256 DIFFERENTIAL CALCULUS The equation of a tangent to the curve at the given point (now the origin) will be (B) * = 1& By (1), p. 76 Let y = mx be the equation of a line through the origin and a second point P on the locus of (^4). If then P approaches along the curve, we have, from f<7) limit m = -- dx Let be an ordinary point. Then, by 155, a and b do not both vanish, since at (0, 0), from (X), p. 255, , . dx dy Replace y in (^4) by mx, divide out the factor x, and let x approach zero as a limit. Then (^4) will become * a -f- bm 0. Hence we have, from (1?) and ((7), ax + by = 0, the equation of the tangent. The left-hand member is seen to consist of the terms of the first degree in (^4). When is not an ordinary point we have a = b = 0. Assume that c, d, e do not all vanish. Then, proceeding as before (except that we divide out the factor z 2 ), we find, after letting x approach the limit zero, that (A) becomes c + dm + em*= 0, or, from ((7), Substituting from (.#), we see that (JO ex 2 + dxy + e^ = Q is the equation of the pair of tangents at the origin. The left-hand member is seen to consist of the terms of the second degree in (^4). Such a singular point of the curve is called a double point from the fact that there are two tangents to the curve at that point.
- After dividing by x an algebraic equation in m remains whose coefficients are functions
of x. If now x approaches zero as a limit, the theorem holds that one root of this equation in m will approach the limit - a-r- 6.
