Deep Water Ship-Wave*. 74:1 



A /f . . f , 1 e cos 6 r 2 cos 20 e^ cos id 



_^co^ ± l^_ ( ^co^/ ± 2)^_ ud . nL | (g2); 



or 



d=^+c/ (83), 



where & and c/ denote finite and infinite series shown in (82). 

 § 45. We are going to make S = ^; and in this case e/can 

 be summed, in finite terms, as follows. First multiply each 

 term by «?"+* «-*-'; and we find 



^=-^+;y +5 [^ cos + 1)0+ |^cos (/ + 2)0 + ftc] 



= ~<S+^)e'"+*j^e-*{RS}^ +1 (l + ^+^V + &c.); 



where q denotes e 1 ^ ; and, as in § 3 above, {RSJ- denotes 

 realisation by taking half sum for +t. Summing the infinite 

 series, and performing \de 9 for the case S=^, we find 



c / = _ c0 + iy + l {RS }^logi±-^f . . . (84), 



1— sjqe 



= — c( ) + i) & +* { life } qp + * loo- - — ¥-. 2 —. y~. — : — f- 



where 



^ = tan ] — — -. -—r-^, Tlr'sstun- 1 - ; TJ (bo). 



l+vecos^fl T 1— v/^cosi^ v J ' 



and therefore 



, i/i _.2vVsinJ0 



-vir — \fr = tan * . 



7 r 1—e 



Hence finallv 



t / =cC;:+i y +} {- CM0 - +i) , log ^i±|^|± 



2^6 cos ^0 + £ 



/•. i \a A ,2*/* sin ^0 1 /0 „ 

 + sin (j+\)0 tan" 1 x —2- I , (86). 



