The curves with which the naval architect has to deal, such as ship's lines, speed curves, curves derived from experiments, etc., can seldom be expressed by means of an equation. This being the case, neither their areas nor their inclinations at given points can be determined with mathematical accuracy. Areas of such curves can be satisfactorily determined by the use of mechanical integrators or by the judicious application of Simpson's Rules or the Trapezoidal Rule to ordinates measured at known intervals. With inclinations, however, the case is different. They cannot be determined mechanically. If tangents are drawn in by eye and their inclinations measured directly, large and irregular errors will occur. So far as I am aware, there is no generally accepted semi-empirical rule for determining inclinations analogous to Simpson's Rules or the Trapezoidal Rule for areas. In seeking such a rule one naturally replaces the actual curve being dealt with by an algebraic curve having so many common points with the first as to closely follow it in the neighborhood of the point at which the inclination is desired.
Consider the parabolic curve:
y = a + bx + cx2 = dx3 +mxn
Here are 2n+ 1 arbitrary constants, a,b, c, . . . m, and hence the curve represented by (1) can be made to pass through 2n + 1 successive points of the non-algebraic curve. Take the origin at the foot of the ordinate from the point at which we wish the inclination. Take n known points at equal intervals in the direction of x positive and n more in the direction of x negative. We shall have then, on substituting in succession the known values of x and y, 2n + 1 equations for the determination of the 2n + 1 arbitrary constants.
The greater the value of n, i. e. the more ordinates used on each side of the point at which we wish the inclination, the more complicated the formula and the greater the work involved in its use. There is nothing a priori to show which formula will give, with the least amount of work, the necessary and sufficient amount of accuracy. This can be determined by applying the formulae to typical curves and comparing the results they give with exact results.
In Table I is given the results of the application of each formula to a curve of sines, the unit interval between ordinates being 10°. It appears that in this case formula (A) is not sufficiently approximate, but that (B) is practically exact and as good as the more complicated formulae (C) and (D). These conclusions are confirmed by the results of the application of the formulae to other known curves.
In Table I the percentages of error resulting from the use of formulae (A) and (B) are contrasted with the percentages of error made by two accurate and careful draughtsmen, who drew the curve of sines and the tangents at the selected points and then measured the inclinations of the latter. The results speak for themselves, and show clearly why "graphic differentiation" is found to be difficult and inaccurate.
For practical use formula (B) can be put in a more convenient form. If T denote the tangent of the inclination, we have by (B).
Table II shows the application of formula (B) to a portion of a waterline of a ship.