IN A RESISTING MEDIUM. 
595 
The remainder of the fraction must be split up into partial fractions 
of the form 
A sn x dn x — A! 
cn x — cn a 
B sn x dn x — B' 
cn x — cn wa 
C sn x dn x —- C 
cn x — cn w 2 a 
To determine A and A’, multiply up by cn x — cn a, and put x = a; 
therefore 
1— cna 
c 3 + A 3 (cn a — cn wa) (cn a — cn w 2 a) 
and 
_ c » ( *'+* )» 
c 3 + $ 
Qc'+Jcf 
K+h_ 
c + k 
( k, + /c __ 
k' + k\ 
/ A' + A 
_ A' + A\ 
\ WC + A 
c+k) 
\w 2 c + A 
c + k) 
(1-®) (l-® 2 ) 
_ c o (A' + A) 3 (1—cn a) sn a dn a 
c 3 + A 8 (cn a-cn wa) (cna~cn w 2 a) 
A sn a dn a 
- sna dna. 
& 
To determine B and B' } multiplying up by cn x—cn wa, and then 
putting x=wa; 
# &'+*)*__ (1-cnwa) 
(k' + kf 
' c 3 + A 3 
(A' + A) 2 
(cn wa— 
cn a) (cn wa- 
—cn w 2 a) 
A' + A 
wc + k 
/A' + A 
k' + /c \ 
(k' + k 
k' + A\ 
\c + k 
w c + A ) 
\w 2 c + A 
wc + Ay 
(w —1) (w —W~) 
= g w (^' + z?:) 2 = - w; 
sn a dn a 
2 >,_ 3 (A' + A) 3 (1—cn wa) 3 (1—cn a) 2 
c 3 + A 3 (cn wa—cn a) (cn coa—cn ora 
3 (A' + A) 3 (1—cn wa) sn wa dn wa 
c 3 + A 3 (cn wa—cn a) (cn wa—cn w 3 a) 
= B sn wa dn wa~ -« sn wa dn wa* 
fit 
