and Infinite Series. 389 
produces a change in the fign of the continual product 
of an odd number of factors, but no change in the fign 
of that of an even number of fadtors. Wherefore, 
changing the fig ns of all the even terms, namely, the 
fecond, fourth, Ac. produces no alteration in the roots, 
but only in their figns, the politive roots being changed 
into negative, and the negative into pofitive. But by 
changing any or all the figns of the odd terms, the equa- 
tion will no longer have the fame roots as before, but 
will have new roots of very different magnitudes from 
thofe of the former, unlefs the fign of the firfl term 
or higheft power is changed alio; but this term is always 
to be fuppofed to remain pofitive, 
3. It alfo follows, that when any term is wanting in 
an equation, or the coefficient of any term equal to o, 
the fum of the negative products in the coefficient of 
that term is equal to the fum of the pofitive products in 
the fame. And if it be the fecond term which is want- 
ing, then the equation has both negative and pofitive 
roots, and the fum of the negative roots is equal to the 
fum of the pofitive ones. But if it be the laft term 
which is wanting, then one of the roots of the equation 
is equal to nothing. And hence arifes a method of tranf- 
forming any equation into another which fhall want the 
fecond term : and to this latter (fate it will be proper to 
F f f 2 transform 
