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(5) Eliminate the constants m and a from
y = m cos (rx + a).
dạy Differentiating twice,
= – qoʻm cos (rx + a).
(7) Eliminate c from the equation X – Y = Ce
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
(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-*
62* + 1
(12) Eliminate the power from the equation
y = (ci' + x*)*.
dy m XY
(13) Eliminate the functions from
y = sin (log x); the result is qila? + x y + y = 0.
Differentiating again and eliminating cot nx by the last equation, we have
(15) Eliminate the arbitrary function from the equation
z = vyp (y). Differentiating with respect to x only,
(16) Eliminate the function p from the equation
y - nx = 0 (x – mx). Differentiating with respect to x only,
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.
*=*** (%) + y*y (%).
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 ď z
- + . = n (n − 1) z;
for the third differentiation
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),
This is the equation of motion for vibrating chords.