It is defined that a right-hand orthogonal system is fixed with uniform flow, and that the directions of reference axes Y and X are fixed with respect to the surface of the earth. Suppose that a turning path is given in the form of X=f (Y), it becomes merely a mathematical treatment to determine the values of turning factors such as the drift angle, the position of a pivot point, and so on. A turning path is usually obtained by drawing a smooth curve which connects the relative positions of a ship to the turning mark. The positions of a ship being observed intermittently, drawing the turning curve may be reduced to the calculation of the curve of regression of X on Y, or γ on θ, in which (γ, θ) is the polar representation for (Y, X). The author's interest is centered in determining the turning factors of a ship whose curvature of the path and speed are not uniform, but here in this paper he reports the results of his attempt to calculate regression curves for the ordinary turning trial data, and their uses. The author intended to obtain the regression coefficients C_m for Laurent series; γ_i=Σ^^∞__<m=-∞>C_mθ^m. From a practical point of view, he limited the range of m to |m|≦3; that is, (1) γ_i=Σ^^3__<m=0>C_mθ^m (2) γ_i=Σ^^0__<m=-3>C_mθ^m (3) γ_i=Σ^^3__<m=-3>C_mθ^m The most suitabie type of equation for the data is deceided upon by comparing the values of Q=Σ__i{γ_i-Σ__mC_mθ^m}^2. Thus, the equation whose Q is the smallest among the three is decided to be the one he seeks after.
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