L 



374 REroKT — 1880. 



quadrant to the opposite side or bottom of the ship and 'would be made 

 to cover one of the divisions. The observed length of this tape being r, 



he integral / -)-r/J wonld give the area. The •work might be still 



further shortened by graduating the tape according to r^ instead of by 



equal divisions. There would then remain nothing but the quadrature. 



This method might be conveniently applied geometrically in the case 

 -of curves having two axes of symmetry, like the ellipse, to which parabolic 

 quadrature is not ap|3licable. This is, however, only one of a very great 

 number of the possible transformations of the independent variable. 



Length of a curve. — Draw a chord between its extreme points, divide 

 the chord into equal parts and draw ordinates at the points of division, 

 .•at right angles to the chord. Draw tangents to the curve where these 

 'ordinates cut it, and let these tangents be produced both w^ays to meet 

 the ordinates at the extremities. Use the lengths of these tangents as 

 ordinates, and integrate them by any method of quadrature, dividing by 

 the number of ordinates. The result will be the length of the curve.* 



Curved Surface. — The only general method of dealing with this is first 

 to construct, and then to integrate between the requisite limits, sec ^ clx dij, 

 where <j> is the angle between the tangent plane and the base, or jjlane of 

 '(x,y). The construction of the term sec <j) for any given point is easy 

 enough, since this is simply the through diagonal of a parallelopipcd, of 

 which the base is given, and the directions of the diagonals of faces are 

 ailso given. The number of ordinates, however, for which this calculation 

 has to be made is large, being in two dimensions, and there is then a 

 sSonble integration to be performed. Moreover the limits are not neces- 

 sarily or usually constant, and then again the methods fail where the 

 surface is parallel to au ordinate, and in either of these cases the surface 

 ihas to be specially cut up, presenting in reality several different pieces of 

 ■work. All this renders it a very laborious task, and unfortunately the 

 integral for the surface does not present any such reductions, when treated 

 by ordinates, as the volume-integral. 



In iron shipbuilding, when the work is complete, and a separate 

 account is taken of every plate, the weight of skin and area of the surface 

 are of course mere matters of addition. But while the design is in draft, 

 it sometimes becomes necessary to estimate the surface in a more summary 

 5nanner. The usual mode is to obtain the lengths of all the level lines and 

 ■transverse sections of the surface, and then to expand the surface on th.e 

 flat by means of two sets of strips of paper, which secure equal lengths 

 for. the sides of the quadrilaterals, the angles being allowed to take iip 

 their own adjustment. This is a very coarse representation, even sup- 

 posing the expansion to be split where the distortion is great, as it usually 

 is where the skin of a ship meets the sternpost. This process is occa- 

 sionally modified by using an orthogonal network on the surface, instead 

 of orthogonally dividing the plan, and that is probably a little better. 

 Another more accurate plan has been given by Mr. Crossland f of the 

 Admiralty. A model of the ship is usually more convenient to work from 



* This method is given by Rankine in his Mnlcs and Tallies, p. 75. It is nothing 

 more than the grapliical quadrature f sec (p . dx. 



t fc'ee the Animal of the Boyal School of Xaval Architecture ("for 1873) pp. 12-Ii. 



