190 
MESSRS. A. E. II. LOVE AND F. B. PIDDUCK 
21. Relation between Pressure and Velocity at a Piston .—The equation in terms of 
cr and u, which holds for the first reflected wave from the left at x 0 = 0, is the relation 
between the pressure on the piston M and its velocity during the time that the wave 
is being generated. It may also be interpreted as the equation of a certain locus in the 
plane of r and s. This locus passes through the point (Ifi, Sj), and we may take its 
equation to be of the form 
-R, 
= B, (a-S.) + (B/£,) + (B.,/2, a ) (s- SJ’ + 
Now if the coefficients B were known, we could determine x 0 , as a function of r and s, 
from the known value, zero, of the function along the locus and the values of its differen¬ 
tial coefficients along the same curve. These differential coefficients also are known 
along the locus. To prove this and obtain formuke for these differential coefficients, 
we write X for x 0 and observe that the equations of Article 6 show that X satisfies 
the differential equation 
A /19X\ = i/F?X\ | 
dll ' 11 du) da II da 
which can be written either in the form 
ffix_ ioax_^x = ( 
ccr 1 or da du 2 
or in the form 
a 2 x 5 /ax 9 X\ _ 
dr ds r + s \ dr ds / 
Further at x 0 = 0 we have 
and 
where dajdu is to be found from the equation connecting r and s. Thus we have along 
this locus 
The equation for X is similar in form to that for Z, and may be solved by Riemann's 
method. When this is done the coefficients B in the equation of the locus may be 
determined by identifying the values of X at r = R x with those given for x 0 by the 
formulae for the first middle wave. 
