; 
ON THE MOTION OF PROJECTILES. 235 
••• (i) 
y + Sco 2 y + 2 co 3 x = 2 mf sin 0, 
. \ x tY + 3o) 2 if — 2coh) = 0, 
x Yl + 3coV v - 2 (a 3 y = 0. 
Whence 
x Yl + 6g)V v + 9co 4 ^ 4- 4co 6 ^=4a) 4 /sin 0, 
or 
(i> 2 4- 4o> 2 ) (D 2 + a) 2 ) 2 # = 4<b 4 /sin 6. 
The solution of this equation is 
x = A 1 sin 2 cot 4 - A 2 cos 2cot -f A 3 sin oot + A i cos cot + A b t sin cot + A 6 t cos cot 
+■£#.(4). 
Cu 
For determining the six arbitrary constants the initial conditions, 
substituted in equations (1) and (2), and their differentials, give the 
following equations 
2ooA^-\- ooA 3 ~{- Aq= u\ 
80 )/!^ + coA 3 A 3 Aq — 3 uy 
'S2coA 1 + coA 3 A 5 A 6 — 5 v) 
Hence 
+ coA i = (0 (a - 4 sin O^J 
2 + cod 4 - 2 J 5 = - co (a- — t sin o'j - 
= -3co 4 sin 0^j - Ac 
CO A 
AcoA 
16coA, + coAa — 4 A 
A 1 — A 2 — A 3 — 0, 
A i = a- — 2 sin 6 ; J 5 = &> 4 sin 0^ + y; A 6 —u. 
Also we have 
^■'+3a) 2 i? 
~2o 3 ~~ 
Differentiating the value found for x and substituting, this re¬ 
duces to 
y — - ^l 4 sin cot + A b t cos <»£ — ^ 6 £sin cot. 
The equation (3) for ^ integrates by inspection, and so we have 
x = (a- 4 sin 6 ^ (cos cot + cot sin cot) + ut cos cot + vt sin cot 4 - ^ . 
y— - ^ sin 6 ^ (sin cot — cot cos cot) — sin cot+vt cos cot . 
z~cAwt-\ft 2 cos$ . 
(5) , 
(6) , 
in 
