PEOFESSOE CAYLEY’S EIGHTH MEMOIE ON QTTANTICS. 
529' 
<p= — f, x= — 76ff, y— — 4Tf ; the cusp, shown in the figure out of its proper posi- 
tion (observe that for x= — 76§y, we have for the asymptote y= — 40fy, so that 
the distance below the asymptote is =§y; Professor Sylvester’s value y — — 25 
for the ordinate of the cusp is an obvious error of calculation). 
<p= — s, gives x— — oo , y=—co , point at infinity, the tangent being horizontal. 
The class of the curve is =4. 
280. The node-cusp counts as a node, a cusp, an inflexion, and a double tangent; the 
node-cusp absorbs therefore (6 + 8 + 1 = ) 15 inflexions, and the other cusp 8 inflexions; 
there remains therefore (24—15 — 8=) 1 inflexion, viz. this is the inflexion at infinity, 
having the line infinity for tangent ; there is not, besides the tangent at the node-cusp, 
any other double tangent of the curve. 
281. The form of the Bicorn, so far as it is material for the discussion, is also shown 
in the Plate, fig 3, and it thereby appears that it divides the plane into three regions ; 
viz. these are the regions PQR and S, for each of which <p(x, y) is = — , and the region 
TU, for which <p(x, y) is = + ; that is, for PQR, and S we must have J=— , and for 
TU we must have J = + . Hence in connexion with the bicorn, considering the line 
y= 0, we have the six regions P, Q, R, S, T, U. It has just been seen that for P, Q, R, S 
we have J = — , and for T, U we have J = + ; and the sign of J being given, the equations 
^=2_L--J , y=^, then fix for the several regions the signs of 2 n L — J 3 and D, as shown 
in the subjoined Table ; by what precedes each of the six regions has a determinate 
character, which for R, S, and U (since here D is = — ) is at once seen to be 3r+2«, 
and which, as will presently appear, is ascertained to be 5 r for P and r+4 i for Q and T. 
282. We have thus the Table 
P, D = +, J=-, 2 n L— J 3 = + }5r, 
Q, D = +, J = -, 2 n L— J 3 = — 1 
!r+4?, 
TTY T~\ I X . OUT XI I I 1 
T, D= + , J=+, 2"L— J 3 = + j 
R, D= — , J= — , 2 n L— J 3 = ’ 
S, 1)=--. J=-, 2 n L-J 3 =+[3r+2?*; 
U, D= — , J= + , 2 n L-J 3 =:+| 
so that we have the kingdom 5 r consisting of the single region P, the kingdom r+4? 
consisting of the regions Q and T, and the kingdom 3r+2« consisting of the regions 
R, S, and U. 
283. Fora given equation if D is = — , the character is 3r+2+ if D= +, J=+, the 
character is r+4« ; if D= +, J= — , then, according as 2 n L— J 3 is = + or is = — , the 
character is 5 r or r+4*. But in the last case the distinction between the characters 5 r 
and t-\-M may be presented in a more general form, involving a parameter arbitrary 
between certain limits. In fact drawing upwards from the origin, as in Plate, fig. 3, the 
lines x — 2y=0 and x-\-y= 0, and between them any line whatever x-\-yjy= 0, the point 
MDCCCLXVIl. 4 c 
