218 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
reaches tlie middle particles, i.e., when the progressive wave has receded to x 0 = \c, 
we solve the equation (x 0 +h) (at+h) = (/i + ^c) 2 , giving t = 0-0014785 sec. The 
velocity of the projectile at this time is 275-4 m./sec., its displacement 21 -4 cm. The 
pressure falls from 5151 -6 kg./cm. 2 at the junction to 4598-7 kg./cm. 2 behind the projec¬ 
tile. In the first middle wave trial and error begins. At the breech u = 0 and <r 0 /o- 
is found by trial to give the correct value of t. For intermediate points we have theoreti¬ 
cally to find both u and <r by trial to make x 0 and t correct. Actually the smallness of 
u allows us to neglect powers of uja above the second, so that the pressure follows an 
approximately parabolic law. The difference of pressure in the first middle wave is 
quite small. At the breech we have 5170-9 kg./cm. 2 , an increase of only 19-3 kg./cm. 2 
over that at the junction, as against a drop of 552-9 kg./cm. 2 from the junction to the 
projectile. 
(Plate 1 , curve 4). —The first middle wave reaches the projectile at time 
t = Tj = 0-0021170 sec., when the displacement of the projectile is — X x = 42-191 cm. 
and its velocity — U x = 37175-64 cm./sec. = 371-8 m./sec. The remaining constants 
at this epoch are R x = 443,092-1, S x = 480,268-8, S x — 923,360-9. The pressure falls 
slightly from 4169-1 kg./cm. 2 at the breech to 4102-5 kg./cm. 2 at the base of the 
projectile. 
(Plate 1 , curve 5) (Articles 18-25). —The first reflected wave begins at t = T x . For 
the constants we find 
log (A 0 /V) = T99416, log (A,/^ 9 ) = 9'98722, log (- 2 ! Ajtf) = 7'66552, 
log (3! Ag/Sd) - 5-20603, log (-4! A 4 /S x 6 ) = 4*53510, log (5! A 5 /2 X 5 ) = 3'35986, 
log ( 6 ! Afi/Xd) = 2-69991, log (7! A 7 /2 X 3 ) = 1*49726, log (8 ! A 8 /S x 2 ) = 0*02177, 
log (9 ! A 9 /S x ) = 0-29583, 
log B x = 0*33341, log B, = 0-84347, log B s = 1*66011, log B 4 = 2*49429, 
log B 5 = 3-33303, log B 6 = 4-18096. 
Kj = -670-58, log (— Lj) = 3*64091. 
To find when the first reflected wave reaches the middle particles, we know that 
r = Bj along a junction with the first middle wave, and s is found by trial, from the 
formulae of the first middle wave, to give x u = \c. Knowing r and s, t is known : we 
find t = 0-002898 sec. The part of the first middle wave which still remains is treated 
as before. The pressure falls from 3316-0 kg./cm . 2 at the breech to 3304-3 kg./cm . 2 
at the junction. A long process is required to find the pressure in the first reflected 
wave. Writing </> = (r —Bi)/-i au d S = (s — S x )/— 1? at the base of the projectile 
([> = B X ()+By 2 -f.,. B, ; (k is a known function of S. We expand the formulae of 
Article 20 to give t+h/a, x 0 and (Z — K 1 — L 1 n )/~ 1 explicitly in terms of <j> and S, and 
try different values of S until t has its required value 0-002898. Then the pressure at 
