544] 
ON A PENULTIMATE QUARTIC CURVE. 
527 
to z — — oo, the lower loops lengthening out and finally becoming each of them a pair 
of parabolic branches parallel at infinity; and then reappearing at z = + oo, again move 
downwards, each loop becoming in this case a pair of hyperbolic branches touching 
two asymptotes at z = — oo, and then again on the opposite sides thereof at z = + oo, 
and coming down as a single branch to touch the double tangent 3 which is now 
above 4. Secondly, the double tangent 4 may come to coincide with the horizontal tangent 
2, at the instant of coincidence being a tangent of four-pointic contact; and becoming 
afterwards (being as before above 2) an ordinary double tangent with two real points 
of contact; viz. instead of a simple loop at C we have a heart-shaped loop. 
But to investigate whether the two cases actually happen, and in what order of 
succession, we require 
the expressions 
of z for the several lines in question; we find, 
without difficulty, 
for line 
1, 
01 1 + 2V 
where X 4 = — 2f+ V{4/ 2 — 2 (a + h)\, 
» 
2, 
1 
* 2_ 1-2\ 2 ’ 
„ X 2 = 2/+V(4/-2(a + A)[, 
1 
^ a — h 
» 
3, 
* 3 " I-2X3’ 
1 
^ a — h 
4, 
* 4 ~T^2X 4 ’ 
” **-2 W(a)-f}> 
where X 4 , X 2 , X 3 , X 4 are 
all positive. 
Observe that in the limiting case — /= \/(a), 
where, instead of the loops at A, B, we have cusps; z u z 2 , and 0 4 are (in general)