370 



REPORT 1880. 



If tlie interval is other than unity, accoiint must be taken of it, multi- 

 jiljing the stim for lij dx by the interval, and the sums for lanj dx and 



By- d.v by the square of the interval.* 



When the integration has to take place in two dimensions, as in cal- 

 culating the disjolacement and mechanical centres of ships, a much greater 

 saving can be effected by systematic arrangement.-j- 



It is sometimes more convenient to compute the area of a curve from 



the polar expression / — ^ ''"<-?^ ; but this needs no special remark. 



The rectification of a curve is but a particular species of quadrature, 



being either / ^/ (Jx^ + dy-) or / (si^c </j.(7,i',) where ^ is the anglebetween 



the tangent and the axis of abscisste. Its arithmetical treatment presents 

 no special feature, unless it be that the secant of ^ has to be obtained from 

 ordiiiates. This is fully treated of in Cap. IV. Sec. 6 of this report. 



The quadrature of a curved surface by means of ordinates rests on 

 a similar principle, namely, on the quadrature of the double integral 



f j sec (j) dxdy where f is the angle between the tangent plane, and the 

 plane of the base (xy). The greater part of the work turns upon the 

 determination of sec ^ = . /| ^ + (jY + (tY } ' '^^^ ^°''®' ^^® 



* See some examples of integTations differently arranged for other purposes, in 

 various papers in the Transactions of the Institution of Naval ^»r7«Yec^s— especially 

 vol. ii. p. 163, vol. V. p. 9, vol. vi. p. 51. 



I For examples of a ' displacement sheet ' see Shiplnnldinfi, Theoretical and 

 Practical, by Napier, Rankine, Barnes, and Watts (Mackenzie) p. 46 ; and T/ieoretical 

 Xai-al ArcMtecture, by Thearle (Collins) pp. 50-58 and Table I. For a similar .sheet 

 suited to the application of Woolley's rule see Trans. I.X.A., vol. ^dii. p. 213. Many 

 other interesting examples of carefully arranged integration will be found scattered 

 throw^h the Trans. I.X.A. and treatises on naval architecture, such as Scott Russell s, 

 and tlie works alreadj- quoted. 



