round us, it is a mere fiction to suppose than an explorer is bound to avoid every field that has been explored before. My object is simply to place myself—and I may be allowed to add, to place this Society-rightly in regard to the matter. March 1856. EDWARD SANG. Description of a New Form of the Platometer, an Instrument for measuring the Areas of Plane Figures drawn on Paper. By JAMES CLERK MAXWELL, Esq., Trin. Col., Cambridge. 1. The measurement of the area of a plane figure on a map or plan is an operation so frequently occurring in practice, that any method by which it may be easily and quickly performed is deserving of attention. A very able exposition of the principle of such instruments will be found in the article on Planimeters in the Reports of the Juries of the Great Exhibition, 1851. 2. In considering the principle of instruments of this kind, it will be most convenient to suppose the area of the figure measured by an imaginary straight line, which, by moving parallel to itself, and at the same time altering in length to suit the form of the area, accurately sweeps it out. Let AZ be a fixed vertical line, APQZ the boundary of the area, and let a variable horizontal line move parallel to itself from A to Z, so as to have its extremities, P, M, in the curve and in the fixed straight line. Now, suppose the horizontal line (which we shall call the generating line) to move from the position PM to PN, MN being some small quantity, say one inch for distinctness. During this movement, the generating line will have swept out the narrow strip of the surface, PMNQ, which exceeds P P the portion PMNp by the small triangle PQp. * Read to the Society, 22d Jan, 1855. N M N But since MN, the breadth of the strip, is one inch, the strip will contain as many square inches as PM is inches long; so that, when the generating line descends one inch, it sweeps out a number of square inches equal to the number of linear inches in its length. Therefore, if we have a machine with an index of any kind, which, while the generating line moves one inch downwards, moves forward as many degrees as the generating line is inches long, and if the generating line be alternately moved an inch and altered in length, the index will mark the number of square inches swept over during the whole operation. By the ordinary method of limits, it may be shown that, if these changes be made continuous instead of sudden, the index will still measure the area of the curve traced by the extremity of the generating line. 3. When the area is bounded by a closed curve, as ABDC, then to deter mine the area we must carry the tracing point from some point A of the curve, completely round the circumference to A again. Then, while the tracing point moves from A to C, the index will go forward and measure the number of square inches in ACRP, and, while it moves from C to D, the index will measure backwards the square inches in CRPD, so that it will now indicate the square inches in ACD. Similarly, during the other part of A B P D R the motion from D to B, and from B to D, the part DBA will be measured; so that when the tracing point returns to D, the instrument will have measured the area ACDB. It is evident that the whole area will appear positive or negative according as the tracing point is carried round in the direction ACDB or ABDC. 4. We have next to consider the various methods of communicating the required motion to the index. The first is by means of two discs, the first having a flat horizontal rough surface, turning on a vertical axis, OQ, and the second vertical, 8 P with its circumference resting on the flat surface of the first at P, so as to be driven round by the motion of the first disc. The velocity of the second disc will depend on OP, the distance of the point of contact from the centre of the first disc; so that if OP be made always equal to the generating line, the conditions of the instrument will be fulfilled. R This is accomplished by causing the index-disc to slip along the radius of the horizontal disc; so that in working the instrument, the motion of the index-disc is compounded of a rolling motion due to the rotation of the first disc, and a slipping motion due to the variation of the generating line. 5. In the instrument presented by Mr Sang to the Society, the first disc is replaced by a cone, and the action of the instrument corresponds to a mathematical valuation of the area by the use of oblique co-ordinates. As he has himself explained it very completely, it will be enough here to say, that the index-wheel has still a motion of slipping as well as of rolling. 6. Now, suppose a wheel rolling on a surface, and pressing on it with a weight of a pound; then suppose the coefficient of friction to be, it will require a force of 2 oz. at least to produce slipping at all, so that even if the resistance of the axis, &c., amounted to 1 oz., the rolling would be perfect. But if the wheel were forcibly pulled sideways, so as to slide along in the direction of the axis, then, if the friction of the axis, &c., opposed no resistance to the turning of the wheel, the rotation would still be that due to the forward motion; but if there were any resistance, however small, it would produce its effect in diminishing the amount of rotation. The case is that of a mass resting on a rough surface, which requires a great force to produce the slightest motion; but when some other force acts on it and keeps it in motion, the very smallest force is sufficient to alter that motion in direction. 7. This effect of the combination of slipping and rolling has not escaped the observation of Mr Sang, who has both measured its amount, and shown how to eliminate its effect. In the improved instrument as constructed by him, I believe that the greatest error introduced in this way does not equal the ordinary errors of measurement by the old process of triangulation. This accuracy, however, is a proof of the excellence of the workmanship, and the smoothness of the action of the instrument; for if any considerable resistance had to be overcome, it would display itself in the results. 8. Having seen and admired these instruments at the Great Exhibition in 1851, and being convinced that the combination of slipping and rolling was a drawback on the perfection of the instrument, I began to search for some arrangement by which the motion should be that of perfect rolling in every motion of which the instrument is capable. The forms of the rolling parts which I considered were 1. Two equal spheres. 2. Two spheres, the diameters being as 1 to 2. I de Of these, the first combination only suited my purpose. vised several modes of mounting the spheres so as to make the principle available. That which I adopted is borrowed, as to many details, from the instruments already constructed, so that the originality of the device may be reduced to this principle-The abolition of slipping by the use of two equal spheres. 9. The instrument (fig. 1) is mounted on a frame, which rolls on the two connected wheels, MM, and is thus constrained to travel up and down the paper, moving parallel to itself. CH is a horizontal axis, passing through two supports attached to the frame, and carrying the wheel K and the hemisphere LAP. The wheel H rolls on the plane on which the instrument travels, and communicates its motion to the hemisphere, which therefore revolves about the axis AH with a velocity proportional to that with which the instrument moves backwards or forwards. FCO is a framework (better seen in the other figures) capable of revolving about a vertical axis, Cc, being joined at C and c to the frame of the instrument. The parts CF and CO are at right angles to each other and horizontal. The part CO carries with it a ring, SOS, which turns about a vertical axis Oo. This ring supports the index-sphere Bb by the extremities of its axis Ss, just as the meridian circle carries a terrestrial globe. By this arrangement, it will be seen that the axis of the sphere is kept always horizontal, while its centre moves so as to be always at a constant distance from that of the hemisphere. This distance must be adjusted so that the spheres may always remain in contact, and the pressure at the point of contact may be regulated by means of springs or compresses at O and o acting in the direction OC, oc. In this way the rotation of the hemisphere is made to drive the index-sphere. 10. Now, let us consider the working of the instrument. Suppose the arm CE placed so as to coincide with CD, then O, the centre of the index-sphere will be in the prolongation of the axis HA. Suppose also that, when in this position, the equator bB of the index-sphere is in contact with the pole A of the hemisphere. Now, let the arch be turned into the position CE as in the figure, then the rest of the framework will be turned through an equal angle, and the index-sphere will roll on the hemisphere till it come into the position represented in the figure. Then, if there be no slipping, the arc AP-BP, and the angle ACP=BOP. Next, let the instrument be moved backwards or forwards, so as to turn the wheel Kk and the hemisphere Ll, then the index-sphere will be turned about its axis Ss by the action of the hemisphere, but the ratio of their velocities will depend on their relative positions. If we draw PQ, PR, perpendiculars from the point of contact on the two axes, then the angular motion of the index-sphere will be to that of the hemisphere, as PQ is to PR; that is, as PQ is to QC, by the equal triangles POQ, PQC; that is, as ED is to DC, by the similar triangles CQP, CDE. Therefore the ratio of the angular velocities is as ED to DC, but since DC is constant, this ratio varies as ED. We have now only to contrive some way of making ED act as the generating line, and the machine is complete (see art. 2). |