564 I'HILOSOPHICAL TRANSACTIONS. [aNNO 1789. 



given equation x" — px"-^ + qx"-^ — rx^-i + &c. = o are equal, by finding 

 the common divisors of the 2 quantities a" — pa"-^ + qa"--^ — &c., and na"-' 



— {n— \)pa"'-^ + (w — '2)qar—'^ — &c., and observed if they admitted only 

 one simple divisor, a — a, then 2 roots only were equal ; if a quadratic, 

 d^ — Act + B, then 2 roots of the equation became twice equal ; if a cubic, 

 a^ — Aa^ -{- Ba — c, then 2 roots became thrice equal ; and so on : or, to ex- 

 press in more general terms what follows from the same principles, if the common 

 divisor bea— 6''X a — c* X a — d' X &c., then r + 1 roots of the given equa- 

 tion will be ^, ^ -|- 1 roots will be c, t -\- 1 will be d, &c. ; and it immediately 

 follows, from the principles delivered in the 2d edition of the same book, pub- 

 lished in 1770, that to find when r ~\- I, v -{- l, t -\- 1, &c. roots are respect- 

 ively equal, requires r -f- * -|- ^ &c. equations of condition, which are deducible 

 from the well-known method of finding the common divisors of 2 quantities in 

 this case of a" — pa"-^ + qa"-'^ •— &c., na"-^ — (n -- i)pa''-'^ -\- {ii — l)qa''-'i 



— &c. of the terms of their remainders, &c. 



In the book above-mentioned the equations of condition are given, which dis- 

 cover when 2 roots are equal in the equations x^ — px'^ -^ qx — r = o, x*' '-{■ qx^ 



— rx ■\- s=.0^ x^ -\- qy? — rx'^ -\- sx — t ■=. 0, in the 2 latter equations the 2d 

 term is wanting, which may easily be exterminated; but it may as easily be 

 restored by substituting for q, r, s, &c. in the equation of condition found the 

 quantities resulting from the common transformation of equations to destroy the 

 2d term. 



2. Another rule contained in the same book is the substitution of the roots of 

 the equation wa"-^ — (n — \)pa"-'^ + (^ — l)qa''-'i — &c. = O respectively 

 for a in the quantity a" — pa"-^ -\- qa"-'^ — &c., and multiplication of 

 of all the quantities resulting into each other ; their content will give the equa- 

 tion of condition, when 2 roots are equal. Mr. Hudde first discovered, that if 

 the successive terms of the given equation are multiplied into an arithmetical 

 series, the resulting equation will contain one of any 2 equal roots, and m of 

 the m -\- \ equal roots in the given equation. 



3. If 3, 4, 5, . . r roots of the equation are equal, find a common divisor of 

 3, 4, 5, . . r of the subsequent quantities a" — pa""^ -f- qa"~'' — &c., na"-^ — 

 {n — l^pa"-* -1- (n — '2)qa"-i — &c., w . (n — l)a"'^ — (n — l) . {n — 2)j&a''-3 

 -J- (n — 2) . {n — 3)^a«-4— (n — 3) . (n — 4)ra"-s -j- &c., w . (n — l) . 

 {n — 2)a"-3 — (n — 1) . (n - 2) . {n— 3)pa''-^ -4- (w — 2) . (n — 3) . (n — 4) 

 qa^-s — &c., ... w . (72 — 1) . (n — 2) . . (» — r -h 2)a''-'+' ^ (n ^ \) , {n—2) 



. . (n — ■ r -1- l)pa"-'' -f &c. ; which will probably be best done by dividing all 

 the preceding quantities by the quantity of the least dimension of o, and the 

 divisor and all the remainders by that quantity which has the least dimensions 

 among them ; and so on : there will result 2, 3, 4, . . r — 1 equations of condi- 

 tion ; and in this case it is observed, in the before-mentioned book, that if the 



