186 
MESSRS. A. F, H. LOVE AND F. B. PIDDUCK 
that the corresponding values of r, s, u are R,, S l5 U u where 
Rj — 2^"0) ^5] — 2°"u) U"j {rr 0 ■—'l)) 
and that the corresponding value of Z is Z l3 where 
7 H Jj.il_ y hi h 
A ” ~ l' 1 ’ ^ W1 6 \zj I 
a 
In like manner we see that the advancing front reaches the piston m at the time 
t u where 
, (h+h) 2 h 
ah a ’ 
and that the corresponding values of a, r, s, u, Z, are a-,, r 1} s u u u z u where 
/. yu 
ft' \ x / 1 \ 
7\ — “ 2 cr 0 , — Wi zh), — °"o °T> 
°T — \ i 
Fc + hJ 
The First Reflected Waves. 
18. Conditions determining the First Reflected Wave from the Left. —After the instant 
t — T] the formulae belonging to the progressive wave from the left cease to hold in 
the neighbourhood of x 0 = 0, and a new compound wave, the first reflected wave from 
the left, is generated there and encroaches upon the first middle wave. The junction 
is characterized by the value R, of r. The conditions determining the reflected wave 
are the condition which holds at the junction, where r = R 1; and the condition which 
holds at the piston, where x l} = 0. It is further necessary that x 0 should vanish when 
r = R l and s = Si. 
The condition which holds at the junction is that the value of Z, calculated from 
the formulae belonging to the reflected wave, should be equal to that calculated from 
the formulae belonging to the first middle wave when r = R x . The condition which 
holds at the piston is the equation of motion of the piston, viz., that at x 0 = 0 
du 
M = 
The condition that x l} should vanish when r = R : and s = S x is the condition that 
oZfdar should vanish for these values of r and s. 
To express the condition which holds at the piston in terms of Z we substitute 
p 0 (cr/cro ) 11 for jo, and 
