82 2>. Waring’s Obfervatlons 
poffible roots. I believe alio, that I firft gave a rule in the 
Milcell. Analyt. for finding the number of impoffible roots 
from finding an equation, whofe roots are the fquares &c. 
of the roots of a given equation, which rule in equations 
of luperior dimenfions fometimes finds impoffible roots, 
when Newton’s, Campbell’s, &c. rules fail, and fails 
when they find them ; and alfo a rule for finding im- 
poffible roots from an equation, whole roots are the Iquares 
of the differences of the roots of the given equation ; this 
rule (as has been obfeived by me in the Milcell. Analyt. 
and 1 hilofophicalTranfa&ions) always difcovers whether all the 
roots of the given equation are poffible or not ; and the laff term 
of the refulting equation difcovers alfo, whether o, 4, 8, 12, 
&c. or 2, 6, 10, 14, &c. impoffible too^s, are contained in the 
given equation ; to which may be fubjoined, if the given equa- 
tion has r poffible and n — r~2t impoffible roots, that the num- 
ber of changes of figns from + to - and — to -f in the refulting 
equation will not be lefs than r . , and the number of con- 
tinued progreffies from -f to + and — to — will not be lefs than t : 
whence, if the number of continued progrefifes be A, the 
number of impoffible roots will not be greater than zt\ and 
the number of poffible roots not lefs than n — 2/'. If the 
number of changes of figns be //, the number of poffible roots 
will not be greater than r\ where r' x Ludisthe greatefi poffible 
number which does not exceed b\ and the number of impoffible 
roots not lefs than n - /. Another rule was, I believe, firff given 
by me in the Milcell. Analyt. 1 762, for finding impoffible roots 
by finding an equation whofe roots are z, where x n —px nmmml + 
q.x n ~ z — &c.. = %, and nx"~~' - n - 1 px”~~ z + n — 2 qx n ~ z - &c. = o. 
In 
