A Synopsis of the Principal Formulae and Results of Pure Mathematics

and so on, as far as may be thought

1 1 1

b2 + b + b1 + &c.

[2] Newton's Method. Determine by experiment a number such that one of the roots is > c1, but > c1 +0,1; for æ substitute c1+y, and from the resulting equation in y, obtain an approximate value of y, by neglecting y2 and all superior powers: thus an approximate value of a is obtained. If greater accuracy is required, the same process may be repeated.

Case 2nd.

roots to be <1.

(Bour. 341-9; Lagr. Equ. Num. Chap. 3.)

Suppose the difference between two possible

Determine the inferior limit of the positive roots of the

equation of differences, and let k be the least integer > √1: transform the given equation into one whose roots are k times as great, in which equation if the series of natural numbers be successively substituted for a, one root and only one will lie between any two numbers m, and m+1, and therefore the

corresponding root of the given equation will lie between m

and

m+1

A nearer approximation may be made to this root,

by either of the preceding methods.

The equation must be cleared of equal roots, before this method can be applied.

(Bour. 350-7; L 217-20.)

QUADRATIC FACTORS.

(43.) Every equation of n dimensions has at least one possible

quadratic factor, if every equation of n (n − 1) dimensions has one possible factor either of the first or second degree.

Hence every equation of 2m dimensions may be decomposed into m quadratic factors. (L. C. 38, 9.)

To find the quadratic factors of (x)=0 divide p(x) by x2+px+q, and continue the division, until the remainder contains only the first power of x, and may be represented by

ƒ1(p, q)x+ƒ2(p, q) = 0,

then f(p, q)=0, and f2 (p, q)=0;

from which equations q may be eliminated, and the corresponding possible values of p and q determined as above (Art. 34.) will give the quadratic factors required. (Bour. 338, 9.)

Further investigation of the commensurable divisors of literal and numerical equations. (Clairaut, Elém. d'Alg. P. 3.)

IMPOSSIBLE ROOTS.

(44.) If a + B-1 is a root of an equation of which the coefficients are possible quantities, a and ß being possible, a-B-1 is also a root.

Impossible quantities of all degrees may be reduced to the form 4+ B-1, A and B being possible quantities.

To find the impossible roots: substitute y +/-1 for a in the given equation, and another equation will be obtained

4+B√√√1=0.

Determine all the corresponding values of y and that satisfy the equations A=0, B=0; let these be a1, α, &c. B1, B., &c. then a,+B1√-1, a2+B2-1, &c. are the required roots.

The series of quantities, -46, -46, &c. will all be found amongst the negative roots of the equation of differences: to determine whether c, a negative root of that equation, is one of these quantities, substitute √c for ≈ in A and B, and if it be such, the resulting polynomials in y will have a common divisor.

If the equation of differences has other negative roots besides the above series, it must also have equal roots, which may be determined by preceding methods.

If any series of quantities be successively substituted for a, there can be only as many changes of sign in the results, as the equation has possible roots.

The equation '(x)=0, has at least as many possible roots as (x)=0, wanting one.

(W. 355-62; G. A. Ch. 2; Bour. 385-92; F. 533—5.)

Newton's rule for discovering impossible roots. (W. 363.)

APPLICATION OF THE THEORY OF EQUATIONS TO SURDS. (45.) To extract the cube root of a + √b: assume

then y

a + √b= x(x + √y)3,

· (a2 — b) x * ;

≈ must be so assumed that (a - b)x-c3, c being an integer.