(5) Eliminate the constants m and a from

(6) Eliminate m and a from the equation

(8) Eliminate a and ẞ from the equation.

Substituting these values of yẞ and x-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

and differentiating,

dy

= 1 − y2.

Differentiating again and eliminating cot na by the last equation, we have

(15) Eliminate the arbitrary function from the equation

Multiply (1) by a, (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

This is the reason why both functions disappear after one differentiation. If we proceeded to a second differentiation we should find

(19) Eliminate the functions from the equation

≈ = 4 (x + at) + √ (x − at),

This is the equation of motion for vibrating chords.