182 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
and at the receding front, where s = |<r 0 and o- = |o- 0 + r, we have 
\9 
'Z = -i-H % \ + S (|<r„-r). 
a l \%<r 0 + rj J a ~ 
15. Determination of the First Middle Wave .—To determine Z from these conditions 
we may have recourse to Riemann’s method, taking the curve AC to consist of 
segments of two lines AB and BC, which meet at 
the point B, where r = s — and are parallel 
C b respectively to the axes of s and r. 
We have then 
A 
CP 
Fig. 2. 
This equation is 
J AB 
(f+ 
5Z ^ 
els + j 
| Z| 
/ 3Y 
oV 
) dr 
\ ds 
r + s) 
J 
PA 
V dr 
r + s/ 
if + 
5Z \ 
\ds + 
f Z| 
(SV 
5V\ 
1 dr 
\ 0S 
r + s) 
! 
'bc 
\dr 
r + s) 
[vz] P -[vz] c -1 z(?-—)<*»+f z(A_At 
J cp \ds r + s/ Jpa \dr r + s 
dr 
or 
+ [vz] B -[vz] v -| z(v-—)*+[ 2 (!r-—)*■ = <>> 
Jab \cs r + sj Jbc \cr r + s/ 
Z(r'.s') = [VZL-CVZL + tVZt + f^zg-- Z(W ~—)dr, 
r + s/ 
bc \ or r + sj 
where /. s' are the co-ordinates of P. Also at A we have 
r = K « = s', { = 0, V = ( 1 Z=Mc + h)^U T ^- r ) -1 - (b»-s f ), 
\ r + s J a IA£<r 0 + s7 J a 
so that 
\9 
O’,, 
1 ha. 
[YZ], = i (c+h)*\. ljrj £fr - h (K -o (tedft 
At B we have 
so that 
At C we have 
aO-'-fs'NK + s') 4 Vr' + s' 
Z = 0, 
t S 2 » j 
r = A » = K i = 0. V = (hkf, Z = — 1H 2t {(i-q-,)”-1 + - (fr.-e), 
\ r +s J ci [\2°'o _ tn/ J ci 
[YZ] C = — J,H — 
__W+^Yl . H/i _ a / icr 0 + 7 A 5 
al(r / + P) 5 (K + / > ') 4 V r' + s' ) J + a ^ 0 M r' + s' 
so that 
