138 
MOTION OE A PROJECTILE 
and therefore 
1 = 
a — 
2 + 2 a 3 + V(3 + 3a 3 ) 
_ (i + 
2 + 2 e 2 \ A} 
since 
_ V3\/(4 + « 2 ) r j , TV 
-T++T- (4 + X), 
Therefore 
p — a — 
+ 
1 - & _ «2 
1MV3 4x/3(4 + a 3 )' 
1 + a 2 1 
V3n/(4 + « 2 ) ^ + X 
giving p in terms of x. 
Integrating this equation, 
y — ax — j 
1 + a 2 
/ 
V3 \/ (4 + < 2 3 ) «/o A + X 
dx 
w* r 
aX 4yP Jq ATX’ 
an elliptic integral of the third kind, giving y in terms of x. 
If, however, a — 0, then b — 1, a = 2 K, and A = 0: and 
^_i /' x (1 — cn x) 2 
sn x dn x 
e?x 
1 — cn x 
- l X 
0 (1 + cn x) (&' 3 + ti* cn 2 x) 
1 -0 
sn x dn x dx 
dz , 
(L + 2 ) O ' 2 + & 2 * 2 ) 
putting cn x = z; and performing the integration, 
1 + « 
1 — 2 k' 2 
§ = i log - i log (*-» + W) + (gp- tan-i “ , + 0 
w 
1 + cn x \/3, t / \/ 3 —-1 \ ir 
2 0 2dn x 2 \x/3 + 1 / ^ ; 
x/3 
24 
where x = Jt/3^» ^ = sin 15°, is the equation of the trajectory when 
the projectile is fired horizontally with infinite velocity, the resistance 
varying as the cube of the velocity, and the terminal velocity being w. 
