368 . BEPOBT— 1880. 



of facility we take for the nature of the curve connecting the points, that 

 is evidently- independent of the law by which the points are assumed, 

 and there is no law connecting the two systems of probability. Any 

 attempt to attain determinateness is therefore of necessity futile. 



This indeterminateness is experienced in practice as well as indicated 

 by theory. One of the commonest modes of ' fairing ' a curve thi-ough 

 given points is by using a flexible batten, or spline, which is pinned down 

 by lead weights to the points through which the curve is to be drawn, and 

 the pen is then drawn along the batten. Now, in practice, it is found 

 impossible to use similar battens for all curves. The batten has to be 

 weakest where the curvature is the greatest, and it is a matter of taste 

 and discrimination to select a batten with the proper taper, and to use it 

 discreetly, so as to get a reasonable and presentable result. The use of 

 moulds or curved patterns is still more a matter of eye. 



In the case of a curved surface such as that of a ship, the problem is 

 rendered somewhat more determinate by the consideration that all the 

 sections, and all their projections, must be fair curves. The two sets of 

 vertical sections, and the water sections, thus correct one another, and it 

 is not an uncommon thing to complete the ' fairing ' by means of diagonal 

 lines. Another mode, nearly equivalent, is to make a model, and to 

 work it until it is not only quite smooth, but until, when it is held up in 

 every possible light, the shadows fall evenly and fairly upon it. This 

 is quite as severe a test as the drawing. Nevertheless, in either case the 

 adjustment is not a matter of rule, but of taste and judgment. Apart 

 from the mechanical skill necessary to produce such, work, there are many 

 people whose perceptions are not sufficiently delicate to appreciate or 

 test it. 



While the problem is thus really and intrinsically indeterminate, all 

 the solutions being strictly secundum quid, instead of being general, the 

 difficulty is by no means beyond the reach of practical skill in the most 

 useful cases. A comparatively small number of sections in two dimensions 

 will enable two experienced draughtsmen to produce a couple of ships 

 which shall differ very little in size or shape when they come to he 

 built. 



It may be worth while to repeat that the indeterminateness really 

 turns upon the want of any arithmetical comparison between two inde- 

 pendent systems of variation of error, or of any analytical means of com- 

 bining them so as to give a single determinate result.* 



* On this subject see Mr. G. H. Darwin on 'Graphical Interpolation and 

 Integration,' Messenger of Mathematics, January 1877, p. 134, and the same author 

 on ' Fallible Measures of Variable Quantities,' Philos. Mag. for July 1877. In the 

 former paj^er Mr. Darwin gives a simple proof that the use of tlie trapezoidal rule 

 gives a less probable error for the area of a curve, when the ordinates are taken as 

 having each the same possible numerical error, than is given by the higher parabolic 

 rules. The arbitrariness of tliis assumption as to the law of error should not pass 

 unnoticed. 



See also a paper by Mr. Eckart in the Transactions of tJie Institution of Naval 

 Architects, vol. xiii. (1872) p. 318 and plate xv., for an example of a fair curve drawn 

 through a series of points whose positions require correction. See, further, a paper 

 by Dr. McAlister in the Quarterly Journal of Mathematics for this year (1880), on 

 the use of the Geometrical Mean for giving the most probable result. This is 

 equivalent to using the logarithms of the terms, instead of the actual terms, in the 

 equation of probability. 



