ON A PENULTIMATE QUARTIC CURVE. 
528 
[544 
positive; A 3 = oo, and therefore z± = 0 ; that is, the line 3 coincides with AB, ceasing 
to be a double tangent; there is in this case the one double tangent 4. 
First. z 3 becomes infinite for 1 — 2A 3 =0; that is, a — h = —\/(a) — / or —f=\/(a)+(a—h); 
viz. for — f= \J(a) + (a — h) — e, we have z 3 = — oo , and for — /= y (a) + (a — h) + e, we 
have z 3 = -f go . 
Secondly. The lines 4 and 2 will coincide if 
a — h 
— 2/ + V{4/- — 2 (a + h)}, 
L 2 W{a)-fl 
X 4 (>4-4/) =-2 (a + h), 
that is, if 
or, substituting for \ 4 its value, 
(a - h) [(a - h) - 8/ {/(a) -/}] + 8 (a + h) {/(a) -/} 2 = 0, 
{3a + h — 4//(a)} 2 = 0, or 4/ /(a) = 3a + h, 
the condition is 
(observe that, f having been assumed negative, this implies — h > 3a). That is, 3a + h 
being =— but not otherwise, the double tangent 4 will, for the value — /= - ^. -, 
come to coincide with the line 2 ; and for any greater value of —/ will be as before 
above line 2, (being in this case an ordinary double tangent with real points of 
contact) as appears from the form, i/ 2 = 0, of the foregoing equation for the determi 
nation of f 
The passage of the line 3 to infinity, and the coincidence of the lines 4 and 2 
may take place for the same value of f viz. this will be the case if 
V(a) + (a - h) = 
3a — h 
that is, if 
4V(a) ’ 
7a + 4 *J(a) + h {1 — 4 V(a)} = 0 or — h = • > 
or, a being small, for the value — h= 7a approximately. If — h is less than the above 
value, then ^ ess than V(a) + a — h, or —/ increasing from /(a), the coinci 
dence of the lines 2, 4 takes place before the line 3 goes off to infinity: contrarywise, 
if — h is greater than the above value. 
In any form of the curve (i.e. whatever be the value of / in regard to a, h), if 
we imagine a, h indefinitely diminished, the lines 1, 2 and 4 will continually approach 
G, and the curve will gather itself up into certain definite portions of the lines 
x = 0, y = 0. Thus any secant through A (not being indefinitely near to the line AC), 
which meets the curve in real points, will meet it in two points tending to coincide 
at the intersection of the secant with the line x = 0 ; analytically there are always two 
intersections real or imaginary which (the secant not being indefinitely near the line 
AC) tend to coincide at the intersection of the secant with the line x = 0; and we 
thus see how we ultimately arrive at the line x = 0 twice repeated; and similarly for 
the line y = 0.