THE AERODYNAMICS OF A SPINNING SHELL. 
363 
equivalent to the equations of type y ; the errors in neglecting further terms may, 
therefore, be determined by comparing the solutions of equations (3.204), (3.205), &c., 
of type a with those of equations (3.404), (3.405), of type y (assuming p constant 
in both cases). Equations (3.7041), (3.7043), (3.7044), (3.705) can be made to give 
the following approximation to the true value of s in terms of T and a :— 
_L_ 
47T 2 /a 2 T 2 
{l — i (2s + l) a 2 }. 
This is valid so long as (*l) a is so small that a 4 may be neglected, and (2) 5 — 1 is 
positive and large compared with a 3 . Comparing this with the corresponding first 
approximation (4.113), we obtain the error in the value of s due to the neglect of the 
terms in (l— l), l ", &c., in equations (3.202), (3.203). The relative value of the error 
is given in the following table :—- 
oc 
s = 1-1. 
s = 2. 
s = 3. 
10 ° 
0-012 
0-019 
0-027 
5° 
0-0030 
0-0048 
0-0066 
2-5° 
0-0007 
0-0012 
0-0016 
In analysing the jump card trial, whenever the error from this cause is appreciable, 
the results have been corrected by determining the values of 5 from the chart 
described in § 3.7. 
It appears also from the solution of the equations of type y that when s < 1 the 
initial angular motion is still periodic, but no longer of the nature of a small oscillation, 
since the period is a function of the amplitude and tends to infinity as the initial 
disturbance tends to zero. 
In using equations (3.202), (3.203) to obtain the particular integral, the order of 
magnitude of the various terms is different. The term is now the most 
important, while n is 0(l/Q) and m is 0(1 /Q 2 ) with the notation of § 3.6. Most of 
the terms neglected are then O (l/Q 4 ) compared to the principal term, and completely 
insignificant, but B nl9\ a is O (l/O 2 ) and would affect the third term in the expansion 
for tj. Its effect however is completely negligible. 
4.31. The Equations of Motion of the Centre of Gravity .—These equations may 
be treated in a similar manner. In obtaining the complementary function, y and z are 
small compared to m and n (see equations (3.624), (3.625)), k/Q being initially less than 
0*01. As regards the differential equation for u (3.2141), the effect of neglecting 
the terms arising from the variation of It with S has been discussed in § 4.22 ; no 
