then, letting 



x = - %, x n = H, AX'(- JO = X'{- %), AZ'00 = - X'(%), (60) 



the first terms in the expansions for C(f\) and S (/A) become 



} ' AX'(x k ) sin/Xz* and — Y^ AX'(z fc ) cos/Xx*. (61) 



A ^ /x ^J 



The effect of corners in Y(y) will not show up until another partial integration is 

 performed and thus their effect on the wave resistance will be of a higher order. The 

 term Y( — H/L) in the expansion of W(f\ 2 ) will ordinarily be zero, but would 

 presumably be kept if one were dealing with a sharply cut off strut. Forming P 2 + Q 2 

 for the case where X is smooth and Y( — H/L) = 0, one finds 



(62) 

 I X"G0 + X' 2 (- JO - 2X'(J0X'(- JO cos/X + ( - 



/ 4 x 6 / \/x 



Then the resistance coefficient C l0 = ^/^ pc 2 £ 2 is 



8 F 2 (0) ( Z" 00 1 



C = \ [X' 2 O0 + X' 2 (- JO] / d\ -. (63) 



tt P ( J i X 4 V\ 2 -1 



cos fX ) /I 



2X'O0Z'(- JO / rfX - — = > + - ). 



x 4 Vx 2 -i) \/ 3 



For large values of /, one has the asvmptotic expansion (see, e.g., Erdelyi [1], pp. 46- 

 51) 



' = = J- ,, 



cos/X J 7T 



i" X 4 Vx 2 -l 1 2/ 

 the first integral in C w has value %. Hence, and returning to the usual Froude number, 



16 



C w = — F 2 (0)[X /2 (J0 + X' 2 (- %)}F 4 



3tt (65) 



2 

 F 2 (0)X'(J0X'(- J0^ 5 cos (F- 2 + Jfcr) + 0(F-«). 



V 



In case ^(j:) has corners, the expansion becomes 

 16 ^ 



-F 2 (0) > [AX'fe)] 2 ^ 4 



3tt ^^ (66) 



/^ 5 cos (F- 2 |.t !: - av! + Ktt) + 0(F- 6 ) 



V 



2 y-, AX'(K)AX'(- K) 



^J 



7T 



The foregoing is only a caricature of Kotik's result, which does not assume an "ele- 

 mentary" ship and carries the expansion much further. 



122 



