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:: substituting &c., and ordering the terms, we finally get a differential equation

dy d'y dy

+ = 1 - f'x,... (a) dis

d.ro dx3 whose integral, which can be found only in certain particular cases, will give an equation between x, y, expressing the nature of the curve required.

To prepare the equation (a) for these particular cases, let dy

then
doy

dp =P;

pdp

Hence by substituting, dit

dir? dx dy equat. (a) becomes

y

p'dp + p = 1- f'x ....(6)

dy which is integrable whenever f(x) is such that its derived func

dp tion fox is a function of p, and of yp

dy

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1+p

с

and p =

y2

y

dy' for p, &c., we easily

dy

pdp (1 - ) pdp
y 1-p + p'-p
:: ly = lc 1. 11 + p = l.

Ni+pa ::1 + pa =

dy co - 4 =

dx

y i. dx =

ydy

so :.X - C = V c? y’, which reduces to yo = c - + 2 c'x – xu the equation to a circle.

Let the origin of abscissæ be that of the ordinates; then putting I = 0, we have also y = 0, and c = c.

:. y= 2 cx 2......(1) the equation to a circle, whose radius is c, which evidently satisfies the conditions of the problem. Again, since p - 1 is also a factor of the equation (e), we have

dy p-1 = 0, or p =

1.

dx ..y = x + С.......(2) the equation to a straight line inclined to the line of abscisse at an angle of 45°, which also affords a solution to the problem,

Ex. 2. Let N,N' be separated by a constant interval, or let fr = a.

Then f'x = 0, and equation (6) becomes
-pdp + p = 1. Hence
ay

dy
y

1-p3
ly = lc - . 1. (1– pø) = 1.

(1 - p3)} : 1 =

whence substituting y3

dx

p’dp_ which gives

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get

ydy

(y3 —c)}

which being integrated (if possible) will give the relation between x and y expressing the nature of the curve required.

For a solution of this problem, and of some others of the like nature, by the method of Finite Differences, the reader may consult a very excellent work entitled A Collection of Examples of the Applications of the Calculus of Finite Differences, by J. F. W. Herschel, A. M., &c. p. 127.

In p. 129 of that work, the author appears to have made an oversight. By the assumption of a=b=1, he makes the subnormal ydy

and consequently y' constant, which is contrary to the question. dx Hence the common parabola does not satisfy the conditions of the problem.

Mr. Herschel's method, however holds good in other cases ; for instance, (we suppose the reader possessed of the book) when

f (y.dy) = V a?+2yoy, we have

a + 2yqy = (y + qy) = y + 2yoy + Q’y, and :: dy = jayo. Hence x =

ydy

ydy Фу

V awhich, as we have already shewn, gives the equation to a circle.

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64. Let A'P, (Fig. 45), a parabola whose vertex is A, roll upon another AP, equal and similar to the former; then supposing the vertices A, A' to have coincided at the beginning of the motion, required the curve described (1) by the focus of A'P – and (2) that described by the vertex A'. (1). Let AN = x', PN = 4'. Then, the curves having the same tangent at P, viz., PT, and equal ordinates and abscissæ, AA' is I PT. Hence S, S' being the foci S'M = y and AM = x, we have

SB = BS, ST = TS, 2 STB = BTS', &c., and putting SA = A'S' = a, we get = SM cot. MS'S = (x +a). cot. PTN d.r'

dx
= (.c + a). = (x + a).
dy'

d. ↑ 4ax

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.. t' =
aya

(1)
la +x)'
Again, y = TM. tan. 2 PTM

tan. PTM = 2TM.

1 – tan.' PTM Hence a' = + x - a

(2)
2(x + a)
and equating values of x', we get after proper reductions

y(x – a) = x + ax?
:: y = x + 2ax + a'

:: y = x + a ... (6) which is the equation to a straight line inclined at an angle of 45° to the line of abscissæ, &c. Secondly, let AM' = x, A'M' = y, &c., as before; then

y = x. cot. M'A'A= x

aʼx – as,

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ayo

(1)

x2 Again, y = TM'. tan. 2QTM. Whence as before,

y2 + x2

2x

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the equation to a cissoid, the diameter of whose generating circle is

the latus rectum of the parabola. 2

20 or

The above process will apply to all equal and similar curves, the contact being always at the same points in both of them. The general theory however, of roulettes, as they are termed, is best explained in Lacroix, Vol. I. p. 430.

PROPERTIES OF CURVES.

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Let ri, 722

65.

Po be the values of y, which satisfy the equation

y" – (a + bx) y^-! + (c + dx + ex®) y2 - = 0, and S, S, .... S, the subtangents of the curve expressed by its corresponding to these values of y; then

7°
+ + = a constant quantity.
S

S
For, by the theory of equations, r, + ro + ... rv = a + bx

df)

dra

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..

+

+

=b,

dx

dix

dx

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ara

and S,

r, dx
S
r_dx

r,dir
SA

by the common difdr. ferential expression for the subtangent. Hence, by substitution, +

=b, a constant quantity. SA

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+

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72

66. The axis minor of an Ellipse is a mean proportional between the axis major and the latus rectum.

For, supposing x to originate at the centre, we have by the equation to the ellipse

bo

(a? – x2) a?

.

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