494 XII Elliptic Functions
If we putb^2 =a^2 ( 1 −k^2 ),wherek( 0 <k< 1 )is theeccentricityof the ellipse, this
takes the form
s(Θ)=a∫Θ
0( 1 −k^2 sin^2 θ)^1 /^2 dθ.If we further putz=sinθ =x/aand restrict attention to the first quadrant, this
assumes the algebraic form
a∫ Z
0[( 1 −k^2 z^2 )/( 1 −z^2 )]^1 /^2 dz.Since the arc length of the whole quadrant is obtained by takingZ=1, the arc length
of the whole ellipse is
L= 4 a∫ 1
0[( 1 −k^2 z^2 )/( 1 −z^2 )]^1 /^2 dz.Consider next Galileo’s problem of the simple pendulum. Ifθis the angle of de-
flection from the downward vertical, the equation of motion of the pendulum is
d^2 θ/dt^2 +(g/l)sinθ= 0 ,wherelis the length of the pendulum andgis the gravitational constant. This differ-
ential equation has the first integral
(dθ/dt)^2 =( 2 g/l)(cosθ−a),whereais a constant. In facta<1 for a real motion, and for oscillatory motion we
must also havea>−1. We can then puta=cosα( 0 <α<π),whereαis the
maximum value ofθ, and integrate again to obtain
t=(l/ 2 g)^1 /^2∫Θ
0(cosθ−cosα)−^1 /^2 dθ=(l/ 4 g)^1 /^2∫Θ
0(sin^2 α/ 2 −sin^2 θ/ 2 )−^1 /^2 dθ.Puttingk=sinα/2andkx=sinθ/2, we can rewrite this in the form
t=(l/g)^1 /^2∫X
0[( 1 −k^2 x^2 )( 1 −x^2 )]−^1 /^2 dx.The angle of deflectionθattains its maximum valueαwhenX=1, and the motion is
periodic with period
T= 4 (l/g)^1 /^2∫ 1
0[( 1 −k^2 x^2 )( 1 −x^2 )]−^1 /^2 dx.Attempts to evaluate the integrals in both these problems in terms of algebraic and
elementary transcendental functions proved fruitless. Thus the idea arose of treating
them as fundamental entities in terms of which other integrals could be expressed.