664 
PROFESSOR SYLVESTER ON THE REAL 
two detached pairs of infinite branches lying in opposite quadrants ( 73 ). The value for 
t 2 , it will of course be seen, corresponds to —2 for (p, and gives, as it ought to do, the 
position of the cusp. 
(99) Finally, for sections parallel to the plane of the discriminant and lying below it, 
making D=—k 2 , 
4-2 1 + 1 ? 
3-p’ 
we obtain in like manner 
T. 3F-1 7 (/ 2 +l) 4 ^ (f-l)(15P-l) h (^+1)(15^-I) 
x — Kl 4 {bt‘ 2 +\f' x~t{t*+\)( 5 t*+\y v ' z-typ-iyw+iy 
v F (* 2 -l)(15* 2 -l)(* 2 +l) 3 ,(15<*-1)(<* + 1) 8a: A* (**-l)(£ + l)* 
8X ~ 4 (5t* + l) 4 ’ (5^+l) 2 ’ lz~ 4 (5* 2 +l) 2 
When f=-T 5 there will be an ordinary cusp, and when f= 1, §^=0. 
There will therefore be three branches, — one corresponding to the values of t between 
— sj and -\-\/ T 5 , the other two to values of t between these limits and — and -f- 
infinity respectively. The middle branch passes through the origin, where it under- 
goes an inflexion, and comes to a cusp at a finite distance from the origin in two 
opposite quadrants. The connected branch at each cusp crosses the axis of x, sweeps 
convexly towards the axis of z, arrives at a minimum distance from it, and then goes off 
to infinity. 
The value for ^ corresponds to — f for <p, and gives, as it ought to do, the cusp- 
node. In fact the values <p= — f, <p= — 2 correspond respectively to a cuspidal and to 
a cusp-nodal line in the limiting surface whose sections we have been considering. 
When the cutting plane is that of D itself, the section becomes a double cubic para- 
bola and a single cubical parabola crossing each other at the origin. 
( 73 ) Let e be an infinitesimal, and 0 2 =l-f s; then 
fe= 4(4+fi«y^— 82, 
lx 
C 1 (2 + £)s 2 C 2 V 1 £ 2 
Hence at either cnsp the branch the further removed from the axis of x corresponds to the values of Q 2 be- 
tween 1 and oo, and the inferior branch to its values between 1 and -l; so that the order of continuity of the 
five branches of the curve may be read as follows from the infinite point in the higher branch of the upper 
pair to its cusp ; thence to the infinite point in the connected branch, which is contiguous to the infinite point 
in the opposite extremity of the middle branch ; thence along this branch to its contrary infinite extremity ; 
thence to the infinite point in the upper branch of the inferior pair ; along that branch to its cusp ; and thence, 
finally, along the lower branch to infinity. 
