IN A RESISTING MEDIUM. 
591 
and similarly 
^ w2 a fic-\-k cn x - cn (/ 3—iy ) 
We can prove that 
1—cn x 
ft + iy = wa, ft — iy = w 2 a; 
for 
//3 * • \ QiC—k' , a . N <j) 2 C—k' 
cn (/5 + ey) = VT , cn (ft—iy) — 
we + k' w # ' ore + A* 5 
and, therefore, if we put 
cn + iy) ~A + iB> 
then cn (ft + iy) cn (ft — iy) — A 2 + B 2 , 
_ ct + k'c + Jc'Z 
“ c 2 —£c + F ’ 
Also 
A -\-iB~ 
we — k' (coc — k') (a > 2 c-\-k) 
o)C “j~ k 
c 2 —£c + /£ 2 
_ c 2 + h (#~*)c - M + iy 3 (£' + *) c 
_p //3 
and therefore j? = 1 t±ML=^^L±£££ 
(c 2 — kc + 
Therefore 
and 
sn 2 (ft + iy) = 1 — (A + iB) 2 
= l-A 2 + B 2 -2iAB; 
sn 2 (ft + iy) sn 2 ((3—iy) 
= (1 -A 2 +B 2 ) 2 + 4>A 2 B 2 
= 1 — 2 (A*-B 2 ) + (A 2 + B 2 ) 2 
= 6 
* + l°-s*+\ 
(c 2 — + k 2 ) 2 
