whence, by solving for x,

x=

p-qy q'y-p'

;

(iv.)

and if r/s and r'/s' be the convergents to (ii.) corresponding to the partial quotients a-1 and a respectively, then

Replacing x in (v.) by its value in terms of y in (iv.), we obtain

p-qy

=

r'(p-qy)+r(q'y—p')

q'y-p' s' (p − qy)+s(q'y — p')'

a quadratic equation in y.

Cleared of fractions, equation (v.) is

s'x2+(sr') x − r = 0;

and, since (s—')2 — 4(—rs') is positive and -r is negative, its roots are real, one being positive, the other negative. Hence the values of y, as shown by equation (iii.), are both real.

from which, by writing down the successive convergents, we derive

Thus x and y are the roots of the quadratic equation

t2-2t2= 0,

and since x must be positive and y negative,

... x = 1 + √3, y = 1 − √3.

Ex. 2. Express the positive root of x2 - 2x — a2 = 0 as a continued fraction, a being a positive integer.

If a, -ß denote the roots of this equation, a its positive, —ẞ its negative root, then

x2 − 2 x − a2=(x − a) (x + ß),

and this identity makes it at once clear that x2 - 2 x - a2 will be negative for all positive values of x less than a. Hence the largest integer less than a is the largest integral value of x that will make x2 - 2x - a2 negative. This value is easily seen to be 1+ a, and we therefore write

a2 - 2a+1 — a2 = (a − 1 − a) (a − 1 + a) = 1,

In the larger text-books of algebra it is shown that any quadratic surd can be developed into a recurring continued fraction. [See Treatise on Algebra, Art. 367.]

Express the positive root of each of the following equations as a continued fraction:

* The asterisk indicates the beginning of the recurring part.

22. Show that the negative root of the equation x2-2x-a2=0, expressed as a continued fraction, (a = a positive integer) is

23. Express both roots of the equation x2 - ах a2b20 in the form of continued fractions, a and b being positive integers.

be used to denote the cross-product differences

bc' - b'c, ca' - c'a, ab' — a'b,

the values of x and y derived from the simultaneous equations

These cross-product differences whose matrices have two rows and two columns, are called determinants of the second order.

Ex. 1. The condition that the equations

ax + by = 0, a'x + b'y = 0,

may be simultaneous in x and y is

a b

= 0.

a' b'