6 
IV. 
We will now show the method of getting equation (7) from Euler’s 
equations. 
Refer the projectile and principal axes Ox , 0y\ Oz' (Ox is axis of 
figure, Fig. 4) to rectangular co-ordinates 0x v 0y v , 0z v of which 0x l 
coincides with the tangent, and Oz is in the vertical plane passing 
through the trajectory. 
Fig. 4. 
The intersection ON of the planes y 1 Oz' and y\Oz\, being perpendi¬ 
cular to Ox and Ox i, represents the direction of K. 
Calling the angle NOy', £, the projections of K on Ox , Oy', Oz' 
are : zero, K cos £, and K sin £. 
Therefore the equations for the rotating projectile are* 
B^ + (B-A) pr 1 = K oos £ 
( 12 ) 
B M ' = K sin£ 
We will refer the motion to axis of figure x and two rectangular axes 
y and z perpendicular to x, which do not share in the rotation round 
axis. The axis y coincides with ON ; y and z will be at each moment 
the principal axes of inertia, q' and r' are the component angular 
velocities about Oy' and Oz '. 
The rotation of a body round any axis is equivalent to three success¬ 
ive rotations round three axes not parallel to one plane. We may take 
for these axes either the principal axes cc, y , z, or ON = Oy , Ox and 0x\. 
* N. Majevski’s “ On the solution of problems of direct and vertical fire.” 
