19(5 
MESSES. A. E. H. LOVE AND F. B. PIDDUOK 
25. Third Method of determining the Coefficients A. —Another nearly equally effective 
process for finding the coefficients A 3 , A 7 , ... , is founded upon an expression for t, valid 
at the piston M. 
The equation of motion of the piston shows that at x 0 = 0, 
cu _ qcr 11 
dt 10Hcr 0 10 
and the differential of u is always 
du 
dx 0 
dx 0 + ~ dt 
ot 
so that, at x 0 — 0, t can be expressed as a function of s by the equation 
10H 
cr, 
10 
a 
11 
{(B 1 -1)+2BA + 3B 3 ^+ 
and thus t — T x can be expanded in powers of S or (s — Si)/Si. Also, since t = ?Z/3w, 
and T, is the value of t given by putting r = B x and s = S x in the formula for the first 
reflected wave from the left, we have 
t - T , = ■) 
V 
V 8 
n 
+ 45 
F/ 3) (2r) F 1 (3) (2R 1 ) 
CT 
-10 
f F, M (2r) _ F, W (2R,) 1 f F, w (2r) _ F,'»(2R,) 
' "i .6 -y 5 
cr 
V B 
- 1 ! 
cr ° 
so that a different form of expansion can be obtained for t—T v By equating coefficients 
of different powers of $ in the two forms of expansion we obtain again a series of equa¬ 
tions giving the values of A 6 , A 7 , ... , successively. The results may be recorded as 
follows :— 
The equation for A 6 is 
105{-9(B 1 +l)++2B 1 ++] -105 U8(B|+l)AA+2B,qA 
+ 45+7 (B 1 +l)AA + 2B 1 fiA[- 10 {- 6 (Bi + l)i^T L ‘ +2B.51A' 
V 6 
“*1 J 
— l 5 
+ +5(B 1 + l)5i+ + 2B 1 5i+} = -Xo|^)°(B 1 -l). 
