656 
PROFESSOR SYLVESTER ON TfEE REAL 
Again, 
%j-x+5=^±((2^-2^+2<p)+(^-^+2<f-4))+5 
= 5 ±- 2 ( 3 ^- 6 ?- 4 )+ 5 : 
Hence when <p= — 1, for which value of $ x and y both become infinite, %y— #+5 = 0 ; 
hence the straight line 2y— #-j-5=0, represented by AN in the diagram, will be an 
asymptote to the curve ( 70 ). 
If now we draw the straight line 2 y — #=0, represented by OB in the figure and join 
OC, the curvilinear triangle OCM will be completely under OC, and the curvilinear 
infinite sector XOP completely under OB. 
(88) What we have to prove is, that so long as p lies between 2 and 1, so long may 
A+^JD be substituted as a criterion in lieu of A, it being remembered that A only 
plays the part of a criterion when D is positive and J is not positive. Hence, since when 
J=0 A+^JD and A coincide, we have only to show that, so long as D is positive and 
J is negative, A+y.JD and A will bear the same sign for all such values of J, D, L as 
constitute a facultative system, i. e. coordinates to a facultative point in space. 
Now at any facultative point G (the function of the amphigenous surface), or say 
rather G(J, K, L)>0, or G^l, >0, and consequently considering 5^ ^ as 
the coordinates of a plane curve, the line G^l, 5, =0 (the sign of J being fixed) 
will separate those points for which J, K, L constitute a facultative system from those 
already seen for tlie dexter Lorn. We see .also that y 2 becomes indefinitely greater than y l , so < that it, is the 
value of <p near to — 1 which gives the inferior branch ; and consequently the superior branch of the sinister 
horn belongs to the range from — | to 0, and the inferior to the range from — -J to — 1. 
( 69 ) It may further be noticed that each horn so called is a true horn, being destitute of any point of contrary 
flexure, except at infinity ; for otherwise we should have 
d ty- 
<Py__d<P' dx dtp (<p + l)Y _ 
da ? dx dtp dx ^ ' 8(4<p +3) ’ 
which implies tp=0 or <p = — 1, for each of which values of <p x and y become infinite. It will be seen here- 
after that it is only for the value corresponding to <p=0 that there does exist at infinity a point of inflexion. 
( 70 ) The two points where the asymptote cuts the curve will be found by writing 
P + 1 
which gives 
1+ </5 
P =-= 2 
The superior sign corresponds to a point x, y in the inferior branch of the dexter horn, and the lower sign, for 
which <p>- — -J, to the superior branch of the sinister horn. It is easy to see that there can be no other asymptote; 
for x, y only become infinite when <p— — 1, or <p= 0; so that if Kx+yy + v is an asymptote, it must contain 
(tp-\-iy, or tp 2 as a factor. The first condition is only satisfied when A : a : v : : — 1:2:5; and the latter cannot 
be satisfied at all. 
