188 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
and the condition that this vanishes when r = R, and s = S, gives 
— U45Al ]+ 945^_ 4 2 0 2^A, + io5?1^ —15ii^ + ^||5 = O r 
thus determining the coefficient A 5 . It is seen easily that 5! A 5 = Q 1 (5) (2R J ). 
lllo UUUCLIillJULIllg rllG AAJclllL JLcIl u xA.5. J. U IS StJtUl f ctMlJ UJld U D i 2 A5 = 
Now when cZ/ro- = 0 the differential equation for Z shows that 
0 2 Z 0 2 Z 
Also we have in general 
a-x 2 du 2 
^=105 F "^ tl) -105 ^^ +J 5 F “'h + " l -10 F "^ +B) + ^ 
OU 2 cr 9 <t s o-' cr b 
™ = -945 F - ,1> <y») +a 4 5 P <3 h +it) -420 F '° 1 h + M) 
\ y 
ccrcll 
, lQ5 W*(<r + u) 15 W»i* + u) F/ 6) (o- + n) 
cr 7 <r 6 a 5 
and therefore when PZ/ro- = 0 we have the equation 
(T^{ 105F/ 2) -105<rF ® + ±5<t 2 F 1 w -1 O^F/ 5 ’ + ^F®} 2 
-{945F 1 (1) -945o-F ] (2) + 420<r 2 F 1 (3) -105o- 3 F 1 (<) +15 ( r 4 F 1 (5) -a- 5 F 1 (6) } 2 
+1 OH (<r 0 10 /rt){105F/ 2 ' — 105o-Fj (3) + 45cr 2 Fj (4) — 1 OtFF/'^ + eFF/ 65 } = 0 
as well as the equation 
9 4 5 F, — 9 4 5 <tF: n) + 4 2 0a 2 F' (2) -10 5 o- 3 F, (3) 
The equation expressing the condition which holds at the piston is linear in F/ 6 ’, 
and therefore can be solved for F, (i;) without ambiguity. As it holds for r = lb and 
s = Si, it determines the coefficient A, ; . The equation in question holds for all values 
of <7 and u for which the equation expressing the vanishing of cZ/ca- holds, and it can 
therefore be differentiated totally with respect to cr. u being treated as a function of <r 
in accordance with the equation cZ/oo- = 0. This process yields an equation which 
determines the coefficient A ; without ambiguity. A second differentiation yields an 
equation from which the value of the coefficient A s may be found. By proceeding in 
this way we may obtain as many of the coefficients A as we wish. 
This method of determining the coefficients A,,, A 7 , ... is not very well adapted to 
numerical computation, and other methods will be explained presently. 
20. Determination of the First Deflected Wave from the Right. —The junction of the 
reflected wave and the first middle wave is characterised by the value of s. The 
conditions determining the reflected wave are the condition which holds at the junction, 
where s = s u and the condition which holds at the piston, where = c. Further, 
x 0 must be equal to c when r = r x and s = s,. 
