INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 249 





^7 = Va 2 cos 2 </> + b* sin 2 </> = 0A(c/>, K), 



to the modulus /c, the eccentricity of the ellipse. 

 Then s = a E(f>, where Jo A</> d<j) is denoted by E<j> 

 in Legendre's notation of his standard E. I. II; 

 it is tabulated in his Table IX alongside of F<j> 

 for every degree of the modular angle 6, and to 

 every degree in the quadrant of the amplitude c/>. 

 But it is not possible to make the inversion 

 and express <p as a single-valued function of </>. 



FIG. 2 



11.31. The E. I. II, E(t>, arises also in the expression of the time, /, in the oscil- 

 lation of a particle, P, on the arc of a parabola, as F<t> was required on the arc 



of a circle. Starting from B along the parabola 

 BAB', Figure 3, and with AO = h, OB = b, 

 BOQ = 0, AN = y = h cos 2 (t>,NP = x = b cos 

 (/> and with OS = 2h = b tan a, OA r = SB 

 = b sec a, the parabola cutting the horizontal 

 at B at an angle a, the modular angle, BRA'B' 

 is a semi-ellipse, with focus at S, and eccen- 

 tricity K = sin a. 



(Velocity of P) 2 



= (b 2 cos 2 + 4^ 2 sin 2 < cos 2 0) 



a 2 (i sin 2 a sin 2 <) cos 2 </> ( -^- 1 = 



cos 



if 7 denotes the velocity of P at A , an 



= a. 



Then with 5 the elliptic arc^^, 



and so the point R moves round the ellipse with constant velocity V, and ac- 

 companies the point P on the same vertical, oscillating on the parabola from B 

 to B'. 



In the analogous case of the circular pendulum, the time t would be given 

 by the arc of an Elastica, in Kirchhoff 's Kinetic Analogue, and this can be placed 

 as a bow on Figure i, with the cord along AE and vertex at B. 



Legendre has shown also how in the oscillation of R on the semi-ellipse BRB' 

 in a gravity field the time / is expressible by elliptic integrals, two of the first 

 and two of the second kind, to complementary modulus (Fonctions elliptiques, 

 I, p. 183). 



