306 
]yiE. A. CAYLEY’S ]ME3IOIE UPOX CALSTICS. 
for the points of intersection with the axis of x. If this equation has equal roots, there 
will be a double point on the axis of x, and it is important to find the condition that 
this may be the case. The equation may be written in the form 
(3, 0, -a, 12A, 3a^-48AmX^, 
the condition for a part of equal roots is then at once seen to be 
-(a^-12Am)^+(a^-18A??ia+54A^f=0; 
or reducing and throwing out the factor A^, this is 
2 7 A^ + 2?u( 8)7f — 9a)A— nf — a ) = 0 . 
This equation will give two equal values for A if 
8m^ —9af-\-27a,%7if—a)=0, 
an equation which reduces itself to 
(4m^— 3a)®=0. 
Whence, if 4m^ — 3a be negative, i. e.if a>-^, the values of A will be imaginar}', but 
if 4m^ — 3a be positive, or a<-^, the values of A will be real. If a=-g-, then there 
will be two equal values of A, which in fact corresponds to a cusp upon the axis of x. 
Whenever the curve is real there will be at least two real points on the axis of x ; and 
when a<-^, but not otherwise, then for properly selected values of A there will be 
four real points on the axis of x. 
Differentiating the equation of the curve, we have 
— oc,)x-\- A^^dx-\- {x^ — cc)ydy=9 ; 
and if in this equation we put dx—9, we find ^=0, or x^-\-y^—a=9, i. e. that the points 
on the axis of x, and the points of intersection with the circle 
points at which the curve is perpendicular to the axis of x. To find the points at which 
the curve is parallel to the axis of x, we must write dx=9, this gives 
{oif-\-y'^—a)x-\-4:A=9, 
and thence 
and 
A-\-x\x~m)=9 : 
this equation will have three real roots if A<^, and only a single real root if A>-^? 
4771 ^ 
for A=~, the equation in question will have a pair of equal roots. It is easy to see 
that there is always a single real root of the equation which gives rise to a real value of 
y, i. e. to a real point upon the curve ; but when the equation has three real roots, two of 
the roots may or may not give rise to real points upon the curve. 
