Sec. 493 



MATHEMATICAL LINES FOR SHIPS 



187 



nel are by no means available in sufficient numbers, 

 especially in a national emergency, to draw all 

 the ship lines that need to be laid down. 



There are several other good reasons, both 

 practical and scientific. For a ship of a new type, 

 or of a novel shape, it is still a draw-and-erase 

 process, even for an experienced hand, to lay 

 down the lines of a 3-diml ship surface that will 

 have the proportion and shape characteristics 

 selected by the designer. When these charac- 

 teristics are achieved, the fairing process remains, 

 or the curvatures require to be checked, as 

 described subsequently in this chapter. Assuming 

 a perfect drawing, its dimensions, coordinates, 

 and offsets still require conversion to numbers, so 

 that artisans with rules and scales can build the 

 full-size ship. These numbers have to be "lifted" 

 from the graphic drawing but they are a natural 

 product of the mathematic method. 



The numerical values of those hull coefficients 

 and form parameters which are not used to set up 

 the mathematical equations may be calculated 

 before any mathematical lines are laid down on 

 paper. The designer may likewise calculate the 

 positions of the various centers of area and of 

 volume in which he is interested, to insure that 

 they fall in the proper places. Actually, the hull 

 parameters are selected by the hull designer 

 while the subsequent calculations and the drafting 

 work are performed by computing-machine opera- 

 tors and di'aftsmen. 



As an example of what can be done with mathe- 

 matical lines in an intensely practical case, the 

 shape of the large blisters added to the U. S. 

 battleships of the New Mexico class in the early 

 1930's was delineated by D. W. Taylor's mathe- 

 matical method, to be discussed presently. In 

 some respects this was a more difficult job than 

 laying down the lines of the whole ship in the 

 first place. 



Entirely apart from the shipbuilding aspect, 

 mathematic delineation is invaluable when pre- 

 paring the lines of a series of models in which some 

 parameter is to be varied systematically from 

 model to model. 



The development of mathematical formulas 

 and methods for representing the principal lines 

 or the surfaces of ships has been somewhat spas- 

 modic and is still far from a logical or practical 

 conclusion. A brief history is given here of the 

 outstanding events in the development, together 

 with a summary of the results achieved to date 

 (1955). ' 



49.3 Existing Mathematical Formulas for De- 

 lineating Ship Lines. The use of mathematical 

 formulas for calculating the offsets of ship lines, 

 or better, for delineating what may be called 

 mathematical ship surfaces, is not necessarily 

 tied to the mathematical calculation of resistance 

 due to wavemaking and other causes, discussed in 

 Chap. 50. To be sure, many of the calculations 

 for pressure resistance due to wavemaking have 

 been carried out for ship forms whose waterlines 

 and transverse sections could be expressed by 

 mathematical equations. These equations may, 

 however, be used for establishing the lines without 

 a subsequent attempt to calculate any element of 

 the ship resistance. 



It appears to have been in the minds of the 

 earliest workers in this field that the use of 

 mathematical equations to derive the usual 

 offsets would also serve to achieve the hull 

 proportions and parameters desired by the 

 designer and to tell him whether his volume and 

 area centers would be where he wanted them. 



"The oldest writer on forms of ships who has given an}' 

 well-defined system of laj'ing down lines was probably 

 the distinguished (Swedish) naval architect Chapman, 

 who proposed to use a system of lines composed of para- 

 boUc curves adapted to the intended size and proportions 

 of the vessel" [Thurston, R. H., "Forms of Fish and of 

 Ships," INA, 1887, Vol. 28, p. 418]. 



Chapman's work in the 1760's or 1770's was 

 followed in the early 1790's by the first recorded 

 systematic tests on models, conducted by Mark 

 Beaufoy and others. These models were geometric 

 shapes and could be said to have had geometric 

 or mathematical lines. From the 1830's to the 

 1870's mathematical curves such as the versed- 

 sine curve of diagram A of Fig. 24.G, the cycloid, 

 the trochoid, and the streamlines around a 

 Rankine stream form were proposed and actually 

 worked into the lines of ships of that day by 

 J. Scott Russell, James R. Napier, W. J. M. 

 Rankine, and others. Arcs of circles and possibly 

 of eUipses as well have been worked into ship lines 

 since time immemorial [Narbeth, J. H., INA, 

 1940, p. 147]. 



John W. Nystrom, in the 1860's, expanded 

 Chapman's use of the paraboHc trace and de- 

 veloped what he called the "Parabohc Ship- 

 building Construction." He utihzed parabolas of 

 varied order, with fractional as well as integral 

 exponents, to make up both waterlines and sec- 

 tions. His method, described in the Journal of 

 The Franklin Institute [Jul-Dec 1863, Third 



