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the equation to a cycloid referred to its vertex.

If a 1, the curve is an epicycloid.

If a2 > 1, the curve is a hypocycloid.

(25) In speaking of the cycloid I mentioned a property belonging to it which was discovered by John Bernoulli, viz. that if BC (fig. 21) be any curve, the tangents at the extremities of which are at right angles to each other, and if this be developed, beginning from C, and if the involute CD be again developed, beginning from D, and so on in succession, the successive involutes approach continually nearer and nearer to the cycloid, and ultimately do not differ sensibly from that curve. The following demonstration of this remarkable proposition is taken from Legendre, Exercices de Calcul Integral, Vol. 11. p. 541.

Draw the successive tangents MP, PN, NQ... which will be alternately perpendicular and parallel to the first, from the nature of involutes. Let be the angle which MP makes with with the line AB, and put

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and so on in succession.

arc EF

=

=

Then if we were to draw tangents,

making angles = do with the other tangents, we should have

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MP CM = x,

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...

PN = DP = b − ≈, QN = EN = x', &c.

dz dx' do

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From the first we have = fæde, which ought to vanish

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Now the last terms in both of these expressions continually diminish, and if n be made sufficiently large they may be neglected. This may be seen by considering that since

x<a, f"d0"x is less than "də"a, or

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π

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and as the greatest value of is, the denominator, when n

2

is great, far surpasses the numerator, and the term diminishes continually as n increases. Neglecting then the last term and making 0 =

π

= a in the second series, we have

2

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From this it appears that when n is large the series

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may be considered as a recurring series formed from a fraction of which the numerator is a polynomial in y of a finite number of terms, and the denominator is

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and if f(y) be the numerator, we may assume

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by the theory of rational fractions.

y" in

f(y)

cos (ay1)

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Now the coefficient of

is supposed to be b) when n is large; and

on the other side the coefficient of y" is

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If n be indefinitely increased, this is reduced to N1, which is independent of n. Therefore b) is independent of n when n is very large: hence

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These equations belong to a cycloid, in which

the radius of the generating circle.

proposition.

is

Thence follows the

(26) Find the surface, such that the intercept of the tangent plane on the axis of ≈ is proportional to the distance from the origin.

The intercept of the tangent plane on the axis of is

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hence we have

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- px qy = n (x2 + y2 + x2)§.

The integral of this equation is

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(27) Find the surface in which the co-ordinates of the point where the normal meets the plane of ay are proportional to the corresponding co-ordinates of the surface.

The equations to the normal being

x' − x + p (x' − x) = 0, y' −y + q (≈' — ≈),

we have when ′ = 0,

x' = x + pz, y' = y + qz,

therefore x + pz = mx, y + qx = ny.

Substituting these values in

dz = pdx + qdy,

and integrating, we find

≈' = (m − 1) x2 + (n − 1) y2 + C,

which is the equation to a surface of the second order.

(28) To find the equation to the surface at every point of which the radii of curvature are equal and of the same sign.

The conditions that this should be the case are

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Integrating these as ordinary equations and replacing the arbitrary constants, in the first equation by an arbitrary function (Y) of y, in the second by an arbitrary function (X) of x, we find

1 + p2 = Yq®, 1 + q2 = Xp2.

From these we find

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and q ought by their nature to satisfy the equation

which in the present case is

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Now whatever be the form of the functions X and Y, this equation is of the form (x)=√(y), and it can therefore subsist only when each side is equal to a constant.

this constant be represented by

(1 + X)−4

2

; then

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2

Let

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b-y,

(1 + X)

(1 + Y)

a and b being arbitrary constants. If from these we take the values of X and Y and substitute them in those of and q we have

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Р

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q { r2 − ( a − x )2 − (b − y)2 } } '

Putting these values into the formula

dx = pdx qdy,

and integrating, we have

(x − a)2 + (y − b)2 + (≈ − c)2 = r2,

-

which is the equation to a sphere.

Monge, Analyse Appliquée.

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