Landweber 



terms of the unit vectors jSjp* egp and np. This requires that the 

 direction cosines of Cjq, e2Q, and tIq relative to ejp, egp and np 

 be calculated for each combination of P and Q; i.e. , ^n(n + i) 

 tables of direction cosines. Furthermore, if (x , y , z ) and 

 (xq, Yq, Zq) are the coordinates of P and Q in_a Rectangular 

 Cartesian coordinate systenn with unit vectors i , j , k, then 



^PO = ^^"^Q " ''P^ ^ ^^^Q " V "^^<^Q - =^p) 



and the expression of Vp(i/rpj^ = rpg/rpQ in terms of e,p, egp and 

 np requires that n tables of^ direction cosines of the latter set of 

 vectors with respect to the i , j , k system also be obtained. These 

 direction cosines and the components of rpQ can be readily deter- 

 mined if the equations of the surface are given in the form 



x=F(|,Ti), y=G(e,ri), z = H(e,Ti). (51) 



A procedure for solving (49) and (50) by iteration is suggested 

 by the following modifications: 



f (Vq - Yp)„ X Vp (-^y e,p dSQ + up^n*. r (^p + n^) • Vp (^) dS^ 



= 4TrUT • iip (52) 



^ (Yq- Vp)nXVp(~)- e^pdSQ+v r (^p + n^) • Vp (-^) dS^ 



J 3 \^Pq/ ^S ^ PO ^ 



= 4TrUT . "igp. (53) 



For ship forms the foregoing procedure can be used to deter- 

 mine the velocity and vorticity distributions and the streamlines and 

 vortex lines on a double ship model at zero Froude number. At non- 

 zero Froude numbers, a similar pair of integral equations can be 

 derived, but these would be considerably more complicated because 

 of the contributions of the wave potential to the velocity on the body 

 surface S. 



X. CONCLUSIONS 



It has been indicated, on the basis of the limited available 

 boundary-layer data on actual ships and ship miodels, that the various 

 integral methods, with or without the small cross-flow assumption, 

 and ennploying streamline coordinates, are of dubious applicability to 

 ship forms because three additional assumptions concerning the 



468 



