Landweher 



Another consideration is that the streamline pattern on a ship 

 form is a function of four parameters, the Froude number, the 

 Reynolds number, the trim angle and the draft-length ratio. Thus, 

 if streamline coordinate were to be used, it would be necessary to 

 calculate a great many coordinate systems. It appears to be more 

 practical to select a unique coordinate system which depends only upon 

 the geometry of hull and is independent of the above four parameters. 



If it sufficed to study thin boundary layers, there would be a 

 free choice of orthogonal surface coordinates on the hull surface. 

 But the boundary layer near the stern cannot be considered thin, and 

 a continuation of the boundaiy- layer calculations into this region 

 could not be undertaken with the equations for an orthogonal coordi- 

 nate system unless the surface coordinates had been selected to be 

 lines of principal curvature. 



VII, DETERMINATION OF LINES OF PRINCIPAL CURVATURE 



First suppose that the equation of the surface S is given by 



F(x,y,z) = (1) 



where (x,y,z) are the rectangular Cartesian coordinates of a point 

 P on S» Let ds = i dx + j dy + k dz denote a vector_elernent of arc 

 along one of the lines of principal curvature, where i , j, k are unit 

 vectors along the x,y,z axes. Then 



grad F = VF = Tf^ + JF^ +kF^ (2) 



is a vector along the normal at P and 



dVF = di . VVF (3) 



Is the change in this vector along the normal in moving an increment 

 ds from P to P' along a line of principal curvature. It can be 

 shown [ 29] that the normals to S at P and P' intersect if and only 

 if "Js is an element of arc of a line of principal curvature. This 

 Implies that the vectors 



ds, VF and ds • WF 



are coplanar, and hence that 



di . VF X (di • VVF) = , (4) 



458 



