334 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
we find that (3.401) reduces to 
(3.402) |-B jcf 2 + 0 /2 sin 2 <5 + 20' (<fcos (p — ^'sin 8 cos 8 sin <p) + 6' 2 (l — sin 2 8 sin 2 <p) 
[ d cos 8 • « , 1 
This is the equation corresponding to the equation of energy. The first component 
of (3.105) corresponds to the equation of angular momentum for a top, and reduces in 
the same way to 
(3.403) ~ {AN cos 8+B<p' sin 2 8} + 2B6f<f sin 2 8 sin <p — B0" sin 8 cos 8 sin <f> 
CiL 
— AN O' sin 8 cos <p + B0 /2 sin 2 8 cos <p sin cp — 0. 
Equation (3.201) remains unaltered. Over the range of the jump card experiments 
a mean value of 8'/O' is 50. We shall therefore regard it as legitimate for our present 
purposes to neglect all terms containing O'. On integrating the resulting equations 
we obtain 
(3.404) |-B (T 2 + cp' 2 sin 2 <5) + [ /xd cos 8 = TBE, 
Jo 
(3.405) AN cos ^+B<// sin 2 8 = BF, 
where E and F are constants of integration. In (3.405) it is assumed that N is 
constant. If /u is constant these equations are of the same form as those of the 
motion of a top. In the more important applications to the jump card trial which we 
shall make of (3.404) and (3.405), n will be treated as a variable function of 8, and also 
of v. 
§ 3.5. The Additional Force Components H, J and K. 
It is now necessary to consider the effect of the additional force components, 
mentioned in §§1.1, 1.12, and denoted by H, J and K. These have so far been 
omitted from the general equations for the sake of simplicity. The couples H and J 
will affect the angular motion of the axis, and the force K will affect the motion of 
the centre of gravity. For algebraic convenience we define new variables h, y, X, by 
the equations 
H = hBiv, J — ANy sin 8 , K = mNvX sin 8, 
where w is the total angular velocity of the axis of the shell. The force components 
may then be represented, in the notation of § 3.1, by the following vectors :— 
H by the vector —hB {[A . A'J —( A . ©) A + 0} ; 
J by the vector ANy (A cos 8 —X) ; 
K by the vector mN vX [A . X ]. 
