ON THE MOTION OF ELONGATED PROJECTILES. 
191 
The second term may be put into the form 
- x [.Py + (R-P) sin 6 sin . w] 
+y [Px+(R - P) sin 6 cos . w] 
which by (iv) and (v) is equal to 
integrating we have 
i/H', 
dT 
— - yH= a constant L say. 
d\fs 
or 
Equation (iii) tells us that 
<j) + cos 6 =constant, n say .(vii) 
Hence our previous integral may be written 
Ayjr sin 2 0+On cos 6- yH= L .(viii) 
The final integral is the equation of energy, 
P+ Mg (z -p cos 6^j=E a const. 
Now squaring and adding (iv), (v), (vi), we get 
P‘2 [tf+yt+P] + (R - P)2 W z+2P . R- P w . w =H 2 + (. K-Mgtf 
P [P ( u 2 + v 2 + w 2 ) + (R - P) w 2 ] + R . (R - P) w 2 = H 2 + (K - Mgt ) 2 ... (ix) 
But multiplying (iv), (v), (vi), by sin 6 cos yfr, sin 6 sin \Jr and cos 0 
and adding we get 
or 
Pw + (R- P)iv=H sin 6 cos ^ + (.K - Mgt ) cos 6 
Rw=Hs\tl 6 cos + Mgt) cos 6 .(x) 
Hence by (ix) 
PR [P (u 2 +v 2 ) + Rw 2 ] + (R - P) [. H sin 6 cos \]s + K- Mgt cos 6] 2 
= R\M 2 + (K-Mgt) 2 ] 
Now the energy equation is 
(P \u 2 +v 2 ] + Rw 2 } + A (\js 2 sin 2 6 + S 2 ) + On 2 + 2 Mg cos o'j = 2P; 
(0 2 + \js 2 sin 2 6) + IMP Rg cos Q'j 
(R - P) [AT sin 6 cos yfs + (K- Mgt ) cos 0] 2 
= (2 E- On 2 ) PR-R [ff 2 + (K-Mgt) 2 ] .(xi) 
. APR 
