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(5) Eliminate the constants m and a from y = m cos (rx + a).

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(7) Eliminate c from the equation a-y-ce ̄ ̄. Taking the logarithmic differential and eliminating,

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(8) Eliminate a and ẞ from the equation

(x − a)2 + (y - ß)2 = r2.

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Substituting these values of yẞ and a -a, we have

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in which a and ẞ no longer appear.

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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;

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(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,

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This is the condition that a curve in three dimensions

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

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(15) Eliminate the arbitrary function from the equation

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This is the differential equation to conical surfaces.

(18) Eliminate and from the equation

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then

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Multiply (1) by x, (2) by y and add,

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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

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

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(19) Eliminate the functions from the equation

≈ = (x + at) + f (x − at),

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This is the equation of motion for vibrating chords.

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