336 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
methods, but the results are of no importance as it is impossible at present to solve 
these equations unless these terms are neglected. 
§ 3.6. The Approximate Solution of Equations of Type a. 
3.61. The Nature of the Solution Required. —The system of equations (3.501), 
(3.502) are linear differential equations with respect to the time of the second order, 
the coefficients being regarded as known functions of the time t. Since these 
functions are in practice empirical and by no means simple, an exact solution is 
impossible. To simplify the discussion we write 
s = A 2 N 2 /4B/x, n x/B = Q*/&s, 
so that equations (3.501), (3.502) become 
(3.611) (,+cf)-(*'Q-A) ^ (,+<*)- •> = ; 
(3.612) ?-(K-im)r,/c = 0. 
If the terms in £, h, y be omitted from (3.611), and s, N and id are assumed constant, 
the equation reduces to that for the small oscillations of a top in the neighbourhood 
of the vertical. 
The coefficient s is the stability factor as defined in § 1.31. In order to be able to 
apply the approximations on which (3.611) and (3.612) are based, we shall find that 
it is necessary to assume that the shell is more than just stable, e.g., s> 1-1. 
We proceed to develop an approximate solution of the equations on the assumption 
that Q is large. If we ignore the dimensions of the various terms, and take the 
unit of time as 1 second, then Q is in practice greater than 100 (radians per second), 
all other terms being of the order unity. This is really equivalent to assuming that 
all the ratios k/Q, h/Q, ... , which are of no dimensions, are small. It will be found 
necessary to assume further that all derivatives with respect to the time are of order 
unity in units of 1 second, e.g., that k, k", s', Q'..., are of order unity. These 
conditions are satisfied in practice. As a result, we can say that k/Q, s'/Q... , are 
small quantities of the first order, and k/Q 2 , s"/Q 2 , 6'\/Q 2 ... , are small quantities of 
the second order. For simplicity, we shall throughout ignore dimensions, and denote 
such terms of the first order by O (l/Q), and terms of the second order by O (l/Q 2 ). # 
The arithmetical values of the various terms are investigated in detail in § 4.3 below. 
The above facts indicate the lines on which an approximate solution is to be 
sought— we require the asymptotic expansion of the solution (or its leading terms) 
for large values of the parameter Q. Methods of obtaining such expansions have 
* In practice the spin N, and therefore ti, decreases slightly along the trajectory, hut the diminution 
is not sufficient to affect the assumption that Q is large. 
