42 Dr. Allman’s Methods of clearing Equations of 
The imaginary values of higher surds of prime indices, 
when found, would be still more complicated : and it is not 
very easy to find, for example, the imaginary values of a surd 
of the iith power,* 
* The imaginary roots of the yth power of unity are, 
/. . I ^ y/7-2»V'-3 . * * y/7 + 2W-S 
(3»40 - 
3 / 74-21-v/ — 3 
+ i_V'-3 
(S>6;) 
i-V— 3 ^ /— 7 + 2i\/— 3 . i + \/-3 
12 
12 
— 21 + 1 
■2lV — 3 
+ t + v' 
7 + 2Iy/~3 
+ ^ + ^-3 ’ / 637-h 147 v^- 3 i-y/-3 ^ / 637-147^/- 3 
2 2 2 ^Sr 2 
as may be found by solving the equation x* x -{■ i z=: 0 
Here, in justice to Dr. Waring, I must observe, that the application of his method 
to the extermination of the higher surds of prime indices, may, in all cases, be brought 
within the condition of solving an equation, whose dimension is half the index of the 
surd diminished by unity. For any equation, of an even dimension, which has the 
coefficients, at equal distances from the middle, equal, the signs being either alike, or, 
as they recede from the middle, alternately opposite and alike, may in effect be reduced 
to half its original dimension. In that case, half of the roots of the equation are the 
reciprocals, or the negatives of the reciprocals, of the other half. An equation, whose 
roots shall be the respective sums of these pairs, will be of half the dimension of the 
proposed equation. Thus, if 
