(5) Eliminate the constants m and a from y = m cos (r x + a). dy Differentiating twice, po-m cos (rX + a). dat d'y + por y = 0. d.ro (6) Eliminate m and a from the equation y = m (q? – X*); dy the result is XY + -Y dæ 2 ( = 0. r (7) Eliminate c from the equation X – Y = CE -4. Taking the logarithmic differential and eliminating, dy X – 2y + y dc = 0. (8) Eliminate a and B from the equation dy Differentiating, (x – a) + (y - 3) = 0. dx Substituting these values of y-ß and x - a, we have This is the expression for the square of the radius of curvature of any curve. (9) Eliminate m from the equation (a + mß) (– my) = mg; the result is аху + (Baca – ay – 73) ? -days Bxy = 0. dx dix 2 (10) Eliminate a, b, c from the equation x = ax + by + C, y being a function of æ. Differentiating two and three times with respect to t, dy 6 d x3 d x3 = 6 This is the condition that a curve in three dimensions should be a plane curve. (12) Eliminate the power from the equation m y = (a“ + x*). (14) Eliminate the exponential and circular functions from Y = = 6" * sin n x. Taking the logarithmic differential 1 dy = m + n cot nr. y dx Differentiating again and eliminating cot nix by the last equation, we have (15) Eliminate the arbitrary function from the equation wyd (y). and therefore X dx da = 0. (16) Eliminate the function ø from the equation y - nx = 0 (x – mz). Differentiating with respect to x only, dz ds dx n This is the differential equation to conical surfaces. = nx. dy Multiply (1) by 8, (2) by y and add, dx dx then + y da This is the differential equation to all homogeneous functions of n dimensions. It is to be observed that the two arbitrary functions are really equivalent to one only, for the original equation may be put under the form y y y + % = 2 () This is the reason why both functions disappear after one differentiation. If we proceeded to a second differentiation we should find 0 (x + at) + y (x – at), x and t being variable, This is the equation of motion for vibrating chords. (20) Let 3-4(";") 2.vy + (x + the result is = 0. (21) Eliminate and yr from the equation XQ () + yy (2), dz dix' dz or :{1 – «^' (*) – yy' (x)} = $(*). dx |