388 Dr. HUTtoN on Cubic Equations 
coefficient of the firft term or higheft power is + i, 
then the coefficient of the fecond term is equal to the 
fum of all the roots with contrary figns ; the coefficient 
of the third term is equal to the fum of all the produces 
made by multiplying every two of the roots together; the 
coefficient of the fourth term, to the fum of all the pro- 
duffs arifing from the multiplication of every three of 
the roots together ; &c. and the laft term, equal to the 
continual product of all the roots; the figns of all of 
them being fuppofed to be changed into the contrary 
figns before thefe multiplications are made. All this is 
evident from the generation of equations. And from 
thefe properties of the coefficients the following de- 
ductions are eafily made. 
3. If the figns of all the roots of an equation be 
changed, and another equation be generated from the 
fame roots with the figns fo changed; the terms of this 
laft equation will have the. fame coefficients as the for- 
mer, only the figns of all the even terms will be changed, 
but not thofe of the odd terms : for the coefficients of 
the fecond, fourth, and the other even, terms, are made 
up of products confifting each of an odd number of fac- 
tors; while thofe of the third, fifth, and other edd terms 
are compofed of products having an even number of 
faCtors: and the change of the figns of all the factors 
3 produces, 
