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(5) Eliminate the constants m and a from

y = m cos (rx + a).

dạy Differentiating twice,

Ice,

= – qoʻm cos (rx + a).
I

da
Multiplying the former by me and adding,

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(7) Eliminate c from the equation X Y = Ce
Taking the logarithmic differential and eliminating,

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Substituting these values of y-ß and x – a, we have

in which a and B 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 + mB) (23* my®) = ma'; the result is

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(10) Eliminate a, b, c from the equation

x = ax + by + C, y being a function of x.

Differentiating two and three times with respect to x,

Eliminating b, we have

This is the condition that a curve in three dimensions should be a plane curve. (11) Eliminate the exponentials from

Ele + 6-*

6-
Multiply numerator and denominator by em, then

62* + 1
€?*1'

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(12) Eliminate the power from the equation

y = (ci' + x*)*.
Taking the logarithmic differential we have

dy m XY
da na + 20°

(13) Eliminate the functions from

y = sin (log x); the result is qila? + x y + y = 0.

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

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(15) Eliminate the arbitrary function from the equation

z = vyp (y). Differentiating with respect to x only,

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(16) Eliminate the function p from the equation

y - nx = 0 (x mx). Differentiating with respect to x only,

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whence

de m

d x

+

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1.

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3

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whence means that = 1.

This is the differential equation to cylindrical surfaces. (17) 1

(17) If = -4 (-9), by the elimination of the function we find

(2. – a) * + (y – 5) = x-e.
This is the differential equation to conical surfaces.
(18) Eliminate p and y from the equation

*=*** (%) + y*y (%).
Differentiating with respect to X,
(1) d = n**** (%) – y a**$* (*) - TV (*)
Differentiating with respect to y,
(2) ed = ** * (T)+ng='y (*)*** (*).
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 should find ď z

ďx

- + . = n (n 1) z;

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for the third differentiation

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=n(n-1)(n-2),

dx dy dys and so on to any order. See p. 26. (19) Eliminate the functions from the equation

= 0 (x + at) + 4 (x – at), x and t being variable,

* = 0" (x + at) + y" (x – at),

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This is the equation of motion for vibrating chords.

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