372 EEPOET— 1880. 



by their sum. This, added to the length previously laid off on the first 

 ordinate, gives the area np to the third ordinate, and is set off upon 

 that, and so on. This gives the ordinates of a new curve,* which repre- 

 sents double the area of the first curve up to any given ordinate, original 

 or interpolated. The curve passes through the foot of the first ordinate, 

 because there is no area until that has been passed. If the curve is 

 incoiiveniently tall, it must be reduced by dividing all the ordinates in 

 the same ratio. 



If the parabolic method is preferred, we divide the base into an even 

 number of intervals, and draw ordinates. Join the heads of all the odd 

 ordinates by right lines cutting the even ordinates (produced if necessary) 

 and divide the portion of the even ordinates included between the curve 

 and the chord into three equal parts. Then the distance from the base 

 to the point of division nearest the curve gives the area comjirised between 

 the adjacent odd ordinates. That is to say, it is the length of a strip, 

 whose base is the double interval, and whose area is equal to that of the 

 >corresponding curved area. The length thus obtained on the second 

 ordinate, is set off on the third : the length similarly obtained on the 

 fourth ordinate is added to that on the second, and the joint length laid 

 off on the fifth. We thus obtain ordinates for a curve of areas ; only the 

 scale is one-fourth of what would be obtained by the previous process 

 ajjplied to the same curve. 



It is very important to keep an accurate account of the scale. This 

 is best written along each curve : thus 



curve of lengths, one inch representing (say) 2 feet 

 curve of areas, one inch representing „ 8 square feet 



curve of volumes, one inch representing ,, 8 cubic feet 

 reduced curve of volumes, one inch = ,, 128 cubic feet. 



The additions are best performed by setting off the lengths in suc- 

 cession on a straight-edged strip of paper. If only the total area is 

 required, the whole operation can be performed upon the strip. 



If moments are required, the fii'st thing is to construct a curve repre- 

 rsenting the moments of the ordinates. Start from the foot of the first 

 ordinate (whose moment about itself is zero), take the head of the second 

 ordinate, double the third, treble the fourth, and so on. Integrate the 

 curve thus obtained, and we get a curve of moments, any ordinates of 

 which represent the moment of the area of the original curve up to that 

 ordinate — the moment being taken about the first ordinate. t The mo- 

 anent may, of course, be taken about any other ordinate ; but the new 

 ordinates on one side of the selected ordinate must be set off below in- 

 stead of above the base. The scale may be reduced at the first operation 

 by taking the multipliers 0, n, 2n, 3n, 4<n, &c., where n is a fraction, 

 instead of using 0, 1, 2, 3, 4, &c. 



If the curve of areas be again integrated from the foremost end, the 

 complete integral represents the moment of the original curve about the 

 final ordinate. This is a consequence of the formula (easily obtained by 

 integrating by parts) 



x/y J.ii = / /y (^^ f^'^ + l^y (^^ 



'* See chapter vi. for some observations on the mode of drawing these curves. 



t The moments must be taken about an ordinate, not about the base. If moments 

 •about the base are wanted, a fresh set of ordinates must be taken parallel to the 

 hase. 



