ON THE MOTION OF ELONGATED PROJECTILES. 
193 
and 
P [ Psin 0+s"r] + (P - P) [(Pcos 0 + af'r) sin 0 cos 0 
4- (P sin 04-s" t) sin 2 0] = P P sin 0 - Mgr, 
A\f^"r cos 2 fi + Cn sin p=Cn sin 0. 
Hence we derive 
iV' 4 - (P - P) (#" cos 2 04- z" sin 0 cos 0) = 0, 
y'=o, 
Pz" 4- (P - P) (#" sin 0 cos 04-s" sin 2 0) = - JP?, 
r=o, 
. •. P \x" cos 0 + z" sin 0] + (P - P) (a?" cos 0 + s" sin 0) = - JP? sin 0, 
R-P 
x" = - p^r dP? s i n /3 cos ft 
so that 
z"=- 
Psin 2 0 + Pcos 2 0 
~“PP 
Substituting in the equations of § 3 to the second power of t re¬ 
membering that y" = 0 and = 0, and avoiding when possible useless 
terms, we get:— 
°=p|t t 2 + ( p-P) 
«= p h T * 
^ U cos 0+ x"r + t 2 ^ (cos 2 0 + 6" t 2 sin 0 cos 0) 
4- ^Psin04-2"r4-|-jr 2 ^si 
sin 0 cos 0 - —j t 2 cos 20 
0 = P^t 2 + (R-P) 
( x"' \ / d" V 
U cos 04- r2 ) (sin 0 cos 0 - r 2 cos 20 J 
4 - ^ U sin 0 4 - |-j r 2 ^ (sin 2 0 - 0 " r 2 sin 0 cos 0 ) 
0 = ^4 iy r 2 cos 2 0 - Cn |y r 2 cos 2 0 
0 = P P^y r 2 cos 0 + (P Psin 0 - J/#r) ^Psin 04-s"r + |y r 2 ^ 
+ A 6" 2 r 2 4- 2 Mg r 2 4- p ~\ r 2 cos 0 ] . 
These five equations may be replaced by 
0 = Paf" 4 - (P - P) [ U6" sin 0 4 - {%"' cos 0 4 - s'" sin 0 ) cos 0 ] 
0 =Py'" 
0 = Pz'" 4 -(P-P) [- U6" cos 0 4- (#'" cos 0 4 - s'" sin 0 ) sin 0 ] 
0 = Ay\r'”-Cn6" 
o 
0 = P P [a/" cos 0 4 - z"' sin 0] 4 -A 6" 2 4- -p g &' cos 0 . 
