sections S 1 and S 2 are at a distance d between centers. Treating £ as a single ship, one 

 may compute, say, 



P{\;d) = // dxdy $ lx (x,y)e" r -v cos vXx + // dxdy $zx(x,y)e vX2 v cos ^Xz (81) 



S 2 



= j f dxdy tix{x,y)e v>fv cos v\x + // dxdy ^(z — d,y)e vX2y cos ^X(a; — d), 



where in the second integral we have shifted to local coordinates centered in 5 2 . 

 Expansion of cos vX(x-d) leads to the following: 



P(\;d) = P^X) + P 2 (X) cos j/Xd - Q 2 (X) sin w2X. (82) 



Similarly, 



Q(X;d) = Qi(X) + Q 2 (X) cos ?Xd + Q 2 (X) sin vd\. (83) 



The subscripts refer, of course, to the individual ship. Substitution in the resistance 

 integral leads to 



4p£ 2 /•- X 2 



R = Ri + #2 + 2 / dX , (84) 



7TC 2 Ji VX 2 - 1 



■ [(PiPa + Q1Q2) cos ^X - (P X Q 2 - QxP 2 ) sin w&] = #! + #,+ Z2 int (d). 

 If the ships are identical, PjQ 2 — QJ** = and 



W /"- X 2 



P = 2Pi + 2 / dX [Pi 2 + Qi 2 ] cos w*X < 4Px. (85) 



7TC 2 Ji VX 2 - 1 



It follows from theorems on Fourier transforms that R int —> as d — > 00. However, 

 one can also find the asymptotic expansion for large d: 



■ipg 2 \2irv 

 R-Ud) « - - V— cos (vd + ^)[Pi 2 (l) + QAl)]. (86) 



7Tf: 2 * d 



Thus, at some separations d, the total resistance is less than that of the ships separately. 

 A similar expansion holds for the case when the ships are not identical. At the other 

 extreme, d = 0, the expression for R verifies the quadratic dependence on beam (a 

 statement which has more mathematical than physical meaning). Kostyukov [1] has 

 also treated the problem of a caravan of n ships, but without the above asymptotic 

 expansion. 



The case of two ships moving abreast, or of one parallel to a wall, is somewhat 

 more complicated. The boundary condition which should be fulfilled on the center- 

 plane sections of each ship is, according to the linearized theory, that <p n = qzc£ is , 

 i z=. 1, 2. However, a source distribution of density — c^ ix /2-w over S i will not satisfy 

 this exactly but will approximate it at small values of the ratio beam/separation (which 

 can presumably be used as a further expansion parameter). The same difficulty does 

 not arise in the case of a ship moving down the center of a deep canal. Both problems 

 are treated by Lunde [1] and the details will not be repeated here. The second one was 

 considered first by Sretenskii [2 and 3. chap. II]. Numerical computations are scanty. 

 However, Sretenskii has computed the resistance for a vertical strut of cross-section 

 £(*) ■=. 5(1 — x 2 /L 2 ), \x\ ^L, for a number of values of gb/c 2 and 2L/b, b the canal 

 width; he gives both tables and a graph of TrR/64pgB 2 L. For the case of the strut 

 Keldysh and Sedov [1] obtain the theoretically interesting result 



limP = 2 P gA 2 /b, (87) 



where A is the cross-sectional area. 



127 



