248 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



With g = In 2 , and reckoning the time / from A, this makes 



in Legendre's notation. Then the angle (/> is called the amplitude of nt, to be 

 denoted am nt, the particle P starting up from A at time / = o; and with u = nt, 



AP AQ AN 



= = &nU = 



DQ PK 



= 



EP NE 



_ duU = 



Velocity of P = n-AB-cn u = VBP-PB', with an oscillation beat of T seconds 

 in u = eK, e = 2t/T. 



11.21. The numerical values of sn, en, dn, tn (u, K) are taken from a table 

 to modulus K = sin (modular angle, 0) by means of the functions Dr, Ar, Br, 

 Cr, in columns V, VI, VII, VIII, .by the quotients, 



eK = 



dn eK C 



VV tn eK = 4 



r = goe 



u = eK. 

 These D, A , B, C are the Theta Functions of Jacobi, normalised, denned by 



B(r) = ^(90 - r) C(r) = D( 9 o - r). 



They were calculated from the Fourier series of angles proceeding by multiples 

 of r, and powers of q as coefficients, denned by 



q = e-"k 



Qu = i 2q cos 2r + 2^ cos 47- - icf cos 6r + . . . . 

 Hu = 2q l * sin r - iq* sin 3^ + 2(f*~ sin 5^ - .... 



11.3. The Elliptic Integral of the Second Kind (E. I. II) arose first historically 

 in the rectification of the ellipse, hence the name. With BOP = (/> in Figure 2, 

 the minor eccentric angle of P, and s the arc BP from B to P at x = a sin <, 

 y = b cos (f>, 



