424 T. H. HAVELOCK 
We now reduce the integrals to a more convenient form. The integration 
with respect to 7 can be expressed in terms of Fresnel integrals ; after some 
reduction we obtain the result 
t 
| LAr dr = p-etwt-0"(gP(pt—q) + gP(q)— He} — 
—p~etrt+9"{q P(pt+q)—qP(q)— die}, (30) 
where 2p? = ky, gq? = g/2y; 
and P(u) = O(u)—iS(u) = i e~iu? dy, 
0 
For the integration with respect to x, we change the variable from « to v, 
given by k = Agu? |-y2t2: and we obtain finally 
t co 
| 7dr i L*xte-2F dic = A, +iB, = k°g-t3 | (A-+iB)v2e-88"* dv, (31) 
0 0 0 
t ce) eo) 
| 7dr | L*«he-2f dice = Ay +i By = hg i (A +i B)v'e-86e* dv, (32) 
0 0 0 
with 
k= 2g/y, B= gflyrPs py=kv—}); Pp, = k(v+9); 
A = O(p,)cos p}+ S(p,)sin pj+ C(p2)cos p3+ S(p3)sin p+ 
+{C(ik)—k-1 sin tk?}(cos pj—cos p3)-++ 
+{S(4k)+k- cos $k?}(sin pj—sin p3); (33) 
B = C(p,)sin p}—S(p,)cos p+ C(p,)sin p3— S(p_)cos p3— 
—{S(4k)+k-1 cos +k?}(cos p?—cos p§)-+ 
+{O(4k)—k4 sin 4h?}(sin p?—sin p3). (34) 
7. For the resistance, we consider first the part R,. This could be obtained 
to the second approximation, but it was thought sufficient meantime to 
examine only the first approximation. The general effect of the second 
approximation is known in the case of uniform velocity; it consists in 
increasing the value somewhat at lower speeds and diminishing it slightly 
at higher speeds. From some rough calculations it appears that the effect 
in the present case would be similar; but for a general idea of the effect of 
acceleration upon R,, which reduces to the wave resistance for uniform 
velocity, it is sufficient to take the first approximation. From (17) and 
(28), we have 
Ry = InpygtattA, = 128ngpa2kB2(a2/f2) | Avite-88* dv, (35) 
0 
in the notation given in (33). 
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