318 PHILOSOPHICAL TRANSACTIONS. [ANNO 1782. 



ing written for x, the equation becomes 8 + 20 — 64 + 36, which is evidently 

 = O; consequently 2 and 2 are roots of it. 



Otherwise 2, the value of x given by the theorem, being written for it in the 

 quadratic equation 3.r 2 + lOr — 32 = O, the result is 12 + 20 — 32 = O. 



Or, dividing the given cubic by the quadratic [x — 2)' = x 1 — Ax + 4, the 

 quotient is x + Q; therefore the 3 roots are 2, 2, and — 9. 



Theorem 2. — If the biquadratic equation x 4 — px 3 + qx* — rx + s = has 



. , 1 2r — '2pq pr — \6s 4b — 2fl , r 



two equal roots, make a = ' , b = , c = +, and d = , 



1 Spp - Sq 3pp - Sq 3 4a + 3p' 4 a + 3p' 



and you will have x = . 



' A — C 



The investigation of this is given by Mr. Hellins. 



Exam. If the equation x 4 >j< — Qx 2 + Ax + 12 = o has equal roots, it is 

 proposed to find them. 



Here = 0, q = — Q, r = — 4, and s = 12; hence a = ■ ' x — — =■— -; 



1 ' ' ~" — 8 x -9 3 ' 



— 16' x 12 — 8 4 x T " + 18 - 1 I - 4 3 , D - b 



"B = ~ ¥x _ 9 = — ; c = 4xV = --; d = T = -; and _- 



= !^~-Ti = — 5~r~5^ = ^ = 2; wn ' c h being written for or, the equation be- 

 comes l6 — 36 + 8 + 12=0; therefore 2 is one of the roots. 



Theorem 3. — If the sursolid equation or 5 — px 4 + qx 3 — rx" 2 + sx — t = o 

 has two roots equal to each other, and you make a = ——J"l B = P r ~ — f 



1 ' ipp — lOy' 4p/> — 10j' 



25? — p« ,0b — 3q 5c + 2r _ s b — e f + c 



C : : \pp - 10?' D "" 5a"T^' E "" 5A~+lp' F : " TT+Tp' ° ~ 7^~d' H = a^d' * = 



B ~ " . and k = , then shall one of the equal values of x be = H ~ K . 



a — c. a - g' > I — G 



The investigation of this theorem is altogether similar to that of the last. 



Exam. Given x b + x 3 — x 2 + 0-0Q433 = 0, to find x, two values of it being- 

 equal to each other. 



Here p = O, q = \, r =1,5 = 0, t = — 0-0Q433, and we get 

 A = — 1*5 E = — 0-4238 1 = — O-OQ/2 



B=0 F = k = — 0-185 



c= +0-2358 o=- 0-2231 and h_-k 



d = + 0-4 H = — 0-1241 1 - g 



It has indeed been supposed, that the number of equations that have equal 

 roots is but small, and consequently that the chief use of the rules for finding 

 their roots, is to get limits and approximations to the roots of equations in ge- 

 neral. That use, it must be allowed, were it the only one, is sufficient to pay 

 for investigating them. But if the equations that have equal roots should here- 

 after be found not so few as has been generally received, then the use cf the 

 above theorems will become more extensive. 



