Roots of Equations . 375 
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and, in a, % rises to 6 dimensions ; in / 3 , to 8 dimensions ; and, 
in y , to 10 dimensions, &c. 
The second remainder is (Ca - BjQ. a-f — ay.A)y m -z 
+ (D 7 — B y .a + fiy — A) y m ~ 4 -j- ; or, by substitu- 
tion, m A* y m ~z -{- nA z y m ~ 4 -j- and, dividing by A 2 , the di- 
mensions of % in m, are 15; in tz, 17, Let 7r, k, <r, r, &V. 
be the dimensions of % in the first term of the 1st, 2d, 3d, 4th, 
5th, &c. remainders ; then 
7T = 6 
* = 1 5 
£ = 2K 7 T — |— 4 
<r = 2^> — j* — {— 41 
r = 2 cr — ^ T 4 
the increment of the m — 1 term of this series is ym ~f- 1, and 
therefore the m — 1 term itself is 2 m .m — 1 + m, or m . 2 m — 1. 
Now, in the m — 1 remainder, y does not appear, and, in that re- 
mainder, z rises to m . 2 m — 1 dimensions ; if then, this remainder 
be made equal to nothing, and a value of % determined, the last 
divisor, y =+z Z, where Z is some function of z, is known ; and 
this is a common measure of the two equations y m -J- by™—- 1 
+ cy m ~ 2 -|- &c. — o, and A y m ~ l By 7 *- 2 -j- Cy m ~z = o ; 
consequently, y =fz Z = o ; and y = =±= Z ; hence =±= V~y, or 
v, = =±: s/ ±Z; therefore, by the solution of an equation of 
m .9.m — 1 dimensions, two roots, % =±= ✓ =±= Z, of the original 
equation, are discovered. 
Cor. 1. Since two roots of the proposed equation are z , 
and z — v f x 1 — 2 zx -f- z % — v* = o is a quadratic factor of 
that equation. 
Cor. 2. In the same manner that the two equations 
