS OR EQUATIONS. 
655 
each of them be zero. Hence there is a cusp at the point where #=— 1, y= 1( 66 ), and 
again at the point where 
y= (M = _ 25 . 
8X256 __ _ m~l U 
X— -L + gl-XOg— <0275 y— (5)21 
(87) When ^>=0, and also when <p= — 1, x and y each become infinite; when <p=+co , 
x and y each become unity. 
As <p passes from -f-oo to 0, by is always negative, and x always positive ; so that there 
will be one branch of the curve (CMP in Plate XXV.) extending from x=—l to 
x— -J-co , for which y commences at y— 1, which cuts the axis of x when <p= 3, i. e. 
x= — fy( 67 ), and which, for the remaining part of its course, lies completely under the 
axis of x, becoming infinite when x becomes indefinitely great. 
Again, as <p passes from — oo to — 1, bx remains always positive, but by is negative so 
long as <p<— 2 vanishes when <p=2, and ever afterwards continues positive. Thus 
there is a second branch, COQ, which starts from the cusp C, touches the axis of x at 
the origin, ever afterwards remaining positive, and increasing up to positive infinity. 
Since when <p=oo , ^=co , the tangent at C is parallel to the axis of y, and conse- 
quently the two branches which start from C lie on the same side of the tangent, so 
that the cusp at this point is of the second or ramphoidal kind ; in Professor Cayley’s 
nomenclature a cusp-node, and equivalent to the union of a double point and a cusp 
of the first kind. 
There remains to account for the values of <p in the interval between 0 and — 1. 
Throughout this interval y and x remain both of them negative, and — ^”t^( 68 > 69 ) 
is always positive. 
There will thus be two branches, in each of which x and y increase simultaneously 
in the negative direction, coming to a cusp necessarily of the first kind at the point 
x— — 76-ff-, y—— 25, one branch corresponding to the values of <p from — f to 0, the 
other to the values of <p from — f to — 1, both of them lying completely under the axis 
of x, and becoming respectively infinite at the extreme values of <p (0 and — 1). 
(66) ■\y] iere this branch, cuts the axis of y we haye <p 3 — <p 2 -\-2<p — 4=0, of which the real root will be a trifle 
less than -|. 
(«7) From this it is easily seen that, whatever may be supposed to be the inclination of the axes x, y, the 
ds 
curve in question is rectifiable by means of elliptic functions; for ^ will be expressible as a rational function of 
<p and the square root of a quartic function of <p. The same conclusion will hold for the curve obtained by 
making J constant when J, together with any invariant of the eighth and any of the twelfth order, are taken as 
the coordinates of the amphigenous surface. 
(68) Fo ascertain which range of <p gives the superior and which the inferior outline of the sinister horn, 
let <p=s, an infinitesimal; then (p*-\-<p 3 =£ 3 , and the other value of <3 is — 1 — y, where ij=s 3 . Hence the two 
values of y corresponding to <p nearly zero and <p nearly —1 respectively will be 
12e 12 , _4(-l-») 4 
v,= — — =r= s- and y„= i 
-? 1 £ 3 £ 2 £ 3 £ 3 
Thus y x is negative for s positive or negative, but y 2 is positive in the one case and negative in the other, as 
