342 MESSES E. H. FOWLEE, E. G. GALLOP, C. N. H. LOCK AND H. W. EICHMOND: 
be seen from the following considerations. The motion in a slightly different plane 
trajectory would be obtained by omitting all terms in ? ? from the equations of motion 
of the centre of gravity, and ignoring the equations of angular motion. Equation (3.612) 
then reduces to = 0; this represents a trajectory which only differs from a varied 
plane trajectory on account of terms omitted in § 3.21, whose retention renders the 
equation non-linear. The value of % in (3.627) gives the alteration, through the 
change in direction of projection, of the first term in rj. 
3.65. In the usual practical case, the initial conditions take the form 
£o = o, ri 0 = a , rj o = bQ, 
where a and b are arbitrary complex constants. It is desirable in such a case to know 
the degree of importance of the three standard solutions. 
The initial values of the standard solutions (retaining the highest order terms only) 
are as follows :— 
m=l, £ = 0(l/Q), 
I2=h & = 0 ( 1 / 0 ), 
% =0(1/Q), &=1, 
i = 0 ( 1 / 0 ), 1=0, 
11 = 2^ ( 1 + °), 
12 — (l —v), 
iz = 0 (l/Q), 
M = 0 (i/o). 
The constants K l5 K 2 , K 3 are determined by the equations 
+ = a, 
K-i^r + K-2>?2' + + n — bQ, 
(3.6501) 
Ki^i + Kg^ + Kg^-j-^ — 0. 
Retaining 
only the highest order terms these reduce to 
(3.651) 
Kj + K 2 = a, 
(3.652) 
^i(l+a) Kj +%i (l o') k a = b, 
(3.653) 
K 3 + 0(l/Q) = 0. 
It follows at once that K 3 % is completely negligible compared to and K 2 >/ 2 , and 
that in investigating »; we may ignore the third solution (and the particular integral) 
altogether. On the other hand, the contributions of all the solutions to f are of the 
same order of importance. We shall therefore take as the solution satisfying the 
most general initial conditions— 
(3.654) 
= K,% + K a fl SJ 
