Again, taking the equation to the ellipse given by (24), if is the acute angle between the x-axis and the tangent at (x, y), ds may be expressed in terms of 7 and dr, as follows;

dy

da

T

b2x
;

=

a2 y

a2 sin T

b2 cos T

{a2(sin 7)2+b2(cosr)2}

the radius of curvature of the ellipse at (x, y),

, y =

; (31)

{a2(sinr)2 + b2(cos T)2 } }

Now ds = pdr; consequently replacing p by its equivalent

8

(35)

(36)

the positive sign being taken, so that s increases as 7 increases: consequently

which is another expression for the length of an elliptic arc.

(37)

159.] The definite integrals given in (32) and (37) bear a remarkable relation to each other depending on the equation connecting and T. Equating the two values of a given in (30) and (34), we have

Now, multiplying this equation by a e2, and taking the integral of it for the limits 7 and 7, and taking 4 and 4, to be the values of 4, at the points to which 7 and 7, correspond, we have

T

Of the two definite integrals in the right-hand member, the latter, by reason of (37), represents the length of the arc contained between two points to which 7 and 7 correspond, and which arc is measured from the minor axis towards the major axis; and by reason of (32) the former with a positive sign represents the length of the arc contained between two points to which and , when they are equal to 7 and 7 respectively, correspond, and which arc is measured from the major axis towards the minor axis, because s increases as 4 increases; so that if σ and s denote respectively the former and the latter definite integrals in (40),

8—σ = ae2 {cos & cos T-cos o cos To}.

(41)

Let x and x be the abscissæ to the extremities of s, and έ and So the abscissæ to those of σ; then x = a cos, x = a cos do, ૐ = a cost, &。 = a cos To; so that (41) becomes

0

(42)

that is, the difference between the length of two elliptic arcs, determined under the preceding conditions, is expressed as a function of the abscissæ of their extremities. The discovery of this theorem is due to Fagnani.

This theorem is exhibited geometrically in fig 9; let PP, be the arc whose length is 8, P and P, being the points to which T and 7 correspond: viz. Po To 0 = 70, PTO = 7; and let RON = 4, RO ONO = Po, where = T, Po = 0; and from R and R let the ordinates RN, R, No be drawn, cutting the ellipse in q and Q.; then the arc Q Qo =σ; and oм=x, ом= x; ON = έ, ON = §。; and therefore from (42)

If Þ。 is at B, that is, if the arc is measured from the the minor axis, then 7, 0, x = 0,

at A; in which case,

Το

(43)

extremity of

фо

= α, and

the abscissæ of the points P and Q being connected by the equation

(38), which in terms of x and έ is

a+-a2x2-a2 2 + e2x2 2 = 0.

(45)

From o draw a perpendicular oz on the tangent to the ellipse

at the point P which is (x, y); then

that is, the difference of the arcs into which the elliptic quadrant is divided is equal to the difference of the semi-axes.

160.] Before we leave the subject which is treated of in the present section, it is desirable to say a few words on certain forms of a general character of those curves which being expressed in terms of x and y, are susceptible of rectification; that is, to consider under what circumstances we can find a general integral of ds2 = dx2+ dy2.

Now a general form which evidently satisfies this equation is ds2 = (dx cos a-dy sin a)2+(dx sin a+ dy cos a)2, where a represents an arbitrary angle: and it is satisfied by

whence integrating

ds
0 =

dx cos a―dy sin a,

dx sin a+ dy cos a;

s = x cos a―y sin a+ƒ (a),

0 = x sin a+ y cos a+ $(a);

where f(a) and

(46)

(a) are two arbitrary constants of integration. Now to combine these so that they may form an envelope and thus a curve, let us take the a-differential of each; then

where f represents an arbitrary function; and hence we have

x = sin a f'(a)+cos aƒ"(a))

y = cos a f'(a)—sin aƒ”(a)
8 = f(a) +ƒ"(a);

the values of which manifestly satisfy (1).

(48)

There are also other general forms which satisfy (1): such as

x = 84(a)+a,

y = 8{1-((a))2 } } +↓ (a),

denote arbitrary functions: and taking the a-dif

where and

0 = −1— {y'(a)}2 + {y—¥ (a)} \'(a),

which are plainly equivalent to those by means of which the equation to an evolute is determined from that to the involute : and which is accordant with the fact that all evolutes are rectifiable.

It is worth observing, that if the equations to a plane curve in terms of x and y are of the form of the second and third in (47), or of the first two in (48), the length of the curve is given by the remaining one of each group.

SECTION 2.-Rectification of Plane Curves referred to Polar Coordinates.

161.] Let Fr, 0) = 0 be the equation to a plane curve in terms of polar coordinates; and let it be required to determine the length of the curve between the points (r,, 0), and (ro, 0): Now if ds is an infinitesimal length-element, by Vol. I, Art. 269, {dr2 + r2 do2 } };

ds=

the integral being taken between the limits assigned by the conditions of the problem.

If the equation to the curve admits of being put into the form r = f(0), then

dr = f'(0) d0;

And if the equation to the curve is put into the form 0 =ƒ(r), so that def'(r) dr,

8 =

[** {1+r2 (ƒ′(r))2}1 dr.

(51)

162.] Examples of rectification of plane curves referred to polar coordinates.

Ex. 1. The spiral of Archimedes measured from the pole.

Let the equation to the curve be r = a0; so that dr = a d0;

Ex. 2. The Logarithmic Spiral.

Let the equation to the curve berao; then since

(52)

(53)

dr

log a

ΤΟ

=

a result which immediately follows from the fact that the curve cuts all its radii vectores at a constant angle, and therefore that the difference between any two radii vectores is equal to the projection of the length of the curve between the corresponding points on a line to which it is inclined at the constant angle.