271 T. H. Havelock. 
We then interchange the order of integration with regard to « and p, and 
integrate first with respect tox. The second term of (7) thus becomes 
ale —HD—1KoD 
QireMe'* | é dp, (9) 
ottpti(f—y) 
with f — y> 0. 
By a comparison with (1), we see that the real part of (9) is the velocity 
potential of a line distribution of doublets along the line y = f, extending over 
the negative half of that ine. The magnitude of the moment per unit length 
at the point (—p, f) is 2x9Me~“?, and the axis at that point makes with Ox 
an angle xp — % — 4n. 
It is necessary to retain the quantity » while manipulating the integrals, 
but we may put it zero ultimately and we have the following result :—The 
image system of the doublet M at an angle « to Ox and at depth f below the sur- 
face consists of a doublet M at the image point at height f above the surface with 
the axis making an angle x — « with Oz, together with a line distribution of 
doublets to the rear of the image point of constant line density 2x9M and with 
the axis at a distance p in the rear making a positive angle «op —« with the 
downward-drawn vertical. 
It is of interest to note how the parts of the image system contribute to the 
surface elevation. From the preceding equations we obtain 
2M (f cos « — x sin «) eneash 7 
s 2 2B (fteosia Saisie) oMet dx, (10) 
ah ef? 0 K— Ko big 
where the real part of the second term is to be taken. 
The integral in (10) is transformed by contour integration, treating x positive 
and « negative separately; when pw is made zero ultimately, the complete 
expressions are 
fab Re 2M Wyiees a—2 sin «) 4M ip m COs (nfSa)e sin (mf—«) e-™ dm, 
i a + f? m + Ko" 
forxz > 0; and 
en — MMe a Sin 4) f dreyMew* sin (xox + «) 
+ 2h | m COS 2A (mf + «) e™ dm, (11) 
m* + K 
forz <0. ‘ 
The first term in each case represents that part of the local surface disturb- 
ance due to the doublet and the isolated image doublet. The remaining terms 
268 
