Summation of Series. 469 
co-efficients of the terms of the refulting equations ch ange con- 
tinually from + to — and — to4, the roots of the given equation 
are all poffible, otherwife not; and in a paper, inferted by me in 
the Philofophical TranfaCtions for the year 1764, in which is 
found from this transformation, when there are none, two or 
four impoflible roots contained in an algebraical equation of 
four or five dimenfions; it 1s obferved, that there will be none 
or four, &c. impoffible roots contained in the given equation, 
if the laft term be + or —; and two, &c. on the contrary, if 
the laft term be = or +. Thefe obfervations and _ tranf- 
formation have been fince publifhed and explained in the Berlin 
Ads forthe years 1767 and 1768, by Mr. pp La Grance. 3d. 
In the Mifcell. Anal. an equation is transformed into another, 
whofe roots are the fquares, &c. of the roots of a given equation ; 
and it is aflerted, that there are at leaft fo many impoflible roots 
contained in the given equation, as there are continual pro-= 
erefles in the refulting equation from + to + and ~to-. It is 
afterwards remarked, that thefe rules fometimes find impoffible 
roots when Sir Isaac NewrTon’s, and fuch like rules, fail; and 
that Sir Isaac NewrTon’s, &c. will find them, when this rule 
fails. This rule may fomewhat further be promoted by firft 
changing the given equation, whofe root is “, into another 
whofe root is /—1%*3 but, in my opinion, the rule of Har- 
RioT’s, which only finds whether there are impoffible roots 
contained in a cubic equation or not, 1s to be preferred to thefe 
rules, which, in equations of any dimenfions, of which the 
impoffible roots cannot generally be found from the rules, {e]- 
dom find the true number. 4th, It is remarked, that rules 
which difcover the true number of impoffible roots require 
immenfe calculations, fince they muft neceffarily find, when 
mou. LXXI1V. Hhh the 
