150. From Art. 149 it follows that :
(1) Changing the signs of an odd number of factors of numerator or denominator changes the sign before the fraction.
(2) Changing the signs of an even number of factors of numerator or denominator does not change the sign before the fraction.
Consider the expression
{1}{(a - b)(a - c)}+{1}{ (b - c)(b - a +{1}{(c- a)(c - b) }
By changing the sign of the second factor of each denominator, we obtain -{1}{(a - b)(c - a) }-{1}{(b - c)(a - b)} -{1}{(c - a)(b - c)}
Now it is readily seen that the L. C. M. of the denominators is (a — b)(b — c)(c — a), and the expression
{—(b — c) — (c— a) — (a — b) }{(a — b)(b — c)(c — a) } ={— b+c — c+a — a + b }{(a-b)(b-c)(c-a) } =0.
151. There is a peculiarity in the arrangement of this example which it is desirable to notice. In the expression (1) the letters occur in what is known as Cyclic Order ; that is, b follows a, a follows c, c follows b. Thus, if a, b, c are arranged round the circumference of a circle, as in
the annexed diagram, if we start from any letter and move round in the direction of the arrows, the other letters follow in cyclic order, namely abc, bca, cab.
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The observance of this principle is especially important in a large class of examples in which the differences of three letters are involved. Thus we are observing cyclic order when we write b — c, c — a, a — b; whereas we are
