his wants, in the advanced stages of mathematical pursuit, shall call it

in requisition.

The author desires to acknowledge the effective assistance which he has received, in revising the work and superintending it through the press, from Mr. J. J. Elmendorf, to whom it is indebted for many valuable suggestions.

Simple Equations containing one unknown Quantity

Examples in simple Equations.

Cases of Impossibility and Indetermination in simple Equations containing one unknown

Quantity

Simple Equations containing two or more unknown Quantities

Examples

General Formulas of Elimination

Problems producing simple Equations

. 142

143

173

Negative, indeterminate, and infinite Solutions

Discussion of Formulas furnished by general Equations of the first Degree, with two

Problems solved by Quadratics involving two or more unknown Quantities

Decomposition of Trinomials of the second Degree into Factors of the first Degree

Indeterminate Analysis of the second Degree.

Maxima and Minima.

The Modulus of imaginary Quantities

Method of Mourey for avoiding imaginary Quantities

PERMUTATIONS AND COMBINATIONS.

Derivation of Formula for higher Orders of Differences by the Method of undetermined

Coefficients

296

Definitions.

GENERAL THEORY OF EQUATIONS.

NATURE AND COMPOSITION OF EQUATIONS.

Page

. 302

If f(x) be divided by x-a, the Remainder will be ƒ(a)

The first Member of an Equation divisible by the Difference between the unknown
Quantity and a Root

Every Equation has a Root

An Equation containing one unknown Quantity has as many Roots as there are Units

in its Degree

Relation between the Roots and Coefficients of an Equation.

Equations whose Coefficients are whole Numbers; that of the highest Power being

Unity, can not have Fractional Roots

302

. 303

. 303

308

309

312

Changing the Signs of the alternate Terms changes the Signs of the Roots

Surds and impossible Roots enter an Equation by Pairs

All the Roots of an Equation must be of the Form atbi

The Roots of two conjugate Equations will be Conjugates of each other

314

DEPRESSION OR ELEVATION OF THE ROOTS OF EQUATIONS.

Equations whose Roots are those of the proposed, increased or diminished by a given

Quantity.

Numbers between the Roots substituted for the unknown Quantity give results alter-

nately Positive and Negative

NUMBER OF REAL AND IMAGINARY ROOTS IN AN EQUATION.

Herne's Method of resolving numerical Equations of all Orders

Examples

To transform an Equation into another whose second Term shall be removed

To transform an Equation into another whose Roots shall be the Reciprocals of those

of the proposed

.

To transform an Equation into another whose Roots shall be any Multiple or Submul-

tiple of those of the proposed

To transform an Equation into another whose Roots shall be the Square of those of the

proposed

To transform an Equation into another wanting any given Term

To transform an Equation into another whose Roots are the Squares of the Differences

of those of the proposed.