250 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



11.32. In these tables, E(f> is replaced by the columns IV, IX, of E(r) and 

 G(r) =-(90 - r), defined, in Jacobi's notation, by 

 E(r) = zn eK = </>- eE 



G(r) = zn (i - e)K, r = goe. 



_g 

 This is the periodic part of after the secular term eE = u has been set 



aside, E denoting the complete E. I. II, 



The function zn w, or Zu in Jacobi's notation, or E(r) in our notation, is 

 calculated from the series, 



co . co 



sin 



2mr. 



This completes the explanation of the twelve columns of the tables. 



11.4. The Double Periodicity of the Elliptic Functions. 



This can be visualised in pendulum motion if gravity is supposed reversed 

 suddenly at B (Figure i) the end of a swing; as if by the addition of a weight 

 to bring the centre of gravity above O, or by the movement of a weight, as in the 

 metronome. The point P then oscillates on the arc BEB', and beats the elliptic 

 function to the complementary modulus AC', as if in imaginary time, to imaginary 

 argument nti = fK'i: and it reaches P' on AX produced, where tan AEP' 

 = tan AEB-cn (nt'i, AC), or tan EAP' = tan EAB-cn (nt' , AC'); or with nt' = v, 

 DR' = DB-cn (iv, AC'), DR = DB-cu (v, AC'), with DR-DR' = DB\ EP' crossing 

 DB in R f . 



en (iv, K) = 



cn (v, AC') 



/j cr\ i <TI if i 



/ \ V Oil \ l/j /V J . f v 



,. _ dn(p, AC') i 



11 ( ' ** " cn (v, AC') = sn (K f - v, AC') 



where K' denotes the complementary (quarter) period to comodulus 

 If m, m' are any integers, positive or negative, including o, 



sn ( + imK + im'lK 1 } = sn u 



cn [u + ^mK + im'(K + iKj] = cn u 

 dn ( + 2mK + qm'iK') = dn u 



11.41. The Addition Theorem of the Elliptic Functions. 



sn (u v) = Sn ^cn^dnz;sn2;cn^dn^ 

 i AC 2 sn 2 u sn 2 



en 



cn M cn v T sn M dn u sn 

 i AC 2 sn 2 u sn 2 w 



dn ( ) = n ' M n ^ T ^ 2 sn u cn M sn v cn 

 i - AC 2 sn 2 u sn 2 



