592 
MOTION OP A PROJECTILE 
Similarly 
dn 2 (fi+iy) = k' 2 + k 3 (A +- iBf 
= £' 3 + * 3 (A^-B-) + IWAB-. 
and dn 3 (/?+ iy) dn 3 (J3—iy) 
= (/fc' 3 + £ 2 (^ 3 -B 3 )} 3 + ^A*B* 
= i' 4 + 2»' 3 (^ 3 — B^ + k^+B 2 )* 
( c -i'/ 2 )V+i < v' 2 + |) 
= (c 2 - Jcc + A- 2 ) 2 
Therefore— (Durege, Elliptische Etmctionen, p. 286)— 
cn 2/3=cn (/3-My-f/3 —^’y), 
_ cn (/3 + iy) cn (J3—iy )—sn (J3 + *y) sn (j3—iy) dn (/3 -f- iy) dn (fi—iy) 
1 —k 2 sn 2 (/3 + iy) sn 2 ((3—iy) 
(ft-kc+k*) {# + k'c+k'*) -V6(c 3 -„4v 2 ) 
(c 2 -fc + F) 3 -6F(c 3 +i CN /2+i) 
”E O 
= 1^-, after reduction, 
c + /c 
= cn a. 
Similarly 
cn2iy= 
(c 3 -fe+F) (c 2 + £'<;4*' 2 ) 4v% 8 - i^/2) 
( c 2_fc + ^7_ 6F (c 2 4 i (V2+i) 
and, therefore, dividing numerator and denominator by the common 
factor c + ic } 
cn (2y, V) = 
c 3 + l c V 2 -I c V 6 + i c -^ 2 , 
c 3 + | cV 2 + | cV6+ 1 c | V 2 
1 — cn(2y, k') __ 4 
1 + cn(2y, k') c 3 + 3 c y 2 + l c _^2 
