XV.—A New Method of Investigating Relations between Functions of the 
Roots of an Equation and its Coefficients. By J. Douctas HamittTon 
Dickson, M.A., Fellow and Tutor of St Peter’s College, Cambridge. 
(Received January 6, Read January 19, 1880.) 
n.n—1 
m2” 
degree, NEwTon’s rule for a superior limit to the number of its imaginary roots 
depends upon the changes of sign in the series of functions—called, by SYLVESTER, 
Quadratic Elements— 
I. If aa" +n. ba" "+ cx" +..,.=0 be a rational equation of the nth 
GC: P=a¢, c= bd, 2. 
n+ 1 in number. 
It is a matter of some interest to know the relations in which the quadratic 
elements stand to the roots of the equation. The following method exhibits 
this relationship, and leads to others of a higher class. 
For simplicity, consider the biquadratic equation 
azt+4bz? + 6cx2?+4dx+e=0, 
whose roots area, 8, y, 6; and P,, P,, .... are the sums of the roots one at 
a time, two at atime..... [This system of notation, viz. a, b, ¢,....; 
a, B,y,....; Py, Py,....; will be continued for equations of higher degrees. | 
It is a known result that, for example, 
ee. L, M,N ee Beale 
P,g,7rTiP, Q,R Pp, 7\P, Q 
and also 7 (L+mM+nN , /P+mQ+nR 
pLt+ qM4+7r7N , pP+ cara 
and the theorem may be continued to any extent. 
The symbol 
cal 1 ’ 1 , 1 ’ 1 | 
1,1,1,1] B+ty+6,a+y+8,a+B+5,a+6+y 
= 24 feel 1 ’ 1 }=2Z{(a—)?}, 
1,1] Bt+y+6,a+y+6 
VOL, xXx. PART TI, 6x 
