ON QUADBATURES AND INTEBPOLATION. 37 S 



The interval used in tlie graphical process is quite immaterial provided 

 careful account be kept of scale. It is impossible to pay too much atten- 

 tion to tliis point. 



It is, of course, not necessary that the original ordinates should re- 

 present lengths. They may represent areas, in which case their curve of 

 areas will represent volumes ; or they may represent pressures, in which 

 case, with a suitable interpretation of the interval, the areas will repre- 

 sent work ; or, again, the integral of a curve of temperatures may 

 represent heat. Whether it actually does so, or not, depends upon what 

 is taken for the interval. 



What all these processes effect is mere summation with suitable 

 coefficients. The processes of multiplication or division, except by a 

 small integer, are not conveniently performed in this way. So, although 

 we get out the moments graphically, we must have recourse to arith- 

 metical division, to find the position of the centres of gravity, or of 

 gyration. Similarly, if we want to set off the squares, or the cubes, of 

 the ordinates, we must use a table of squares or cubes, and set off from 

 that. 



It is best to use printed or lithographed sheets divided into squares, 

 the interval being chosen with reference to the work to bo done. In 

 English shipbuilding work, which is usually drawn on a scale of :^-inch 

 to the foot, quarter-inch squares are the most convenient. There should 

 be a thicker rule at every fifth or tenth line, to prevent mistakes in 

 counting. Any sized square will do, only if the rigiit size bo chosen, it 

 saves, at least, one set of reductions. For mere quadratures, it is not 

 necessary that the lines should be exactly at right angles, but in cross- 

 measurements it is inconvenient to have the two diagonals measuring 

 different lengths. 



The intersections of curves drawn to the same scale solve graphically 

 a number of equations, differential and other, which it would be difficult 

 to treat otherwise. There is no difficulty in changing the independent 

 variable. One of the simplest ways of doing this is by the interchange 

 of X and y, by taking a fresh set of ordinates at right angles to the old 

 ones. But as there is no restriction to rectangularity, and as we may 

 measure to an inclined or even a curvilinear base, it is obvious that the 

 range of transformation is very wide indeed. An example of the appli- 

 cation of this to the problem of rectilinear motion in a resisting medium 

 will be found in the ' Phil. Mag.' for June, 1868. Neither of these points,, 

 however, falls strictly within the scope of this report, and therefore it is 

 unnecessary to enlarge upon them. 



As regards the accuracy of these graphical methods, the work, in the 

 Royal School of Naval Architecture was generally done from drawings 

 on a scale of ^-inch to the foot. The displacement got out correctly in 

 two ways used generally to agree within about ^ percent. If it exceeded 

 ^ per cent., it used to be regarded as evidence of a blunder. As regards 

 blunders, graphical processes have the advantage of making these appa- 

 I'ent by a corner in the curves. 



Polar quadrahtre of an area. — It was pointed out to the author by the 

 late M. Normaud of Havre, that the polar quadratui'e could be very 

 rapidly applied to finding the area of transverse section of a ship's hold. 

 For this purpose the angles could be marked on a wooden quadrant held 

 vertically athwartships at the corner underneath a deck, and divided into 

 equal angular intervals, while a tape would pass from the centre of the 



