Wave Pattern of a Doublet in a Stream. 519 
4mk, sec? § e~ f°"? sin (kya sec? 6) 
2 
+ 2 Se e"= mdm, fora <0. (10) 
0 
We have now to integrate with respect to 0, subject to the conditions in (9) 
and (10). The form of the surface is symmetrical with respect to Oz, so we 
may write down only the expressions for y positive ; and we shall put 
x=rcos0’, y=rsin &. (11) 
We find that the value of ¢ can be given by one expression, valid for0< 0’ <n, 
namely, 
ae 2M 
e+ fe 
aL 2M (e sec? 0.40 I Ky sec? 8 cos mf + m sin mf e—m7 008 (6) | an dim 
TC eae m? +- Ko% sect 0 
2 —in 
a sot sect Qe—"f"? sin {gr cos (8 — 8’) sec? 6} dO. (12) 
Cc in 
This expression is exact, apart from the usual limitation that, at the surface 
of the stream, we neglect the squares of the additional fluid velocities. 
4. The first two terms in (12) represent the local effect which is only of 
importance in the neighbourhood of the origin. A few preliminary calculations 
show that, as in the two-dimensional case, it changes from a depression to an 
elevation about the value x,f—=1. Considering the elevation at the origin, 
we have from (12) with r = 0, 
2M , 4 ir . i 
Co = of =f = =|. sec? 0d0 (, ee du, (13) 
where p = kof sec? 0 = 6 sec? 0. P 
The integration with respect to w can be expressed in terms of the logarithmic 
integral, and we obtain ane 
aa ane i {1 — pe? li(e®)} a6 |, (14) 
The integral in (14) was evaluated approximately for certain values of kof 
ranging between 1 and 2. The integrand was calculated in each case for a 
sufficient number of values of p and was then graphed on a base of 0; the 
value was found by taking values from the graph and using Simpson’s rule. 
In this way it was estimated that ¢, is zero at about «jf = 1-4, 
or ¢ = 0-84y/(gf). It was also verified that at lower speeds C, is negative, 
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