346 MESSRS. R. H. FOWLER, E. G. GALLOP, C. N. H. LOCK AND H. W. RICHMOND: 
where J and j are new arbitrary constants; of these j is small if \r/\ is small at 
t — 0.* The motion is a combination of the following components :— 
(1) A uniform rotation about the origin, represented by the term e m . 
(2) A damping of the amplitude, represented by {a^crf e~ q \ 
(3) An oscillation of period determined by p 2 whose phase is continually changed 
by the factor {j—q 2 )- The values of $ and <p are given by the equations 
(4.031) § 2 = £J 2 (<r 0 /<r) e~ 2?1 {cosh 2 (j — q a ) - cos 2 p 2 ], 
(4.032) ^ = $ {) +p l + arc tan {coth ( j—q 2 ) tan_p 2 },t 
So long as {j — q 2 ) does not change sign, the average rate of increase of <f> over any 
number of complete periods is (p\+p' 2 ). 
Let a and /3 be the successive maximum and minimum values of S (assumed positive). 
In determining the values of a, ft and the corresponding values of t, it is legitimate 
to neglect the changes of q u q 2 , and a, which are very small in a single period p 2 . 
The maxima and minima are then given by putting cos 2 p 2 equal to — 1 and +1 
respectively in (4.031). Writing 
(4.041) aj = J (o-o/cr) 4 e~ 91 cosh ( j-q 2 ), 
(4.042) ft = J (. trja) k e~ 91 sinh ( j-q 2 ), 
so that <*!, ft are defined for all values of t, we have 
(4.051) « = «,(T,). 
for values of T n given by 
(4.052) 
(4.053) 
for values of T' n given by 
(4.054) 
P 2 { T„) = ^( 2 w+ 1 )tt, 
/3 = ift (Tft)|, 
Pa( T'„) = n-K. 
An alternative expression for <]> is then 
(4.06) 
<p — 0 O +_pj + arc tan (~ tan p 2 ) • 
\Pl J 
* The curves of 8 against t appear to have a minimum very near the muzzle of the gun in all rounds 
fired, but it will be seen that, in analysing the results, it is not necessary to assume any definite origin for 
t or 0 . 
t Here arc tan (A tan x) is determined in such a way that it changes continuously as x increases 
indefinitely. 
