Page images
PDF
EPUB

(5) Eliminate the constants m and a from

y = m cos (r x + a).

dy Differentiating twice,

po-m cos (rX + a).

dat
Multiplying the former by pe? and adding,

d'y

+ por y = 0. d.ro

(6) Eliminate m and a from the equation

y = m (q? – X*);
dy

dy the result is

XY +

-Y
dr?
dx

2

[ocr errors]

(

= 0.

r

(7) Eliminate c from the equation X Y = CE

-4. Taking the logarithmic differential and eliminating,

dy

X 2y + y

dc

= 0.

(8) Eliminate a and B from the equation
(ir – a)? + (y - 3)2 = pe.

dy Differentiating, (x – a) + (y - 3)

= 0.

dx

[merged small][merged small][subsumed][merged small][ocr errors][subsumed][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

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

[subsumed][ocr errors][ocr errors][merged small][subsumed]

This is the expression for the square of the radius of curvature of any curve.

(9) Eliminate m from the equation

(a + mß) (my) = mg; the result is аху + (Baca – ay – 73)

? -days Bxy = 0. dx

dix

2

[ocr errors]

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

x = ax + by + C, y being a function of æ. Differentiating two and three times with respect to t,

dy
and

6
dx?
dx

d x3 d x3
Eliminating b, we have

[ocr errors]
[ocr errors]
[ocr errors]

= 6

[ocr errors]
[blocks in formation]

This is the condition that a curve in three dimensions should be a plane curve.

[merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small]

(12) Eliminate the power from the equation

m

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

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

(14) Eliminate the exponential and circular functions from

Y =

= 6" * sin n x. Taking the logarithmic differential

1 dy

= m + n cot nr.

y dx

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

[ocr errors][merged small][merged small][merged small][merged small][merged small]

(15) Eliminate the arbitrary function from the equation

wyd (y).
Differentiating with respect to x only,
dz
= y(y);

and therefore X
dc

dx

da

= 0.

(16) Eliminate the function ø from the equation

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

ds dx

n

[ocr errors]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

This is the differential equation to conical surfaces.
(18) Eliminate $ and y from the equation

[ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors]

= nx.

dy

Multiply (1) by 8, (2) by y and add,

dx dx then

+ y

da 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

y y y

+

% = 2

()

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

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small]

0 (x + at) + y (x at), x and t being variable,

[merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

This is the equation of motion for vibrating chords.

(20) Let 3-4(";")

2.vy + (x +

the result is

= 0.

(21) Eliminate and yr from the equation

XQ () + yy (2),
dz

dz
0 (x) + x Ø (z) + yy' (2)
dx

dix'

dz

or

:{1 – «^' (*) yy' (x)} = $(*).

dx

« PreviousContinue »