4-4 Mr, u ell ins’s Method of finding the 
the co-efficient of the liigheft power of we have thefe four 
equations, in each of which one value of x is one of the equal 
ones fought : 
4-0*6* - o*-4 = o. 
x % - 1-5*** 4- 0*23 <; 8 — 0. 
x * 4-0*4* - 0*4238 = o. 
x ~ — 0*2231* — 0*1 241 = o. 
Now the moft eligible equation is the quadratic ** 4-0*4* — 
0*4238 — o, whofe affirmative root is-v/o^jS] - 0*2 = 0*4811, 
agreeing with the" value of * found above, but true to two 
places lower in the decimal. 
• EX A M P L E II. 
To find the two equal values of at in the equation 
( 64*' - 20X 1, 4-3 = 0. 
The given equation being divided by 64, we have 
- a: 5 -0*3 125**4- 0*0468 75 = 0 ; and then, from the firft of the 
• five equations given above, we get J 5* 4 — 0*625* = o, and x~ 
^0*125 = 0*5. .But 'from the fecond of the equations juft 
mentioned, we have 0*93^5^ - 0*2343^5 = o, or *’ = 
2 34375 _ 0>i2 and < x </c'2 c = o* c. 
0-9375 v a a 
From the foregoing few pages it is evident, that rules may 
• be made for finding the equal roots of equations of more than 
five dimenfions by divifion ; but the operations by them will, 
: in moft cafes, be long and tedious. It is obvious, however, 
rthat. fuch equations may be deprefled to any dimenfion the alge- 
braift pleafes. 
A 
It has indeed beetiTuppofed, that the number of equations 
• that have equal roots is but fmall, and, confequently, that the 
chief 
