IN A RESISTING MEDIUM. 
135 
denoting cos $ or cos am x by cn x, sin $ by sn x, and V (1 — k 2 sin 3 <£) 
by dn x for brevity, according to Grudermann's notation. 
Suppose x = a when z = 1 ; 
therefore 
and 
1 — b _ 1 -f-cna 
6 V 3 1 — cn a ’ 
__ 1 — cna x/3 + 1 + (\/3 — 1) cn x 
1 — cnx V3 + 1 + (s /8 — 1 ) cn a 
_ 1 — cn a k' + k cn x _ 
1 — cn x k‘ + k cn a’ 
since 
and therefore 
k = 
V3 - 1 
2s/2 
— sin 15°, 
V3 + 1 
^2 s/2 
= cos 15°. 
Since 
s/3 + l _7c' _ 
s/3-1 ~ k 
cn ^iK ', we may put 
1 — cn a cn -§ iK' — cn x_ 
1 — cn x cn 4^iK! — cn a ’ 
where 
k' = r ln . 
J J 0 s/ (1 — k' 2 sin 3 <£) 
As a increases from 0 to oo, b diminishes from 1 to 3 —, and cna 
/ 3 _j_ 1 _ 3 
increases from — 1 to- 7 ^- , 3 , or a diminishes from 2K to 
§Z, where 
K 
s/3 — 1 + VI 
dcf) 
— r 
J 0 V (1 — k 3 sin 3 $) ’ 
the complete elliptic integral. 
Since b is the value of 2 when x = 2K, therefore 
1 — cn a k' — k 
b = 
Also, 
z*-b* = 12s/3b 
’ k’ + k cn a 
sn 3 x dn 3 x 
(1 — cn x ) 4 
= 12V3 6 3 X 3 , 
__ snx dnx 
~~ (1 — cn x ) 3 * 
16 
putting 
