(5) Eliminate the constants m and a from y = m cos (rx + a). (7) Eliminate c from the equation a-y-ce ̄ ̄. Taking the logarithmic differential and eliminating, (8) Eliminate a and ẞ from the equation (x − a)2 + (y - ß)2 = r2. Substituting these values of yẞ and a -a, we have in which a and ẞ no longer appear. This is the expression for the square of the radius of curvature of any curve. (9) Eliminate m from the equation (a + mẞ) (x2 - my3) = my2; (10) Eliminate a, b, c from the equation ≈ = ax + by + c, y being a function of a. Differentiating two and three times with respect to x, This is the condition that a curve in three dimensions Differentiating again and eliminating cot na by the last equation, we have (15) Eliminate the arbitrary function from the equation This is the differential equation to conical surfaces. (18) Eliminate and from the equation then Multiply (1) by x, (2) by y and add, 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 |