Dr. Waring on the Method , Sec. i6j 
In the book above mentioned the equations of condition are 
given, which difeover when two roots are equal in the equations 
a? 3 - px z + qx -r*= o, X' + qx z — rx + s = o, x s qx * — rx z -\~sx-~t 
= o, in the two latter equations the fecond term is wanting, 
which may eafily be exterminated ; but it may as eafily be 
redored by fubdituting for q, r , s, &c. in the equation of 
condition found the quantities refulting from the common 
transformation of equations to dedroy the fecond term. 
2. Another rule contained in the fame Book is the fubditu- 
tion of the roots of the equation na n " 1 -n— ipa 4- n — 2 qa n ~ w * 
— &c. — o refpedtively for a in the quantity a n — pa n — 1 + qa n ~~ l — 
&c., and multiplication of all the quantities refulting into each 
other; their content will give the equation of condition, when 
two roots are equal. 
Mr. Hudde fird difeovered, that if the fucceffive terms of 
the given equation are multiplied into an arithmetical feries, 
the refulting equation will contain one of any two equal roots, 
and m of the 7/24- 1 equal roots in the given equation. 
3- If 4> 5. - .r roots of the equation are equal, find 
a common divifor of 3, 4, 5, . . r of the fubfequent quantities 
a 11 — pa n ~ x 4- qa n ~~* — &c., na n ~~ x — n — 1 pa n ~~ x 4- n — 2 qa nr ~* — &c., 
n . ia Tt '~ z — n — 1 . n — 2pa n ~~> 4-72 — 2 . n — — n — 4 . 
n — \ra n ~"$ 4- &c., n 1 . 72 — 2 — n— i.n— 2 . 7 i — q >i pa 
4-72-2 . 72 — 3 . n — &qa n ~~ s — &c., . — l . n — 2.. 
72 — r 4- — 1 .72 — 2 . . 72 ~ r 4. \pa n ~~ r 4 &c. ; which 
will probably be belt done by dividing all the preceding 
quantities by the quantity of the lead dimenfion of 77, and the 
divifor and all the remainders by that quantity which has the 
lead dimenfions amongd them ; and fo on : there will refult 
2, 3> equations of condition ; and in this cafe , it is 
D d 2 obferved, 
