204 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
F 1 (2) (2R 2 ) 
s' 8 
^2 
F t (3) (2R 2 
"S' 7 
^2 
2! A 2 
3! A, 
/2Ri —2R. 
V 8 
2d 
\ t, 
1 4! A, 
2! 2d 
3!A S 
4! A 4 
/2Rx-2R s 
Si 7 
2d 
V s, 
2l 
+ 
2! 2d \ 2 X 
Fd 4) (2R 2 ) _ ' ti Y 14! A 4 5! A 5 /2R,! - 2R 2 
\^2‘ 
S' 6 
^2 
s' / s' b s' 5 \ v 
■ i - / 1 -^i —i \ -^i 
+ — . 
2! V \ 2 
5! A S| 
( 2Rx-2R 2 
2d 
V 2x 
6! A« 
( 2Ri — 2R 2 
V 4 
-a 
l 2 t 
7! A, 
/ 2Ri — 2R 2 
2d 
\ 2 a 
+ 
+ 
+ 
31. Transformation of the Formula for the Second Middle Wave .—In what follows 
we shall disregard coefficients A beyond A 9 ; if it were desired to include further co¬ 
efficients A some of the formulae would require modification, but there is no difficulty 
arising from the convention to stop at A 9 . The most effective transformation of the 
formula for Z in the second middle wave is found by putting for Z B the value derived 
from the first reflected wave from the right, viz. :— 
Z B = ki+li (Ri —Si) + 
T 
ccr, 
10 
+ 
« 0 + ai (2s —2si) + a 2 (2s — 2s t ) 2 + ... 
)■ = Ri, s = St 
so that the terms contributed to Z by Z B come to 
— if j^i (Fi + Sif + 105 (\ —+cioj —105 (Rj + Si) cti 
+ 45 (R! + 5i) 3 2! a 2 — 10(R 1 + s 1 ) 3 3! a 3 + (R 1 + s 1 ) 4 4! a 4 
(i a V f(* 
and then, before putting R, for s l5 or — A 0 , — A 1} ..., for a 0} a u ..., transforming tin 
terms contributed by A 0 , A,, A 2 , A 3 to the form 
i _0_\ 4 
cr dcr) 
+ l i+ 
i {A. + 2A! (r-R.) + 2 a A 2 (»— R 1 ) , +2 , A 3 (r-R,) 
3 } 
'A, 
■56 
TT^f+UO 
Wi+sJ 
f Sj+r ' 
\Ri + / 
■120 
1 si + r V + 35 / Si + ^Yl 
vRi+s-i 
v Ri + Si 1 .1 
.MAu^ij _ 21 (^)q 56 (^)+ 50 (^y +15 (A±r.j} 
+ M t±f *{_ 6 (^)h i7 (ggf -16 (^y +5 (££)} 
_ 2% (K, + s,)» f _ + 3 / Jill. Y -3 ('A+AV + (-fUV 
Ri+ i/ \Ri+sJ \Ri+si/ J_ 
