636 
PROFESSOR SYLVESTER ON THE REAL 
When fl= -12, 
■*=*JtS*= 144 x|=192; 
and when 5= — 16, 
2=256x^=3071. 
Moreover, when q is a maximum or minimum, it will readily be found that 
$ 3 +ll0-j-24=O; so that 0= — 3, or Q= — 8. Hence for the value of & found from the 
D 128 . 
above cubic y<192 and p = 1 — - is positive. 
(58) When J=0, v=0 ; and when L=0, v—co . 
For these two cases it will he more simple to dispense with the auxiliary variable 0, and 
to revert to the original equation between J, K, L. 
Accordingly, when J=0, we find 8LK 3 — 432L 3 =0. Hence 
L=0, or K 3 =54L 2 , i. e. V=54L 2 ; 
’ ’ \ 128 / 
so that the complete section of G made by the coordinate plane J becomes a straight 
line, viz. the axis of D, and a semicubical parabola whose axis is the negative part of D . 
When J is very nearly zero, v becomes a positive or negative infinitesimal in the equa- 
tion 0 4 -f 40 3 =i\ 
One real root of this equation is 0= ^ j . 
The other is — 4+S, where (4( — 4) 3 +12(— 4) 2 ]B=v, 
Now 
64 
K 3 _/my 
L 2 v + 4 / ^ + ' ~(0 + 4) 
The first value of 6 gives K 3 =54L 2 to an infinitesimal pres ; the other value gives 
or, to an infinitesimal pres, 
J 3 
so that D passes from +oo to — oo , i. e. yj passes through 
zero. 
(59) In the annexed figure ( 48 ), the plane of the paper repre- 
sents the plane of D, i. e. the plane for which D=0 ; JOJ is 
the axis of J, OJ being the positive and OJ the negative 
direction ; LOL is the axis of L, OL being the positive and 
OL the negative direction. In order to avoid any appearance 
of an attempt at a practicably impossible accuracy of drawing, I use straight lines to 
( 48 ) I shall refer, when I have occasion to do so, to this figure, which contains a synopsis of the whole theory, 
under the name of the Dial figure. 
