Page images
[ocr errors][ocr errors]

.-. f + ,= fsi, e-n = -i?V2;
.-. fl + ,'* = (2o)#.

Accordingly the evolute is another hypocycloid, the radius of whose base-circle is twice that of the original circle, and whose cusps are on lines bisecting the lines joining the cusps of the original curve; see fig. 107. Similarly may the evolute of this new hypocycloid be found, which will be another hypocycloid, the radius of whose base-circle will be 4 a.

Theoretically, the equations to the evolutes of all curves may be found by means of equations (33), but the difficulty of elimination is in all cases, save in two or three besides the above, so great as to be beyond the present powers of analysis.

290.] We proceed now to discuss general properties of the evolute, of which the current coordinates are £ and r\. From equations (32), we have

[ocr errors][ocr errors]

whence, squaring and adding,

(£-X? + (r)-y)* = p*. (35)

Multiplying the former by dx, and the latter by dy, and

adding (£-x)dx+{r,-y)dy = 0. (36)

Again, multiplying the former by d2x, and the latter by d2y, and adding,

(f - x) d2x + {n - y) d2y = ^ (d2x dy - d2y dx),

= ds2; (37)

since by (5), p (d2x dy d2y dx) = ds3.

Now the relation between (35), (36), and (37) is very remarkable; for (36) is the differential of (35), and (37) is the differential of (36), the differentiations being calculated on the supposition that x and y vary, while £, r), and p do not change. And what geometrical fact is hereby implied? The following: (35) is the equation to a circle of which p is the radius, and the coordinates to whose centre are £ and rj, and of which x and y are the current coordinates. Hence the radius and coordinates to the centre of this circle remain the same, when for x and y we have successively x + dx, y + dy, and x + 2dx + d*x, y + 2dy + d2y; this circle therefore passes through three consecutive points on the curve, and therefore there are three points, and which is the same thing, two consecutive elements, common to the circle and the curve. The circle is for an obvious reason called the circle of curvature. This result is in accordance with the principle of Art. 281; for although nothing was said as to a circle having points common with the curve, yet since f and rj refer to the point of meeting of two consecutive normals, and each normal implies a tangent passing through two points, there must be three consecutive points in the curve for which f, 17, and p do not vary. It is also to be observed, that (36) is the equation to the normal, if £ and rj are its current coordiuates; and thus the centre of curvature is on the normal.

291.] Again, siuce the new curve is the locus of the point of intersection of any two consecutive normals of the original one, if the new curve is continuous, each normal must pass through two points in the new curve which are infinitesimally near to one another. Hence, in the expressions (34), £, p, x, y may all vary simultaneously, and we have , , dy d2y ds d2sdy

and substituting for p and d2s from (5) and (11), wc have and similarly, dr\ ~~^dp

y. (38)

Whence it appears that we may differentiate (34) on the supposition that p, £, and r; vary independently of x and y; that is, the normal passing through (x, y) passses through (f, rj) and (£ + di, t) + dy), though of course, as is plain from the figure 102, the length of p varies.

292.] Squaring and adding the two equations (38) we have rff s + dy2 = dp2. (39)

Let <r be the length of the arc of the new curve, and da an element of it, then by (23), Art. 218,

do-2 = 2 -f dri2;

,'. da% = dp2,

da- = + dp; (40)

And taking the positive sign, that the analytical expression may be accommodated to the curve in fig. 102, where An = a, and the element of the arc at n = da, and where therefore <r and p are increasing simultaneously, we have

da dp = 0;

<r — p = a constant = c:

and therefore the difference in length between the radius of curvature of the original curve, and the length of the arc of the new curve is the same, at whatever point of the old curve the radius of curvature is drawn. Imagine then, see fig. 103, a perfectly flexible and inextensible string to be fixed at a point ((> v) of the new curve, say at n, and of length equal to the radius of curvature of the old curve which abuts there, say equal to pn; then, if the string is wrapped round the curve, say towards A, just so much will be taken off from the string by the wrapping that the remainder will be equal to the radius of the old curve, corresponding to the point in the new curve at which the wrapping ends; and therefore if an inextensible string is unwrapped from the new curve, as e. g. from An, the length of which is exactly equal to the length of the new curve + Ao, which is constant and is the radius of curvature of Op at o, the extremity of it will generate the old curve, viz. Op. It is for this reason that the new curve is called the evolute*, as being that from which the string is unwrapped, and the original curve is called the involute with respect to it.

293-3 It is manifest that the lengths of all evolutes can be determined; that is, the lengths can be compared with straight lines, whence they are said to be rectifiable; for, it appears from what has preceded, that the length of the evolute is equal to the difference of the radii of curvature of the involute corresponding to its two extremities.

Of this we subjoin some examples:

* By French writers the evolute is named dtveloppte, and the Involute dtveloppante.


Ex.1. To fiud the length of the evolute of the parabola, in terms of the coordinates of its extremities.

osi = x, Mp.= y; On = f, Nn = Tj; see fig. 103. Let the equation to the parabola be y2 = 4?ax.

Then by (9), p = ; ;therefore by what has preceded,


The length of An = rad. of curv. at P — rad. of curv. at o;

2(a + x)*

[ocr errors]

since by Ex. 1, Art. 289, x = £-5-?.

Ex. 2. To determine the length of the fourth part of the evolute of the ellipse; see fig. 104.

The length of Ec = rad. of curv. at B'— rad. of curv. at A,

= fjl _ fl by Ex. 1, Art. 283,
o a


[ocr errors]
« PreviousContinue »