590 
MOTION OP A PROJECTILE 
tlie equation of the trajectory may be written 
,x dyi 
= 2c T 
X + A 
X-A , 
a x, 
X 2 - A* 
introducing elliptic integrals of the third kind. 
Now 
sIQ. __ Sn2x <ln 2 X SU 2 a ^ n2a 
~ ~ ~ (1—cn x) 4 " ” (l^iT^ 
(1 + cn x) ( [k ' 2 + k 2 cn 2 x) ___ (1 + cn a) {k' 2 4- h 2 cn 2 a) 
(1 — cn x) 3 (1—cn a) 3 
_ (1 — cn a) 3 (1 + cn x) (&' 2 4 k cn 2 x)—(1 + cn a) {Id 2 + Fen 2 a) (1 — cn x) 3 
(1—cn a) 3 (1—cn x) 3 
and this fraction must be split up into factors (one factor being 
obviously —cnx~) * n or( ^ er ex P ress the elliptic integrals in 
the normal form of the elliptic integral of the third kind, and finally 
to express them by Jacobfis and elliptic functions {vide Cayley, 
Durege, or Enneper's Elliptic Functions). 
Now 
where 
X 2 - A* = 
z* — 1 
12*/3 6 3 ’ 
1 — cn a k' + k cex < 
k' + k cn a 1—cn x 
and the factors of 2 8 — 1 are z— 1, z~~o>, z— w 2 , when <o and <o 2 are the 
imaginary cube roots of unity (w = — J + i |*/3, w 2 = —\—i 
And 
z 
k' + k 
k' + k cn a 
cn x — cn a 
1 — cn x * 
z 
_ c + k cn x — cn a t 
c 1—cn x 
(o (k' + ken a) + k (1 —cn a) cn x — cn Q3 + iy ) 
k' + k cn a 1—cn x 
_ ooc 4/c cnx—cn (fi + iy) 
c 1—cnx * 
cn (P + iy) = 
w (k' + k cn a)— k' (1 — cn a) 
o> ( k' + k cn a) + k (1 —cn aj 
wc — k' t 
we -J- k ’ 
Where 
