. An elementary treatise on the differential and integral calculus. 198 EXAMPLES. therefore it is not a multiple point. When x > or yx Therefore the point is of the first kind, and the tangentsto the curve at the point make with the axis of x angles whose tangents are -f- ^/a and — \Za- 2. Examine x* + 2ax2y — ayz = 0 for multiple points. We proceed according to the second method, as all thecritical points in this example are not easily found by inspec-tion. J = 4^ + «y) = 0; (1) 0 = B(a*-iy) ==; (2) dy 4#3 -f- 4:axy dx ~ 3ay2 — 2ax2 Solving (1) and (2) for x and y, we find Ix = o\ Ix — iaV
. An elementary treatise on the differential and integral calculus. 198 EXAMPLES. therefore it is not a multiple point. When x > or yx Therefore the point is of the first kind, and the tangentsto the curve at the point make with the axis of x angles whose tangents are -f- ^/a and — \Za- 2. Examine x* + 2ax2y — ayz = 0 for multiple points. We proceed according to the second method, as all thecritical points in this example are not easily found by inspec-tion. J = 4^ + «y) = 0; (1) 0 = B(a*-iy) ==; (2) dy 4#3 -f- 4:axy dx ~ 3ay2 — 2ax2 Solving (1) and (2) for x and y, we find Ix = o\ Ix — iaV6\ Ix — —$aV$\\y = 0); \y=-ia)> \y = -%a / Only the first pair will satisfy the equa-tion of the curve, and therefore the ori-gin is the only point to be examined. Evaluating -j- in (3) for x = 0 and y = 0, and representing ~ by p, and -j-by p9 for shortness, we have (3). CUSPS. 199 dy dx V 4^ + ±axy __ 0%atf~—%ax2 ~ 0 12 a;2 + kay + 4a;rp Oft?/? - 4a#24a; + 8rtp + kaxpGap2 -f 6tf?/// — 4a .*. j9 (Gap2 — 4#) when when V = dydx whenSap;0, + V% or — V%. P = °)\y = 0J (X = °) \y = 0l (* = °) \v = 0/ Hence the origin is a triple point, the brandies being in-clined to the axis of x at the angles 0, tan-1(\/2), andtan_1(— V2), respectively, as in the figure. (See Courte-nays Calculus, p. 191; or Youngs Calculus, p. 151.) 3. Examine y2 — x2 (1 — x2) = 0 for multiple There is a double point at the origin, the branches being inclined to the axis of x at angles of 45° and 135°respectively. 4. Show that ay3—xsy—axB = 0 has no multiple points. 111. Cusps.—A cusp is a point of a curve at which twobranches meet a common tangent, andstop at that point. If the two brancheslie on opposite sides of the common tan-gent, the cusp is said to be of the firstspecies; if on the same side, the cusp issaid to be of the second species. Since
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