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Ex. 5. To determine the values of m and n, for which the curves expressed by the equation amy" = "+" are rectifiable. =x+"

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n

(16)

on comparing which with equation (86), Art. 43, the conditions requisite for integration by rationalization are, that either or n 1

2m+should be an integer.

From the former of these conditions we have

2m

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Ex. 6. In the curve whose equation is ay2 = x3, shew that the length of the arc from the origin to the point (x, y)

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Ex. 7. To determine the length of the arc of the catenary, measured from its lowest point to any point on the curve; see

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and taking a general value x, which will refer to any point on the curve, for the superior limit,

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which is the same result as that found in Ex. 4, Art. 293, Vol. I. Hence (20)

82 = y2 — a2;

consequently the arc measured from the lowest point of the catenary is equal to the side of a right-angled triangle, of which y(MP) is the hypothenuse, and a(= мn) is the other side; that is, AP PM. This result is also evident by reason of the relation which exists between the equitangential curve and its evolute, which is the catenary.

Ex. 8. Prove that the whole length of the hypocycloid whose equation is x+y3 = a3, see fig. 10, is equal to 6a.

e* + 1

e-1 prove that

Ex. 9. The equation to a curve being e" = the length of the arc between (xo, yo) and (x, y)

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156.] The process of rectification is frequently simplified by the use of a subsidiary angle, the form of which is suggested by the equation to the curve. The following are examples of this method.

Ex. 1. Let the circle referred to the centre as origin be expressed by the two equations x = a cos 0, y = a sin 0; then a sin o de, dy = a cos e do ;

dx =

..

ds = a do;

8 = a (0-0),

(21)

if s is the length of the arc between the points to which „ and 0 correspond. Thus

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Ex. 2. It is required to find the length of the evolute of the

ellipse defined by the equation (~)*+ (~~)*

= 1.

Let the curve be expressed by means of the two simultaneous equations xa (cos 0)3, y = ẞ (sin 0)3;

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dx=-3a (cos 0)2 sin 0 d0, dy = 3 ẞ (sin 0)2 cos e de,

ds = 3 {a2 (cos 0)2 + (32 (sin 0)2} 1 sin cos e de

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=

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- B2

2

-3 / { a2 + +2 = cos 20 d.cos 20

B2

0

2

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If a ẞ, the result is the same as that given in Ex. 8 of the

=

preceding Article.

Ex. 3. Determine the length of the arc of the parabola, x+y= a1, contained between the coordinate axes.

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.. ds = 4a {(cos 0)2 + (sin 0)4}1 sin ◊ cos ✪ do

a {1+(cos 20)2}+d.cos 20;

4a (sin 0)3 cos e do ;

2

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Ex. 4. The cycloid being defined by the equations

x = a(1—cos 0), y = a(0+ sin 0),

prove that the length of the arc beginning at the vertex

= 4 a sin

a sing

157.] In all the preceding examples of rectification, the value of the definite integral which expresses the length of the curve has been determined by means of the indefinite integral: this however is plainly not possible in all cases, and we are obliged to have recourse to those methods of evaluating definite integrals which have been explained in Chapter IV. The number of such cases is infinite; and consequently I shall consider only those

which have some special interest. The most important perhaps is that of elliptic arcs; on account of the forms of the definite integrals, the geometrical properties derivable from them, the large number of problems whose solutions depend on these integrals, the history of these functions, and the treatises which have been written on them; and especially on account of the large generalizations and developments which double periodic functions, more general than elliptic integrals, have received at the hands of Abel and Jacobi, and which are now being reduced into systematic treatises. As however it is beyond the scope of the present work to give a systematic account of these discoveries and of the properties of these high transcendents, I may refer the student to (1) Théorie des fonctions doublement périodiques, par M M. Briot et Bouquet; Paris, Mallet-Bachelier, 1859: (2) an Appendix by M. Hermite to the new edition of Lacroix's Differential and Integral Calculus; edited by M M. Hermite and J. A. Serret, Paris, Mallet-Bachelier, 1862: (3) Theorie der Elliptischen Functionen, von Dr. H. Durège; Leipsig, Teubner, 1861 (4) Die Lehre von den Elliptischen Integralen und den Theta-Functionen von K. H. Schellbach; Reimer, Berlin, 1864. I propose however to take the simple problem of the rectification of the arc of an ellipse. Let the equation to the ellipse be

:

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and let us suppose s to be measured in such a manner that s increases as a increases; then if e is the eccentricity of the ellipse, and x and x, refer to two points P and Po, see fig. 9, on the curve, a being greater than x, and s being measured from the point nearer the minor axis towards the major axis,

S=

x a2 — e2x2
a2-x2

)

dx.

(25)

Now the indefinite integral of the element in the right-hand member of this equation cannot be determined; (see Art. 80). As however e is less than unity, and x is less than a, we may employ the method of Art. 121, and expand one of the factors of the element-function into a converging series, in ascending powers of and thereby obtain an approximate value of the

ex

a

integral. Let us assume the arc, whose length is required, to

begin at B, the extremity of the minor axis, so that in (25) x = 0; then since

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158.] Although the length of an elliptic arc cannot be expressed generally in finite terms of the rectangular coordinates of its extremities, yet some properties of the arcs may be deduced from the differential of the arc given in (25), and from other equivalent expressions, which deserve a passing notice.

Let the ellipse be defined by the two equations

x = a cos &, y = b sin &;

(30)

then, supposing, as heretofore, s and a simultaneously to increase,

ds=

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— a {1 — e2 (cos p)2}* dp,

8= -a

Φο

Hence the perimeter of the ellipse

= 4a

(31)

(32)

√* {1—e2 (cos p)2}1 dp.

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