ON QUADEATUEES AND INTEEPOLATION. 3^0 



than drawings, although it is quite possible to get the constructioDS 

 without much difficulty from these. Nevertheless, it is a laborious and 

 nnsatisfactory process, and quite inapplicable to complex or highly curved 

 surfaces. It does, however, give a rough approximation, and, as such, is 

 found both useful and necessary by shipbuilders. 



X. — Mechanical Quadeatuees. 



ILectlfication. — The only satisfactory mechanical means of doing this is 

 by running a wheel along the curve, and observing its travel. In the 

 opisometer it is done by starting the wheel from a stop, running it along 

 the path to be measured, and then applying it to the scale of the majD or 

 diagram, and running it backwards until the stop is felt. This saves the 

 trouble of any readings, except the final one upon the scale, and it also 

 avoids all conversion of scale. It may be objected to it that it is a little 

 wanting in minute accuracy, from a small yielding at the stop giving a 

 considerable error at the edge of the wheel. This, however, can easily be 

 tested on a plain scale, and careful use of the instrument with a delicate 

 hand gives very good results. 



There is a more elaborate but very convenient form of the machine 

 sold under the curioi:s name of ' Wealemefiia.' 



hired mechanical quadrature. — If a disk revolve at uniform velocity, 

 and a friction wheel roll upon it, having its axis parallel to the plane, and 

 meeting the axis of the disk, then it is clear that the ti-avel of the friction 

 wheel will be directly pi'oportional to its distance from the centre of the 

 ■disk. If, therefore, the friction-wheel be made to slide upon the disk, so 

 that its point of contact shall be separated from the centre by a distance 

 equal to the varying ordinate of a curve, while the disk rolls along a straight 

 line base, the travel of the friction-wheel will integrate the area of the 

 curve. This is the simplest mechanical integrator there is. It is used in 

 one form as the ' continuous indicator ' in steam-engines, and in another 

 form it is used for integrating the curves of the German tide-gauges. It 

 is also used in the recording part of Moriu's dynamometer, and in Sang's 

 planimeter. 



Javies Tliomsoris integrator. — This ingenious instrument was devised 

 in order to get rid of the sliding which takes place ita the continuous 

 integrator, and in Amsler's planimeter. It consists of a plane circular 

 disk inclined at an angle of 45° to the horizon, and turning freely on 

 an axis normal to its plane. A cylinder with its axis horizontal and 

 parallel to the plane of the disk is mounted on journals in front of the 

 disk, so that they just clear one another. A smooth sphei'e is dropped into 

 the trough between the disk and the cylindei", and the machine is so 

 adjusted that the sphere can just roll over the centre of the disk. The 

 amount of rotation of the cylinder as the disk turns through a given 

 angle, will vary as the distance of the point of contact of the sphere and 

 disk from the centre of the latter. The travel of the sphere laterally is 

 obtained by means of a fork, -which is made to slide in the direction of the 

 axis of the cylinder, and which nips the sphere between two pads or 

 cushions, on which it slips easily. If now a flat templet with a straight 

 base and a curved edge opposite the base, is moved in the plane of the 

 disk and at right angles to the axis of the cylinder, so that the disk, or a 

 pinion in gear with it, rolls along the base, M'hile a pin in the fork-handle 



