to 1 , and then to add 2 to the result; and 1 + ( 3 + 2 ) directs us first to add 2 to 3 , and then to add the result to 1 . Again a numerical example of equation (5) is 2 × ( 3 + 4 ) = ( 2 × 3 ) + ( 2 × 4 ) . We perform first the operations in brackets and obtain 2 × 7 = 6 + 8 which is obviously true.
BnoteB (136).–-This fundamental ratio is called the eccentricity of the curve. The shape of the curve, as
distinct from its scale or size, depends upon the value of its eccentricity. Thus it is wrong to think of ellipses in general or of hyperbolas in general as having in either case one definite shape. Ellipses with different eccentricities have different shapes, and their sizes depend upon the lengths of their major axes. An ellipse with small eccentricity is very nearly a circle, and an ellipse of eccentricity only slightly less than unity is a long flat oval. All parabolas have the same eccentricity and are therefore of the same shape, though they can be drawn to different scales.
CnoteC (204).–-If a series with all its terms positive is
convergent, the modified series found by making some terms positive and some negative according to any definite rule is also convergent. Each one of the set of series thus found, including the original series, is called "absolutely convergent." But it is possible for a series with terms partly positive and partly negative to be convergent, although the corresponding series with all its terms positive is divergent. For example, the series 1 − 1 2 + 1 3 − 1 4 + etc. is convergent though we have just proved that 1 + 1 2 + 1 3 + 1 4 + etc. is divergent. Such convergent