423 
equal Roots of an Equation by Divijion . 
1 . 5* 4 - 4 px* + zqx' - zrx 4 r — c. 
2. px* - 2 qx 3 4 3rx * - qwc 4 5^ = o. 
3 ■ 4 Aa 4 Ba 4 ^ ~ o# 
4. A' 1 4 Da* 4 E.v - F = 0. 
3 • x 4 Ga 4 FI ~ o • 
It is obvious, that, when p vanishes, the work will be con- 
liclerably fhortened ; and when both p and q are wanting, 
though the above formula fails, yet the equal root may be eafily 
obtained from the equation ^>a 4 - iqx 3 4 3 rx 1 - 4JA 4 st — o, 
which in that cafe becomes ^rx* — 4JA4 5^ = 0. Whenever 
s is wanting, F, in the fecond cubic above, will be — o, and 
confequently x may be found from the quadratic equation 
a* 4 Da 4 E = o. But in any of thefe cafes the equal root may 
be found by divifion. However, the operation probably will not, 
in general, be fo fhort as extracting the root of the quadratic ; 1 
will therefore haften to give an example or two of the ufe of 
the theorem. 
Given a s -\~x 3 - a 2 40*09433 = 0, to find x 9 two values of it 
Herep = o, q — 1, r= 1, s = o, t — -0*09433, and we get 
EXAMPLE I. 
being equal to each other. 
A = - i ’5 
B =0 
C = 40*2358- 
D = 40*4 
E = -0*4238 
G = - 0*2231 
H =~- o*i 241 
I = - 0*0972 1 
K = - o*i 85 
The proper -values of the co-efficients being written in the 
five equations before mentioned, and fome of them divided by 
thee 
