NO. IB.] APPENDIX. 141 



00 



ip(a) cos (yjff) da , 



in which ip(a) is a uniform analytic function of a, regular for all values of 

 a ^> 0, and when 



ip'(a) | da 



I 



is finite ; | xp' (a) | being the modulus or "absoluter Betrag" of xp' (a). By 

 partial integration it is easily seen that such a term decreases infinitely with 

 increasing x, and that it in consequence represents a local disturbance in the 

 neighbourhood of the ridge, which causes no resistance. 



The right hand member of (2) is a holomorphic function for all finite 

 values of a; it has no zero for positive values of a if s > 1, and when s< 1 

 a simple zero for one positive value a x , only. It is then easily seen that in 



the former case — 7-r satisfies the above conditions for ip(a) , so that (1) gives 

 q>(a) 



no wave-making resistance. When s <C 1 we may write (1) in the form 



2«r, cos (j^o) 



A_ 



TtD 



(ff a — ff,V(ff,) ^-D 



./ 



ifj (a) cos f j. a J da (4) 



1 2g t 



W-pM (ff 2_ ffi2)9 , >l) . 



where <p' (a) is the first derivative of f(a). 



The function tf> {a) obviously satisfies the above conditions for the second 

 integral in (4) to make no contribution to the resistance, and this integral is 

 therefore left out. As a result of (3), the first integral in (4) gives 



h = -W(a7) sin {5' J i) for -'->° 



fo = + ^W) Si " (I ffl ) for *<°- 



When an endless series of waves which annul the waves for x < 

 are superposed, so that there are no waves ahead of the ridge, those behind 

 the ridge become doubled, and we obtain for x positive, 



D<p' (ff t 



•ZA. . ( X\ 



^) sin p D) 



