594 
PROFESSOR SYLVESTER ON THE REAL 
are real, as should be the case ; for in order that A may be capable of vanishing, g(g 3 — J 7 2 ) 
and q(q 3 — s 2 ) must, by Newton’s rule, be both negative, which could not be the case if 
either g or q were negative; so that g 3 — if' and q 3 — g 2 must have the same signs, in fact 
each must be negative. 
The curve A under consideration has a multiple point of the 4th order of multiplicity 
at the origin, where it is touched by the axis of p. Its distance from the axis for the 
extreme value of p, viz. p = — yo, is 2 qo o- 
It has three real maxima and minima, two belonging to its upper portion and one to 
the lower portion at the points, for which p has the approximate values —rs, — |f, 
and -f( 18 ). 
The curve R, i. e. <r=((j?+l)drQ?+l) 3 ) 2 , has the values 0 and —4 at the origin, a 
cusp at its extremity corresponding to p— — 1 , where both of its branches meet and 
touch the axis of p, and a negative maximum in its upper branch at the point where 
P- — Q- 
At all points within the curve R, s and q are conjugate, and for the points outside real. 
Its lower branch will meet and touch the lower portion of A at the point where jp = — f, 
and its upper branch will intersect and pass out of the upper branch of A at the point 
where ^=— f. The only part of the area A therefore which corresponds to real values 
of g, q, is that which is included between the upper segment of A and the upper branch 
of R, and extends only fromp=0 to^>=— f, i. e. from g^=l to g^=f. Hence we may 
easily find an inferior limit to the values of g and q when the equation (g, q) has two real 
roots; for we have in that case g, q, q 2 —e 3 , s 2 —q 3 all positive. Hence 
*? 5 >s y>£ 3 , rf < g V < y 2 . 
Consequently g, q must each of them always lie between qf, qf ; and since the least value 
of q is g, q must each be always greater than i. e. than ’33499 ( 19 ). 
( 18 ) The large numbers which enter into A may be usefully reduced, and the equation A=0 made more 
2*7 v Qxc 
manageable, by aid of the simple substitutions <r= — - gj-, p = — The equation A=0 then becomes 
(v — 3m 2 + 7 u 3 ) 2 = 2 v ? — 5 u 6 , 
whose maxima and minima will he given by the equation 
(v — 3 v? + 7 w 3 )(— 6 u + 21u 2 ) = 5u 4 —15u s ; 
which, making 1 — 3 u=ca, becomes 
270w 3 -46w 2 -9w+l=0, 
whose roots are all real, and are one just a little greater than — J, another a little less than A, and the third 
a very little less than -Jj- respectively; whence _p=-|(a;—l) will have the approximate values given in the text. 
( 19 ) g : q will have a maximum value, which can he found by writing Se : Sq : : £ : q ; and consequently, remem- 
bering that q=p + l, S=e 5 + ij 5 , <r=S—q 2 —q 3 , 
JS : Sq : ; 5S : 2 q, 
and therefore 
S<r : Sp :: 5 <r+q 2 —q 3 : 2q : : 5(r+p(p + l) 2 : 2(p+l). 
Substituting the values of Sir: Sp in JA=0, and combining the result with the equation A=0, p and c r may be 
.found by the solution of a numerical equation of the 5th degree, and then s and q may be found by the solution 
