252 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



so that 



en v dn v, 



du , 



sn v en v dn v . -n , x 



i - K Z sn 2 u sn 2 v sn v 



i. e (v - u) 



9 - 



- 2 "" Q(v + u) 

 This gives Legendre's standard E. I. Ill, 



M d<f) 



+ n sin 2 </> Ac/>' 



where we put n = - /c 2 sn 2 v = - K 2 sin 2 \(/, 



the normalising multiplier, Jlf . 



The E. I. Ill arises in the dynamics of the gyroscope, top, spherical pendulum, 

 and in Poinsot's herpolhode. It can be visualized in the solid angle of a slant 

 cone, or in the perimeter of the reciprocal cone, a sphero-conic, or in the mag- 

 netic potential of the circular base. 



11.51. We arrive here at the definitions of the functions in the tables. Jacobi's 

 Qu and Hw are normalised by the divisors 60 and ~H.K, and with r = goe, 



D(r) denotes -^r^-, A (r) denotes 



while B(r) = A(go - r), C(r) = D(go - r), and B(o) = 4 (90) = D(o) = C( 9 o) 

 = i, C(o) = 



Then in the former definitions, 



A(r) A (go) ^ 



7^r4 = ^x ; sn u = VK sn eK 

 D(r) D(go) 



B(r) B(6) 



W) = D(o) U = CneK 



CM _ C(o) _ dneK 



D(r) ~ D(o) ' 



Then, with u = eK, v = fK, r = goe, s = go/, 



zn/K = E (s), zn (i -/) K = E (go - s) = G (s). 



