364 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
numerical data are available ; the effect is theoretically second order. The term in 
1— x is obviously negligible. Omitting these terms, the equation can be reduced to 
the form 
where 
u'+au — —gy cos 6 ly 
a = {Rfo + flw, 0<S)} (0 < 0 < 1). 
m* dv 1 
As a rough approximation we may assume that It = kv 2 , so that a = — 2v\/v 1 . We 
then find that 
d u yq cos 6 
dtv?~ v 2 ‘ 
In the case of the complementary function, y consists of a constant term less than 
2 x 1CT 3 , and periodic terms whose period is of order l/Q. 
The former makes a contribution tou/v 1 which is still less than 10 -3 after 20 seconds. 
The latter contributesf a term of order yg/v jQ which is always less than 3 x 10~ 6 . 
In the case of the particular integral y is 0 (l/Q 2 ), see § 4.2. Hence in all cases we 
are justified in putting u = 0, v = v 1} so long as the equations of type a hold at all, 
with the proviso that this conclusion may be at fault if the k of § 4.22 is numerically 
large. 
In reducing equations (3.212) and (3.213) we put x = 1 , cos § = 1 . This amounts 
to neglecting 1—x, 1— cos 8 compared to 1, and is obviously justifiable. We omit 
altogether from (3.212) the terms xd\ + (g/v) cos Q u or — g cos (x/vx—l/v). This 
term is excessively small, but could be retained, if desired. Finally we omit the 
terms in y cos 0 U justifying the omission by the arguments used above for the same 
term in the equation for u. 
§ 4.32. Errors in the Solution for the Complementary Function. —The second term 
R,! in the expansion of R in equation (3.6233) will be taken as representing the 
principal part of the error in the standard solution for the complementary function 
arising at this stage. Its value is 
E. = -iM-« (M-‘). 
where M has the value appropriate to (3.6231). For simplicity in estimating errors 
we may take only the leading term in M so that here 
M = iQV. • 
The values of s determined from the jump card trial and the data of the 50 degrees 
plane trajectory are tabulated in column 2 of Table YIIIb. The value of cr can be 
t This contribution is of the form j" f{t)e int dt, which is of the order (1/12) x (maximum of f(t)) under 
suitable restrictions on f(t), which are satisfied here. 
