w 
410 Dr. WaRING on the — ati 
the roots become equal. In order to this, in the Mifcell. Anal. 
there is found an equation, whofe roots are the reciprocals of 
the differences of any two roots of the given equation; and - 
from finding a quantity (7) greater than the greateft root of the 
given, and ( x) greater than the greateft root of the refulting . 
equation, and fubftituting <, c—A, 7—2A, &c. for x» in 
the given equation ; will always be found the true number 
of impoflible roots. . 5th, In the fame book are affumed two 
equations (nx — 1px"*-4+n—29x"-3 — &e. =o and] 
x" — px"—' + &c. =w), and thence deduced an equation, whofe 
root is w, from which, in fome cafes, can be found the num- 
ber of impoffible roots. 
6. In the Mifcell. Anal. is given the law of a feries, and its 
demonttration, which finds the fum of the powers of the reots 
of a given equation from its co-efficients. Mr. Evier has fince 
publifhed the fame in the Peterfburg Acts. Mr. pp La GRANGE 
printed a property of this feries, alfo printed by me about the. 
fame time; v/z. that if the feries was continued 7 imfinitum, 
the powers would obferve the fame law as the roots, which 
indeed immediately follows from the feries itfelf; but from 
thence with the greateft fagacity he deduces the law of the 
reverfion of the feries (y=a+6x +cx* + dx’ +&c.): it has fince 
been given in a different manner from fimilar principles in the 
Medit. Analyt. 7. In the Mifcell. Analyt. the law of a feries 
is given for finding the fum of all quantities of this kind (a x 
Q" x y' x dx &c. + &c.) where a, 2, y, 3, &c. denote the roots of a 
given equation, from the powersof the rootsof the given equation. 
This law, with a different notation, has been fince publifhed in the 
Paris A@s by Mr. VANDERMONDE; who indeed mentions that he 
3 had 

