(W/7±W/11)2 = 7±2√√/7× 11 + 11 = 18 ± 2/77. Therefore,

√/18+2/77 = √/7 ±√/11.

Whence it appears that an expression, such as √√±√√b, may sometimes be reduced to the form a' ± √√√b', or a ±√b'; and this transforination should be made when it is possible, because in that case there is only one or two simple square roots to be extracted, whereas the expression ab requires the extraction

of the square root of a square root.

119. Having given a quantity of the form a±√b, to discover whether it is the square of an expression of the form a′ ±√b′, ora'b', and to determine that root.

Call p and q, the two parts of which the square root of a + √b is composed; p and q are either both irrational, or one is rational and the other irrational.

and, since P and q are either one or both irrationals of the second degree, p2 and q3, and consequently p2 — qo are rational quantities; therefore, a2b is a rational quantity.

Whence we may conclude, that ab cannot be the square of an expression such as a' ±√b', or va ±√b', unless a2 — b is a perfect square.

Let this be the case, and let a2bc. The equation (3) becomes p2-q2 = c.

Equations (1) and (2) squared, give

p2 + q2 + 2 p q = a + √b,

These two formulas may be verified à posteriori; square both members of the first; then

In a similar way the second formula may be verified.

120. Remark. Since the formulas (6) and (7) are true, even when a2b is not a perfect square, they may still be used to give values for the expressions

a+✔b, and a — √b;
✔

but in that case the expressions are not simplified by such a process, since the quantities p and q are of the same form as the given expressions.

It is only proper to make this transformation when ɑ2 perfect square.

b is a

121. As an example of the use of these formulas, take the numerical expression

94 + 42 √5, or 94 + √8820.

Here

a = 94, b = 8820,

c = √a2 b = √8836 ·

88204,

a rational quantity; whence the formula is applicable, and

and

= ±(√49 + √45)

= ± (7 + 3 √/5),

(7+35)2 = 49 +45 + 42 √594 + 42 √5.

Again, let there be the expression

whence the formula is applicable; and the root required is

√np + m2 — m)2 = np+2 m2 - 2 m √n p + m2. Let it be required to simplify the expression

16+

16 + 301 + √16 — 30 √—I.

By the formulas, we find

therefore

16 +30=I= 5 + 3 √−1,

16301=5—3 √1;

16301+16-30/110.

√16

This last example shows better than all the others the utility of the general problem which we have solved; for it proves that the combination of imaginary expressions may produce real, and even rational quantities.

The formula may be applied by the student to the following

cases,

√28 + 10 √3 = 5 + v3;

√1+4√=3= 2 + √— 3;

bc+2bvbc-b2 + √/ b c — 2 b √ b c — b2 = ± 2b;

√ ab + 4 c2 — d2 + 2 √ 4 a b c2 — a b d2 = √√ a b + √/4 c2 — d2.

Indeterminate Analysis of the First and Second Degree.

Introduction.-When the enunciation of a problem furnishes a less number of equations than it has unknown quantities, the problem is called indeterminate, since its equations may be satisfied by an infinite number of values attributed to the unknown quantities. But it frequently happens that the nature of the question requires the values of the unknown quantities to be expressed in whole numbers; in this case, one of the unknown quantities, to which we may first give a value altogether arbitrary, must receive only entire values, and such that the corresponding value of the other, in each of the other unknown quantities, may be expressed also in entire numbers. Now this condition very much restricts the number of solutions, particularly if we consider only direct solutions, that is, solutions in entire and positive numbers for all the unknown quantities.

The object of indeterminate analysis of the first degree is to resolve indeterminate questions of the first degree in entire and positive numbers. We shall see hereafter the purpose of indeterminate analysis of the second degree.

I. Equations and Problems of the First Degree with Two Unknown Quantities.

122. Every equation of the first degree with two unknown quantities may be reduced to the form a x + by = c; a, b, c, designating entire numbers positive or negative.

We begin by observing, that if the coefficients a and b have a common factor which does not divide the second member C, the equation cannot be satisfied by entire numbers.

For let a ha', bhb' the equation becomes

=

an equation which cannot be satisfied by any set of entire values of x and y so long as c is not divisible by h.

We suppose in all that follows, that a and b are numbers prime to each other. Since if they have a common factor, c must likewise contain this factor, in which case, it might be suppressed in the equation.

123. For the sake of clearness we will first treat of particular equations and afterwards generalize.

Question 1. To divide 159 into two parts, one of which shall be divisible by 8 and the other by 13.

Let us designate by x and y the quotients of the division of the two parts sought by the numbers 8 and 13 respectively; then 8 x and 13y will express the two parts, and we shall have the equation

which, according to the enunciation, is to be resolved by entire and positive numbers for x and y.

We deduce, in the first place, from this equation

or, performing the division as far as possible,

Now, we perceive that the value of x will be entire if we give to

this condition is necessary; so that it is only required that 7-,59

should be equal to some whole number. Let t be this whole number (t is called an indeterminate), we shall then have

Every entire value of t which, substituted in equation (2), will

give a similar value for y, will satisfy the condition that

7-5y

should be a whole number. Thus the two corresponding values of x and y will be entire, and moreover will satisfy equation (1),