IN A RESISTING MEDIUM. 
597 
Also, (Enneper, Mliplische Functionen, p. 215), 
sn a cn a dn a , 
—o-o— dx 
cn^ x — cira 
1 io ff @ ( a + x ) sn ( a + x ) 
2 ® © (a—x) sn (a—x) 
x Z a; 
and therefore 
sn a dn a , 
-—dx 
cn x — cn a 
+ 
©(a + x) 
© (a—x) 
x Z a 
1 — cn (a + x) 1 + cn (a—x) 
1 + cn (a + x) 1 — cn (a—x) 
sn 2 (a + x) 
sn 2 (a—x) 
= l log gfe+4 _ xZa. 
Also 
and therefore 
1—cn (a—x) ©(a—x) 
/ x sn x cln x c | x __ 
cn x—cn a 
log 
1—cn a 
cn x — cn a : 
f. 
x sn x dn x — sn a dn a 
cn x — cn a 
2 
= 1 \ 0 J 1 ~ cna V i 
2 °\cnx—cna/ 1 
1—cn (a—x) © (a—x) 
lo£ 
1—cn a 
•cn (a + x) © (a + x) 
©a ©x 
; + x Z a 
+ x Z a 
& 1—cn(a + x) © (a + x) © 0 
n . ©x i ©A (a + x) ©A(a + x) 
= x z a + log- - a log - J thr j * 
®jja 
1 Ik 1 
the functions © x u, © 2 u, © 3 u being respectively Hu, / - H (u + K), 
and a//c'© (u + K), and therefore such that 
©1U ©211 -i ©.,11 
snu=—, cn u = ——, dn u = 
©u ©u ©u 
(Cayley, Elliptic Functions, p, 156). 
Therefore, in the expression for the trajectory, log ^ disappears, since 
1 + (o + co 2 ~ 0, 
74 
