Without finding q, or qp , however, the following useful relations can 

 ifer: 

 [31a,b]: 



—2 — 2 



be inferred directly from the expressions given for q, and q^ in Equations 



i^ + i^ = 2q2 [e - ig + ^2 (^ _ ig)2]l/2 ^^^^^ 



ij - i| = 2^q2(e - ig) [36b] 



-2-2 ^ ( ■ \ u -- 2 . 1/2 



q-j_<l2 ~ *1 ^^ ~ ^S), hence q q = q (e - ig) I36cj 



That the square root of (e - ig) occurring in q-iq2 really is the one 



1/2 

 denoted by (e - ig) can easily be verified by means of the argument 



from continuity, which holds also for the individual values of q and q . 



Also: 



qj^ + q2 = 2q^ [e - ig + 2C^ (e - ig) ] [36d] 



-3 -3 ^ ^ 



qi ^2 ^1 ■*■ ^2 o p 1 /? 



=-+ ^= = 2q2 [1 + 2C (e - ig) ] (e - ig)^/^ [ 36e ] 



q2 1i q^ ^2 + 



In the further development of the analysis some relations among the 

 exponential-trigonometric-hyperbolic functions are useful. A systematic 

 list of some of the relations for sin iz, cos iz, sinh iz, cosh iz, 

 e— ^ , e— , sin (x + y) , cos (x + y) , sinh (x + y) , cosh (x + y) , 

 sin (x + iy), cos (x + iy), sinh (x + iy), and cosh (x + iy) can be found 

 in an elementary calculus textbook. Here z = x + iy, x and y being any 



real numbers or 0. 



- -2-2-2 



The four possible values of X will then be, since A = - q or q^ , 



A = +_ iq or +^ q . The corresponding values of M are e 1 , 



giwt - iq^x ^iwt + qgX ^iojt - ^2^. 



31* 



