* Mr. Wood ow the 
y m ~\~ b y K ~~ l -f- cy m ‘ a ‘ t 4 * &c. = o, and Ay*— 1 4* B jy’ 3 — 8 -]- C 3 _j_ 
=0, are reduced to one, may any two equations be re- 
duced to one, and one of the unknown quantities exterminated 
also, the conclusion will be obtained in the lowest terms. 
Prop. hi. 
Every equation has as many roots, of the form a =t= s/ z±z 6 a , 
as it has dimensions. 
Case 1. Every equation of an odd number of dimensions 
has, at least, one possible root ; and it may, therefore, be de- 
pressed to an equation of an even number of dimensions. 
Case 2. If the equation be of 2 m dimensions, and m be 
an odd number, then m . 2 m — 1 is an odd number, and con- 
sequently % and v 1 (see Prop. 11.) have possible values; there- 
fore the proposed equation has a quadratic factor, x s — - 2% x 
4- 2*— v*=z o, whose coefficients are possible; that is, it has 
two roots of the specified form ; and it may be reduced two 
dimensions lower. 
Case 3. If m be evenly odd, or ^ an odd number, then the 
equation for determining z, has either two possible roots, or 
two of the form a^±zb s/ — 1, (Case 2.); and v % will be of the 
form c-±zds/ — 1 ; hence, one value of the quadratic factor 
x* — sz x 4- if =so, will be of this form, x z — 2a -[-26 s/ — i.x 
4~ A B 4- CD \/~— 1 = o ; and another of this form, 
a: 1 — 2 .a -r- 2b V — 1. x 4- AB — CD\/~ 1= o; consequently, 
x 4 -~ yax z (2AB 4- 4^ 2 + 4,6*)^*— (4a AB 4-4.6 CD) x 4-A 4 B a 
4- C a D 3 =o, will be a factor of the proposed equation; and 
this biquadratic may be resolved into two quadratics, whose 
