AND IMAGINARY ROOTS OR EQUATIONS. 
637 
denote cubical parabolas, and pay no attention whatever to relative magnitudes, but only 
to the order or progression of magnitudes, using the lines which are drawn in the figure 
not as copies but as symbols of the actual curves which are to be mentally imagined. 
Thus the line JOJ is used to represent the straight line L=0 ; A'OA' the cubical 
parabola J 3 = 27 -2 10 L; AOA the cubical parabola J 3 =2 n L; HOII the cubical parabola 
J 3 = — 27L( 49 ). 
It will be observed that certain combinations of plus, zero , minus , positive and nega- 
tive infinity are placed along the lines and inside the sectorial spaces. The meaning of 
these will be sufficiently obvious from what has preceded. They refer to the signs of 
the two values of D in the surface G for each point in the line or sector along or within 
which they are placed. At every point along the line OJ, ^ has only one value, and that 
positive ; along A'OA', ^ has two values, one positive and the other zero. Along AOA, 
has two values, one positive the other negative. Immediately below LOL two values, 
one + oo, the other finite and negative. Immediately above LOL two values, one — oo, 
the other finite and negative. Along 11011 one value, finite and negative. 
Moreover D has been shown to be never zero, except along A'OA', AOA. Hence it is 
obvious that inside A'O J and the opposite sector D has two values, both plus ; inside the 
next pairs of opposite sectors two values, one plus , the other minus ; inside the next 
pair of sectors also two values, one plus, the other minus ; inside the next pair of sectors 
two values both minus , and in the pair of sectors left vacant, for which v< — 27, it 
has been shown that D becomes impossible. 
( 49 ) It has been shown in the preceding articles that corresponding to the line JOJ and to the line Eton, 
the vertical ordinate D of the amphigenous surface (G=0) has only one value positive for the former, negative 
for the latter ; along the line A'OA' two values, one positive the other negative ; for the space between AOA', 
LOL indefinitely near to the latter two values, one positively infinite, the other negative ; and for the space 
indefinitely near to the same on the opposite of it, two values, one negatively infinite, the other negative. These 
results are collected and represented symbolically in the Table annexed. 
J A' A L n 
+ 0 (+ oo)~^ 
+ — 
0 (-») 
Thus, corresponding to the upper sheet of 0, we have the succession 
+ + 0 (+°°) ~ — 
and to the lower sheet 
+ 0 — — (— °°) — 
the two sheets coming together at a cuspidal edge above JOJ and below IT OH. 
Moreover these are the only positions of the line revolving in the plane of D corresponding to which a change 
in the nature of D can take place, and thus we can without further examination fill up the Table, giving the 
nature of D for the intervening spaces, and may thus obtain the Table embodied in the dial-figure above, viz., 
J A' A L n 
+ + + 0 + (+l>) - 
+ 
+ 0 - - _ _(_co) - 
4 q 2 
