468 T. H. Havelock. 
corner replaced by a parabolic are extending from a = x, to 7 = ag, the point 
«, lying within this range. Considering the effect of this by itself apart from 
any other changes, we see from (5) that the corresponding contribution to the 
surface elevation is now 
IC za 
cn = | Tee —~ I) ai (7) 
were Zs 
We shall use the notation 
Q: (p) = i Qo (p) dp, (8) 
Pot (p) = 1+ Ps (p) = ("Po (p) ap. (9) 
After evaluating (7) for points in advance of x3, between 25 and 3, and in the 
rear Of Xa, we find that we may express (7) in a single expression for all values 
of x, namely 
Co3 = —> (= 494 (072) + £Q, (093) — Po? (koa) + Po (kog’s)}s 
Tg” (Cgz— Lp) 
(10) 
with dz = % — x3 = — qo, 3 = & — %3 = — q’y, and with the convention that 
P, ‘1s zero for negative values of its argument, while Q, is anti-symmetrical 
So that 
QO, (= p) = — %(p). 
The expression (6) is, of course, the limiting value of (10) when 73 — ag is 
small and the points x, and x; ultimately coincide with the point 2. 
Numerical values of the functions may be calculated from their definitions 
as integrals, or from suitable series ; for example, using the expansion of Hy 
as a power series, we have 
me 4 6 
Le i eg ce a ag ica aaa 
DOP) To (Bisse topsite eae nerainG oD 
p 
—_ 7) =i “i p (P = 
Q; (p) = Po (p) + 2.3.4.5 2 e607 neers 
DoS | yeh a 
(12) 
4. The special object in view is a comparison of the relative values of (6) 
and (10). The quantity C may be either positive or negative, and z, may be 
at any point between x, and v3. But to make the problem definite in the first 
place, we suppose C negative and take 7; =, and a, <a,; thus we are 
considering a sharp-angled shoulder on the model, such as Q or R in fig. 1, 
with the smoothing out entirely to the rear of that point. This process, if 
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