366 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND : 
There are also a number of other terms involving c', c" and c' 2 . The terms in c' are 
very small initially and vanish at the vertex, so that they are never likely to become 
important. The other terms in c" are certainly very small provided that s is of order 
unity. Since s varies roughly inversely as the square of the velocity (i.e., 
constant), the magnitude of the terms containing s rises very rapidly in the later 
stages of the trajectory when v becomes small. The first term in > 7 , — AisO'jQ, is 
given numerically in Table VIII., where it appears how rapidly it increases as the 
velocity falls. The values of the second term as given in equation (3.632) are also 
given (Table VIIIb., column 8 ). It appears that the ratio of the second term to the 
first term is always small so long as the first term is small. This term represents the 
effect of the particular integral in altering the co-ordinates in the plane of fire. The 
third term as given above is more difficult to evaluate, and only a rough estimate 
has been made of its value at two points on the 50 degrees trajectory. The 
results are:— 
Seconds. 
Third term/first term. 
Third term. 
t = 0 
-2-02 x lO- 8 
-8-5 x 10“ 7 
t = 20 
- l-94x 10~ 2 
- 7•2 x10~ 4 
The value of the drift as estimated by the first term is therefore slightly too large. 
The first part of the third term, — £syi'\/(iQ) 3 , is of special interest, as it represents the 
sole contribution of the term >/' in equation (3.613) to the value of rj to this order. 
The term rj' represents all that remains in the equations of type a of the 
terms in B neglected in § 3.3 in obtaining the equations of type (3. The initial 
value of — £sij'J(iQ) 3 is only 3'46xl0 -5 of the first term in rj in the 50 degrees 
trajectory, and this ratio does not tend to increase as the velocity diminishes. 
This makes it likely that the equations of type (3 give an accurate solution in all cases 
when the initial conditions are those of the particular integral. 
Returning to the particular integral, we have shown that the third term is only 
0*03 (?) of the first term at the vertex of a 50 degrees trajectory where the velocity is 
as low as 500 f.s. For a trajectory at still higher elevation the minimum velocity is 
lower ; the value of the first term soon becomes too great for the use of approximations 
which neglect 1 — cos §, while the third term can no longer be neglected in comparison 
with the first term. The solution therefore fails when the elevation much exceeds 
50 degrees as soon as the velocity has fallen much below 500 f.s. The true degree of 
approximation given by the expansion can only be obtained in a special case. If the 
terms in rj' in equation (3.613), and terms of the solution containing c', &c., arising 
from the terms in £ are neglected, it may be shown that the error of the expansion at 
