134 
and 
MOTION OF A PROJECTILE 
// 3 \ dz 
i8 s/\* + ±#)J. P - 1) VO 3 - « s ) 
3 r" t 
2 A * 3 -1 
= a h + 2 “ J'{t+w)£ p-i) vp-i 5 ) 
+ i log 
(*-l) 2 
z 3 -M+l 
— V3 tan' 
.j 3£+4 
x/3 
+ ...( 10 ) 
involving elliptic integrals of the third kind. 
Again, 
P -i dp =-^ = d i 
(a — p)z qz 
therefore | = \ f” l ~ I « V~ 
'( 3 'I 
zdz 
\4 + 4 a 2 / 
x/(.s 3 —6 3 ) 
zd-Z 
1 ) \/(.S 3 — 6 3 ) 
= IW3 - i log - V3 tan- ^ 
~ * a y(i?c)/ p -1) vp- « 3 ) ; . (u > 
also involving elliptic integrals of the third kind. 
To reduce equation (9) to the normal form of an elliptic integral 
of the first kind, put 
z — b-b^/S 1 + C0Sc fr , 
1 — COS <p 
and then f ' g dz /3 . — ■ j f ^-—r — . 2 ; 
Jz (z 3 — b 6 ) s/{bs/Z)J Q \/(l — Jr sm 2 <f>) 
where 
x/3 -1 
2s/2 
= sin 15°. 
d<fi 
f * 
Therefore / — 72 • 2 ,x 
J'o V(1 — Jr Sin 3 </>) 
-^vs,y(i ±^)5 
_ (4 + ft 2 )l (4 + 4 ft 2 )! 
31 
= x, suppose; 
and therefore, with Jacobfis notation, 
<£ = am (x, /;), 
1 + cn x 
and 
z = $ ”h b \f 3 
1 — cn x ’ 
