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§ 4. The first analyser on this principle was made for me last summer by Mr. James Hicks. Though suffering from sundry defects (due entirely to my own design) it proved a really useful and workable analyser, but required too much care and patience in use. The construction has now been entirely revised. The present instrument, designed by Mr. Horace Darwin and made by the Cambridge Scientific Instrument Company, is the final outcome. For several suggestions I am indebted to Professor Karl Pearson.
The ruler X X of fig. 1 is a rolling parallel ruler with a rack cut along its front edge (fig. 2). The weight of the
rack is counterbalanced by a block projecting from the back of the rule. Normally this block swings just clear of the paper, but it may be held down when one wants to keep the ruler still.
Corresponding to the disk of fig. 1 we have a series of toothed wheels; the number of teeth in the successive sizes. being 240, 120, 80, and so on. Four of these disks have actually been made; they would probably be workable up to the sixth. The analyser is intended to work to a base-length of 30 centim.; the rack being cut 8 teeth to a centimetre. The ruler itself is made longer for the sake of stability. The largest disk is the simplest. It is a flat disk of brass with teeth cut round the edge. Three windows are through it. One, in the centre, is glazed, and the centre of
the glass is marked (on the side next the paper) with a black dot, the "tracing dot." The two remaining windows are provided with reference marks that give with the centre dot a base-line for setting the disk in any desired position. A small conical hole is made in the top of the disk, on a radius perpendicular to its base-line at a distance 10/π centimetres from the centre. This hole serves to receive the tracingpoint of an ordinary Amsler planimeter, which performs the integration. The smaller disks are built up in three layersa flat bottom plate, the toothed wheel, and a projecting crank, in the top of which is the planimeter hole. This hole is outside the circumference of the toothed wheel, so a crank or some such device is necessary. The crank swings clear above the rack when the wheel is in gear. The windows are arranged as in the first disk.
§ 5. The ordinary pattern Amsler planimeter with an arm about 16 centim. long does very well, but the tracer must be made vertically adjustable. The alteration is easily made without risk of damage. The planimeter remains of course available for its ordinary purposes, and for the determination of the absolute term.
The travel of the ruler is limited in two ways: first by the "reach" of the planimeter, secondly by the risk of running the reading wheel into the rack. Curves to be analysed must consequently be drawn to a moderate scale, but the permissible magnitude varies with the type of curve. If the type be anything like (1) of fig. 3 (sine type), 10 centim.
amplitudes can be taken in comfortably; but if the type be that of (2) in the same figure, a considerably smaller scale
must be used. In cases of physical curves (e. g. E.M.F. curves of alternators, conduction-of-heat curves) one is generally free to choose the limits of the period so as to bring the curve to the desired type.
The accuracy of the instrument will be measured by the accuracy of the planimeter. This cannot be fairly stated in percentages, as an error of unity in the vernier reading is never difficult, and may be anything per cent. in a small total. I strongly recommend drawing curves on cardboard; it is much more favourable to the planimeter than drawingpaper. The following tests may be taken as typical of the results that are obtained with care: the curves were drawn on card :
(1) Actual curve,
3.13+4.60 cos 0+ 1.82 cos 20 +39 cos 30+045 cos 40. Analyser,
3.14+4.58 cos 0+1.84 cos 20 +39 cos 30+042 cos 40. (2) Actual curve (sloping straight line),
6.37 sin 0-3 18 sin 20 +212 sin 30-1.59 sin 40.
6.39 sin 0-3·20 sin 20 + 2.11 sin 30-1.58 sin 40. The units are centimetres.
§ 6. So much for the accuracy and range of the instrument. To get any desired coefficient, the ruler is set with its edge parallel to the curve-base and with the proper disk in gear. Ruler and disk are then adjusted till the tracing-dot stands over the point P (fig. 1), and the base-line of the disk is either vertical or horizontal according as a sine or cosine term is wanted. The planimeter-point is finally dropped into the hole provided for it, and the tracing-dot carried completely round the curve. The resulting planimeter reading will be the area of the curve plus or minus 10 n times the desired coefficient.
Both hands must be used in guiding the disk, as rack and disk have to be held together while the latter is turned. The operator forms, in fact, an essential link of our mechanism which without him is unconstrained. It is this liberal use of the operator that enables me to dispense with slides, carriages, and other expensive things, and thus gain in simplicity.
§ 7. This analyser arose from a simple form of step-by-step integrator or "adder" which may be worth a brief description. The instrument is shown diagrammatically in fig. 4 : it was made from materials at hand and I describe it as made. ABCD is a square sheet of card with a foot-rule glued along one edge AB. A set square FE G can be slid up and down
along this rule of sines S S.
it is provided with a tracer T and some scales The whole sheet of card is guided parallel to
the axis of a by a T-square D AR clamped to the drawingboard if the card is pulled forward by the corner A the friction keeps it set against the square. OP is a planimeter with its pole fixed to the card and its pointer P resting on one of the scales of sines, which must stand parallel to the axis of x. Mark off along the base of the curve to be analysed a number of equal divisions, e. g. 6° each, and erect the ordinates 71, 72, &c. at the centres of each of these elements. Suppose the pointer P of the planimeter to rest initially at the zero of the scale, and the tracer T attached to the set square to stand over the origin of the curve. Pull the card forward, carry T to the top of y1, and then shift P to 6° on the scale. Pull the card forward again, carry T to the top of y2, and then shift P to 12°. Continue this procedure right on to the end of the curve.
P will then have come back to its starting-point on the scale, after describing on the card a certain curve or stepped polygon. The area of the polygon is
71 (sin 60- sin 0°),
+ y2 (sin 12°- sin 6°),
+y3 (sin 18°- sin 12°),
+34 (sin 360°- sin 354°),
Phil. Mag. S. 5. Vol. 39. No. 239. April 1895.
a quantity which (as the elements are small) approximates to
Thus the planimeter-reading after this procedure gives us the first cosine coefficient. I need not enter at length into the mode of getting the others. For the second coefficient one would have to shift the tracer P 12° at a time instead of 6o, and so on.
In my case I actually used steps of 6°, as above; and there were three scales on EFG going by steps of 6°, 120, and 18° respectively for the first three terms, the separation of the scales helping to avoid confusion. The results were good: for example, in one test the actual coefficients were 4.82, 1·09, 009: the instrument gave 4.86, 1.08, 01. The chief objection to such a non-automatic integrator is of course that one is liable to forget to shift the planimeter pointer at some stage of the proceedings. The chief advantage is that your curve need not be drawn to any particular base-length. Whatever the base, it is only necessary to divide it into the proper number of equal parts, and erect the ordinates at the centres of these elements.
Suppose we wished to make the arrangement automatic. We might substitute for the harmonic motion of P along SS a circular motion round the centre of S S. This merely amounts to giving P another harmonic motion (perpendicular to SS), a proceeding which adds nothing to the planimeter-reading if the integration be continued completely round the curve.
But this is not an easy motion to obtain mechanically. The difficulty is obviated at once if we remove the card ABCD altogether and fix the pole of the planimeter in the drawingboard. If we give P the same circular motion as before we have the "disk" analyser, which I described in the first part of this paper. The area of the curve analysed is added to the integral given by the planimeter, but that is all.
Evidently in the instrument of fig. 4 we have only taken a special case in making the scales scales of sines. We might have used scales graduated proportionally to " and got moments. Any other integral could be obtained approximately if a proper scale could be drawn.