Reference / Paper · 1886
Mechanical Integrators, Including the Various Forms of Planimeters (Shaw, 1886)
Hele Shaw's survey of planimeters and integrators of every kind known in 1886, with their theory.
Reference / Paper · 1886
Hele Shaw's survey of planimeters and integrators of every kind known in 1886, with their theory.
MECHANICAL INTEGRATORS INCLUDING THE VARIOUS FORMS OF PLAN M ETERS. I Prof. HENRY S. H. S.", Reprinted from the Proceedings of the Institution of Civil Engineers. / ^47 ^ NEW YOEK D. VAN NOSTRAND, PUiiJblSHfiR, 23 MuBBAY AND 27 Wabebn 1886. mn 19C8 ^''"" " PREFACE. , Mechanical tation aids to mathematical compuhave always deservedly been regarded with interest. Aside from the labor-saving quality which most of them possess, they have a value arising from the fact that they represent thoughts of more or less complexity expressed in mechanism. They are of many kinds, and serve widely The reader will find in different purposes. this essay descriptions of many that are use- ful directly or indirectly to engineers. k Mechanical Integrators. of All measurements are made in terms some fixed unit. The method may consist of a simple comparison of the unit with the quantity to be mensured, but wiien this cannot conveniently be done, some indirect means must be employed. Indirect measurements may be made by measuring some physical effect, the magnitude of which is known to be a function of the quantity to be meas- ured of a ; as, for instance, rod or wire is when the length estimated by its Where, however, the unit in terms of which the measurement has to be made, is what is known as a derived unit, the indirect method generally consists in measuring in terms of the simple units from w^hich the weight. 6 former is derived, and performing, with the results, the necessary calculation. An example of this latter method given in obtaining the contents of an area, by taking its length and breadth is and multiiDlying them together, instead of adopting the tedious process of ascertaining, by direct comparison, bow many times the unit of area would be contained within it. Now, such calculations, even when of so simple a kind as mere multiplication, often become very inconvenient, and a large number of instruments have been designed for performing them by mechanical means. Such instruments may be divided into two classes, one in which the final result of conditions which vary in an arbitrary manner is found, such as the contents of a surface or the work of a motor, requiring a process of multiplication or addition; the other in w^hich the relation or ratio, at any instant, of two such quantities is given, such as space and time in the case of velocity, requiring at each instant a process of division. per is The object of the present pa- to deal with the theory, design and practical applications to engineering problems of the former class alone. It maybe briefly stated that very little has yet been piractically done in the use of the latter. Quite recently, Professor A. W. Harlacher, of Prague, has published an account of the instruments and methods Henneberg and Smreker, gauging the velocity of a river current, the principle of which is the same as that independently adopted by the author and others in this country. The conditions or data above referred to, from which the required result has to be calculated by instrumental means, are obtained in two ways: of Harlacher, for (1) By intermittent or separate obser- vations and measurements. (2) By the continuous motion of a machine in connection with self-record- ing apparatus. The former is the case in measuring an area of country, taking dimensions of a river or embankment section, or ob- 8 taming the forces exerted at different times by a machine or body in motion. The latter is generally given in the form of a graphic record, an important exam- ple of which is the diagram of energy or work taken from a prime mover. In both cases the result, whatever it be, whether boundary, area, volume, work, etc., can be found by calculation, but only with an approximation to the truth, depending upon the extent of the calculation. The reason of this is that the data of calculation, which are taken directly in the first case, or selected from the graphic record in the second, only represent actual conditions more or less closely, according as the number of data so taken is greater or less, and the greater the number the greater is the labor The instru- of determining the result. ments discussed in this jDaper perform such work mechanically, with the great advantages of rapidity of operation, accuracy of results, and without requiring mental effort on the part of the manipulator; and all this, moreover, to a great extent independently of the complexity of the calculation required. All the re- measurement of which will be considered, can be measured graphically. sults the have been made sepbe plotted, whether in the form of a diagr im of energy, or on If the observations arately, tliey can the plan or elevation of an area or section, and the boundary can be filled in with a tolerably close approximation to accuracy. record is, In the other case the graphic or may be, directly given. The subject, as far as the theory of the cal- culation goes, can therefore be studied with reference to such diagrams without the necessity of considering in the first case how they were obtained, and it will be convenient to do this, and afterwards to examine separately various examples of their application. Such diagrams may be drawn upon any kind of surface, and an instrument for dealing with measure- ments upon that of a sphere will be hereA plane surface may, after described. however, be employed upon which to represent all cases of any practical import- : 10 ance, and the question thus arises, What are the measurements of the nature un- der consideration which are required, and which can be obtained from either a regular or an irregular plane of figure? Such measurements are of three kinds (1) The length of its perimeter or boundary. The area of its superficial contents. some point, line, or other figure on the surface, e. g.^ its moment of area or moment of inertia about (2) (3) Its relation to a given line. All these three kinds of quantities can be ascertained by successive operations of addition. The first requires the ad- dition of elements of length, the second may be obtained by adding up successive elements in the form of strips of area, and the third by adding products obtained by multiplying such strips by some quantity, the magnitude of which depends upon the position of the other point, line, or figure in question. Tak^ ing the general case of an irregular fig- 11 ure, it is evident that absolute accuracy can only be obtained when this opera- becomes that of integration or summing up of an infinite series of indefintion itely Instruments for small quantities. performing this operation are therefore mechanical integrators.'' In all called such instruments the rolling action of two surfaces in frictional contact is em'• ployed, for this, as seen, enables will be hereafter the conditions of motion to be continuously varied in a way which could not be effected by mere trains of wheel-work, such as form the mechanism of some kinds of This fact calculating machines. necessitates something than a mere discussion of more the mathe- matical principles upon which the calcu- performed, for though the lations are action of an integrator may be absolutely correct as far as its theory of the per- formance of the calculation is concerned, yet there is always some instrumental depending upon the rolling, and also, as will be seen, of the slipping of the two surfaces in frictional contact. error 12 This error may be exceedingly small. Init of great importance to it is a matter ascertain its exact amount, and the subject will therefore be investigated at under the heading "Limits of Accuracy of Integrators," where an account will be also given of the experi- length, mental results of Professor Lorber, of Leoben; Dr. William Tinter, of Vienna, and Dr. A. Amsler, of Schaflfhausen. In it will be showu that when integrators are examined upon the this investigation mechanical principles of action, they are all found to belong to one of two clasBes. (1) In which the surfaces in question slip over each other. (2) In which only pure rolling motion of the surfaces is assumed to take place. The significance of this mode of clussification is that it not only leads to a cl&ir understanding of the nature of the results to be expected from any particular instrument, and teaches the best method of manipulating it, with regard to its position relatively to the figure to be meas. ured, but it also brings out prominently 13 tlie meolianiral principle uptui wliicb tlie has inventor sometimes, as relied it NVouM appear, unoonsciouHly, for the acruracv of tlie r«->»tiltH »\jnMt« d to be ob- UiintMl. It may be here remarked that the wime priiJcipU', by which an intej^'rator to^'ni]>!iic record, is em- from an au- j)h)yo(l to determine a result may Ih) apphed directly from the machine or bo<ly in motion, such, for into «»btain a continuous result stance, as an ordinary dynamometer or dynamo-electric machine, from which the autoj^niphic record was obtained. Thus, after discussing' the action of intejjrators for dealing' with diagnims, it will (mly be necessary to consider the mechanical detnils of the instruments for direct appli- cation to a will be Continuous machine, and this done under the head of *' Intef^rators." Coming now to the consideration of the actual measurement of the three kinds of (piantities, it will be found that the lirst 18 very simple, and, in fact, the only reas<^)n whv it is not convenient to meas- 14 ure it, by comparing it with the unit of length in the ordinary way, is its contin- uous change of direction. The only mechanical method of rectifying a curve, as it is called, that is. ob- taining its length as a right line, is by rolling a wheel along This wheel is it. connected with a suitable train of wheels for recording the total number of revolutions, and as the first wheel is contains a rolling circle either a of the unit in length or known relation the length of the curve to this unit, or boundary is given at once by the reading of the grad- uated wheels. The use of such instru- ments is very ancient, and Beckmann, in liis " History of Inventions," describes amongst various odometers, one mentioned in the Tenth Book of Vitruvius. Such an instrument is made upon a small scale for use by a draftsman, and in one form it is sometimes termed an " opisometer," in another form the chartometer, or Wealemefna it is also employed upon ; a large scale as a road or route measurer. The same principle has been em- 15 ployed in one of the latest forms of ane- mometer, in which the plane of the wheel always kept coincident with the direction of the wind, while its edge rolls in contact with the recording surface, and measures the total travel of the wind. In this class falls the device suggested in is a letter to Xature by Mr. V. Ventosa, of Madrid, for continuously obtaining the N., E., S. and VV. components of the wind, a device which was independently arrived at by the author in connection with certain mathematical principles referred to hereafter. The object of emj^loying a rolling wheel is merely to enable the direction to be changed so as always to coincide with the curve to be measured, and the jDrinciple is, therefore, that of direct unit measurement or comparison. The fore- going instrument, w^hich evidently belongs to the second class, in which only rolling motion, without slipping, is sup- posed to take place, forms, however, one kind of mechanical integrator. The measurement of the other two 16 kinds of quantities is, as in the case of the first, a problem of addition but instead of being the addition of an infinite ; number of infinitely short lines, an innumber of continually changing finite magnitudes has to be added. This is the same both for instruments required in the second kind of measurement, or *'Area and Planimeters," the third eters." to those required " Moment are sim- the case of cal- referred deal with in to culating the contents of areas finding iu Planim- From the fact that the chang- ing magnitudes pler or kind, theii' than in moments or any mathemat- ical results of an equivalent kind, the dis- cussion of the theory of the two above kinds of instruments is not the same, and will, therefore, be dealt with under two separate headings. Area Planimeters. The area of any plane figure, such as ABDE (Fig. 1), can be obtained in the following manner. lines, Take any straight OX and OY, at right angles to each 17 otber, and parallel to OY ; draw a series from each of straight lines equidistant other, dividing the figure into a number of strips, or elements of area. A series of rectangles may then be found, as shown at AB, the area of which is equal to that of the corresponding strip, so that, by adding the rectangles together, the area of the whole figure is obtained, as adopted in the common method of finding the area of an indicator diagram. The greater 18 the number of strips the more closely will the liei^-bt of the two sides of each tend to become equal to each other, and to*the^height of the corresponding rectangle. Let A. = the width of any element of area such as AB, at a distance X from 0. y= height of the corresponding '^ rectangle. Then ?/ A a; = area of the element AB, and the sum of all such elements of area is When the area of the figure AP>DE. the number of elements is increased in- definitely, this expression becomes / 2/c/ic=area of the figure ABDE, a and h being the extreme values of ic, i.e., the limits of integration. It is evident, therefore, that the area be correctly measured by an instrument in which the recording wheel or will measuring roller proportional to always turns at a rate the ordinate curve, while the y of the body from which it derives its motion moves at a uniform rate along the axis OX. 19 Area plauimeters have been classified accordinjjf to apparently different by which the operation modes of integration is performed but, since the action of them all can be explained upon the foregoing principle of adding elements of area, and, in fact, by means of the same notation, it is not surprising that such classifications are anything but satisfactory. In fact, in one important kind of planimeters, it becomes doubtful to which class they belong, or, whether they should not be placed in two or more classes. It is, without doubt, very convenient to dis; tinguish different planimeters, and, there- names which have been given them will be used but this will not denote any difference of principle, and the fore, the ; be adopted is and depends on mechanical conditions of action. In what follows, one mode of viewing the mathematical operation is adhered to throughout, and it may be stated that the object classification which will that already explained, of the author has been to make clear the principles of action of integrators, rather 20 than to obtain rigid and exhaustive demonstrations of their theory. Planimeters in which Slipping of the Measuring Roller Takes Place. From a brief account of the subject byProfessor Lorber, it appears that the first recorded idea of a planimeter is attributed Munich, who worked it This idea of Hermann's, which was pubhshed in 1814, seems to have fallen into oblivion, for in 1827 Oppikofer, of Berne, constructed a to Hermann, of out with Lammle. planimeter upon similar principles, and it was thenceforth called On the other hand, after his name. Favora gives the priority to Professor T. Gonella, of Flor- ence, who, in 1828, without any knowledge of what Hermann had done, invented and described a very similar instrument. The development of the planimeter seems to have grown out of the instrument of Oppikofer, who, in conjunction with a Swiss mechanic, Ernst, finished a plan- imeter which 1836. won a prize, in Paris, in Important improvements are due 21 22 to Wettli, of Zurich, who, with Starke, in 1849, took out a patent in Austria for the instruments now called the Wettli-Starke Later on, in England, other planimeter. inventors (Sang, Moseley) worked at the but subject, all these instruments de- pended for their action on the same principle, which is as follows Let (Fig. 2) be the plan of a disk : M rolling in contact with a straight guide PQ, which is parallel to OX, and at a distg-nce from it equal to the radius of the disk, so that the plan of the center of the latter always lies in OX. Let m be a roller upon the surface of the disk, gradu- ated and connected with wheel-work and an index, so that the distance turned through over the surface of the disk can be read in revolutions or parts of a revo- The plan of the point of contact (B) of the roller with the disk is always lution. made to coincide with that particular point on the curve which is in the line drawn OX, through C of the disk. The plane of otation of (m) which may be called the at right angles to the center 23 measuring roller, is always perpendicular to the disk M, and the plan of its axis, as shown in the figure, is always parallel to OY, so that, in following the curve, it slips backwards or forwards across the surface of the disk, in a direction parallel to OY. Suppose the disk to roll along .r, eipial to the width of the element AB. PQ for a distance :_. Then if //, = distance of B from OX. li = radius of disk M. of measuring roller m. reading of ;ij = consequent measuring roller for this ?•=: radius travel of disk. Then turned ^ a = throughdistance by a point on the linear .'' disk at the distance y, from the center. 2 7r;v/ J = linear distance turned through by a point on the circumference of ?/i / but since )n rolls on these M distances are equal. 2 nrx Therefore or, ". ?/ ^ = ~Ax, = .'AA^X2;i^; 24 but K — =^ is a constant, which, by taking 27rrll r and unity. R in suitable ratio may be made n^—y^Ax, Then that is, the reading of the roller m meas- ures that part of the area of the element above OX. If the iDoint of contact be made to fol- low round the curve continuously in one direction, then, when the portion of below OX is being AB measured, the disk is moving in the opposite direction along PQ, but, at the same time, the roller is turning in the opposite way relatively to the disk to that which it was doing before, since the point of contact is now The final result of these two below C. opposite motions is to cause the roller to turn, as at first, and so add the result given for CA to what was given for CB. If the motion of the disk ax for the width of AB be now regarded as negative, and —?/2=: distance AC also ??2=i'6ading of roller for this element of area, . 25 then by similar reasoning to that already used, and 71 = 71^ + ^'2~ (Vi +.'/o) A x — y A x- = area of element Tliis reasoning holds for any possible 2)Osition of the roller, or of the axis OX, which may be altogetheivonfcside the figure, as it practically is for the integra_ tion of the portion be found that DHE. Then it will DKH is subtracted, and DKHE is added, so as to give the required actual area DHE. Inasmuch as this reasoning is independent of the actual value of the width of the element, and as the vertical motion of the roller m has no effect in theory upon the distance rolled through bj it, therefore in the limit when ^aTbecomes infinitely small, the actual value of series of infinitely compose the narrow figure x4.BDE the which given by strips is the final reading of the roller when the traverse of the boundary is completed. The Wettli-Starke planimeter (Fig. 3) 26 acts directly upon the exception that is this it is principle, with the disk that moved according to the changes in y instead of the measuring roller, and the following is a description of the best form of this instrument: On a base-plate P (Fig. 3), are three guides SSS, along which a frame carrj^ing the vertical axis of the disk can be M moved to and fro. The disk, which is 27 made and covered with paper, one rectiHneal along the guides, and one of rotation about its axis. The motions are imparted to it by means of an arm (L), which passes through the roller-guides igg) in the frame carrying the disk, and rotates the latter by means of a German silver wire {del ) passing round a cylinder w upon its axis, and attached by the two ends to the extremities of the arm. The measuring of glass has two motions, roller (m) rests upon the surface of the disk, being carried in another frame (B), which is hinged to the base-plate. action is as follows : The the base-plate being placed in juxtaposition to the figure to be integrated, any line parallel to the guides, z.e., to the direction of rectilinear motion of the disk, may be taken as the axis OY; and line OX, drawn through the edge of the roller, perpendicular to OY, may be taken as the other axis. Then, as the pointer ^j> at the extremity the arm is made to pass round the boundary of the figure, the disk wall be turned through a distance proportional of 28 to the travel along OX, while at any in- stant the roller {m) is at the same distance from the center of the disk as the pointer is from OX. If 2/^z=CB=: mean height of element A. x' = = width of element AB. Then, by the reasoning already given? the reading of the roller which the pointer passes over the upper boundary of the element AB, is 71^ — y^Ax, and the final reading of the roller is N=area of the figure A DDE. Hansen, in 1850, still further improved this instrument, and, in conjunction with Ausfeld, introduced a different method of reading the result, and of carrying the frame, this instrument being known as the Han sen- Ausfeld planimeter. Various other instruments of the same kind were shown in the Great Exhibition of 1851, but in all, the motion of the arm carrying that is, the the pointer was " linear motion, which must be possible in every direction, is obtained by compounding two rectilinear movements, at right angles ; '' 29 30 to each other. Such instruments are therefore called " Imear planimeters." Many different forms of linear planimeters have been suggested, but the only modification of the disk and roller which will be worth while to notice is the cone and roller. Let MM', Fig. 4, be the cone cor- it responding to the disk M, and rolling on the edge of its two bases in a direction parallel to OX. Let the roller ni always be in contact with a circle on the cone, whose center B' is at a distance CB' from the apex C of the cone, such that CB' = SB = ?/=mean ordinate ment SB. where the element of ele- AB is being at that instant integrated. Adopting the same notation as before, when the cone has rolled over the surface through a distance Ace, then, whatever be the angle of its apex, the distance rolled through by the roller m is XV n,=y^^xx^^-^. 81 As might have been anticipated, the expression is the same as was obtained in the case of the disk, the hitter being a special case of the cone when the vertical angle is 180°. Thus the cone may be employed instead of the disk, and such an instrument was invented by Mr. E. Sang, who, in 1852, published a description of it, ac- cording to which the action was extremely accurate, but it does not appear to have come into very extensive use. No more instruments of the kind will be described, since they have given place to those in which the arm carrying the pointer turns about a center or pole, and which are, therefore, called "polar planimeters." In the year 1856, Professor AmslerLaflbn invented and brought before the world the now well-known polar planimeter bearing his name, and, since then, no less than twelve thousand four hundred of these instruments have been made and stnt out from his works at Schaffhausen. According to authorities, 32 which Professor Lorber quotes, Professor Miller, of LeobeD, invented independently a planimeter of this kind in the same year (1856), which, being made by Starke, of Vienna, is known as the Miller-Starke planimeter. Previous to this, in 1854, Decher, of Augsberg, as well as Bouniakovsky, of St. Petersburg (1855), had improved upon previously-existing forms of polar planimeter, though it is well to note that the planimeter s already mentioned as sent to the Great Exhibition of 1851 from various parts of Europe, as Italy, Switzerland, France, and Prussia, were all linear, and no mention is made of polar planimeters in the jurors' re- port. The Amsler planimeter Fig. 5. It consists of is shown in two bars, (a) the radius bar, and (b) the pole-arm, jointed at the point C. The tracing point p, which now coincides with the point B of the figure ABDE, is carried round the curve, and the roller m, which partly rolls and partly slips, gives the area of the and by means of the graduated figure ; 33 m V : : 34 dial h, and the vernier y, in connection with the roller m, the result is given corcan The sleeve rectly in four figures. H be placed in different positions along the pole-arm b, and fixed by a screw s, so as to give readings in different required units. A weight at W is placed upon the bar to k'eep the needle-point in its place, but in instruments by some other makers T is a pivot in a much larger weight, which rests on the paper. A recent minor improvement has been to fix a locking spring to the frame, so that the roller can be held when the planimeter is raised for the purjDose of reading it. The theory of the polar planimeter may be simply deduced from that of the disk and roller thus Let Fig. 6 represent the same disposition of the disk figure M with regard to the ABDE, as in Fig. 2. but now let the roller m move round the edge of the disk instead of across from it, its distance OX being always the same as be- fore, viz. 0^ = SB=y^. 35 36 The turning of m ioY a given travel, A aj, of the disk is found thus — draw Iq (Fig 6a) tangent to the disk at m^ so that lq=z A a:, and draw Ik parallel to the axis of rotation of m^ then qk is the distance tui-ned through, and Ik is that slipped through by the edge of the roller m^ when the disk has rolled through a distance A x therefore, using the same notation as be; fore, qkz:z^7rrn^ and i-= Iq '- A £C = ^m^qlk; ^1 ' but in Fig. 6 ?^=.^::.sin OS^. But by similar triangles Therefore leretore or AX '=—, E' n^=y^/\xx 27rrK This is the same result as previously obtained, and it has been given in this way because there is an important class of planimeters to be hereafter described. 37 combining the polar planimeter with the disk and roller, in which a principle is employed which is thus made obvious. Tliis principle is that the turning of exactly the same as if it at the point B, no matter in I tion it m is were in contact what posi- may be along the line through B parallel to OX. The turning of m thus measures the area of the element as long If, however, the as y does not change. value of // changes so that ')n changes its distance from OX, the measuring roller is likewise turned a certain additional amount from this cause but this does not affect the correct reading of the area so long as its first and last positions The reaare equally distant from OX. ; 38 son is, that then the roller has turned as. much in one direction in moving away from OX as it has in moving towards it, and this is the case for the initial and the pointer when the complete travel of the closed curve has final positions of been made. Now, inasmuch as the ve- locity of the edge of the disk is just the velocity at which the center has been shown to move along OX, the disk may The roller is altogether. be removed then moved in contact with the surface and with identically the same amount of turning as before, provided that its plane of rotation is turned so as to make the same angle with OX of the figure (which is now its direction of tion), as it did of transla- before with the direction motion of the edge of the disk in This is the case when it contact with it. is turned through that angle, and then its axis of revolution coincides with the radius S^'. In order to keep the direction of the plane of rotation always at right angles, it is only necessary to have a rod or bar 39 / Vm^ / /<!' 40 S^ capable of turning on a pin at the point S. The pin at S is attached to a sleeve, which can freely move along a guide-bar, whose direction coincides with By employing the bar qSq^ the axis OX. itself as the axis of rotation on which ??i simple turns, the Pig. 7 is planimeter shown in in which the point measuring roller is around the diagram. obtained, of contact of the made to pass The turning of the 771 correctly gives the superficial contents. The roller roller can be moved to any position on the rod, such as shown in dotted lines, Fig. 7, without in any way affecting its resultant turning, and the former point of contact of the roller is replaced by a pointer, which is made to follow the curve instead Professor Burkett Webb has described to the author a planimeter of the roller. of this form used in the United States, known as " Coffin's'' planimeter. If, finally, the point S, instead of moving along the straight hne OX, which may be considered as a portion of a circle with its center infinitely distant, 41 moves along the arc SZ (Fig 7), or any other arc, as, for instance, that with radius TC, the instrument becomes the or- dinary Arasler planimeter (shown in dot- This explanation, so far, is based upon that given by Sir Frederick Bramwell, who has further shown that ted Unes). the change from the motion in a straight path to that in the arc of a circle has no effect upon the ultimate reading when the complete travel around the closed curve has been made, and the arm SB has returned to initial position. The following demonstration of the truth of this appears, however, to have an advantage, in that it follows throughout the operation of integration, especially as recent plan- imeters are more complicated than that of Amsler. 1st. Let point S (Fig. 8), tremity of the radius rod, the broken line at the ex- move along STUVWXYZ, and from these points draw arcs with a radius =:K, cutting the t^t.^, ii^u^, t\u„, common curve in points &c., s and z being tangent points at the end of the curve from the 42 points S and Z. The proof has ah'eady been given that the integration of the complete portion stj,^, taken separately, is given by the reading of the measuring so also are given the areas of the various other portions, t^u^y^t^, &c. If roller the ; separate portions were integrated consecutively, any arc, such as t^t^, would be traversed in both directions by the measuring roller, because it would move one way around in traversing one figure and the apposite way in going around the adjacent one, and the reading due to the arc would be eliminated. Thus the whole curve may be integrated correctly at once without going round each separate portion formed as above, even if the point S at the end of AB moves upon a broken line instead of along OX. Next, substitute a continuous curve TUVXYZ (Fig. 8a) for the broken line. This curve may be supposed to consist of an infinite number of straight portions. The infinitely small portions contained between an arc, as n^u^, and another very close to it, drawn 43 from the beginning and end of these straight portions, may, just as in the case of the broken Hne, be supposed to be in- tegrated separately and with a correct result, which is independent even of a possible crossing of the arcs, as v^v^ and 44 t('^u^. In the same way as before, it may be seen that the arcs u^u.^, etc., need not be traversed and, so long as the point S returns to its initial position, the area of the figure is given by the simple traverse of its boundary, whatever be the curve on which the point S moves, which, in 45 the case of the Amsler planimeter, is a circular arc. Various writers have explained the acthe simple polar planimeter in tion of ways more or less different. these ways, recently given One of by Mr. F. Brooks, of Lowell, U. S., may be alluded to. He shows the area that may be treated as the difference between the area swept out by the line Tp (Fig. 9) and the sector of the zero circle, or circle upon whose circumference the pointer {p) being moved, the measuring roller is not in This is true, both consequence turned. for the outside or inside, if proper signs be taken. Let the element of area 2^9, ^^ passed over, the curve at /> being for a small distance considered concentric with the zero circle, this small area subtending an angle w at the center; then, if values be taken as shown upon Fig. 9, in which CT — radius- bar = (7 ; Cp =:one portion of pole- arm =r^ Gm = i\\e kxeo. 2)q other portion of ; pole-arm = \io {a^ + lr -\-1ab cos 6) —^ro {cr + b'-{-2cb)=2ob {a cos O-c)^ 46 and, by a geometrical construction, the measuring travel of the roller is easily shown to be equal to the same expresMr. Brooks also explains why the sion. area of the zero circle must be added to the reading if the figure to be integrated contains the center T. The following ap- pears, however, to be a planation. still simpler ex- Keferring to Fig. 7, it is evi- dent that going around the outside of the zero circle corresponds to a move- ment taken continuously above the zero line is OX when only the portion above OX In measured. curve this latter case the could never be completely trav- ersed as long as the pointer nioves in one direction. the ends of Suppose, however, that bent round and OX are brought withiu the figure, then the motion in one direction will enable the com- made but only the portion outside the line, i. e., correspond- plete circuit to be ; ing to that originally above OX, will be measured by the roller, and that within must consequently be added to the recorded result. This quantity is evident- 48 ly the area of the zero circle in the case of the Arasler planimeter, which must, therefore, always be added to the result when the center the radius- arm of is within the diagram to be measured. As the Amsler planimeter far as the author is alone, so aware, has been modified to measure the area of any non- developable surface, may be here noticed. this modification The only surface of the kind to which it has been adapted is a spherical one. Fig. 10 shows the instrument, and from that figure it will be seen that the chief alteration is the placing of two joints j j\ one upon the («), and the other upon the pole-arm h, so as to allow the employ- radius-bar ment of the integrator for surfaces of varying curvature. The joints are equidistant from the end of each bar, and exactly opposite to each other— the radius-bar and pole -arm being now of equal and a pin / is placed on («), which fits into a corresponding recess in length, [h), so that when the two arms are closed, they can be together bent at the joints i 49 to the required amount, and, the joints being purposely made stiff, they will re- main at the proper angle when the instrument is used. The joint (J) acts so that the tracing point (p) is always in the place of the axis of rotation of the 50 measuring roller. The theory of the action of this instrument has been fully exj^lained by Professor Amsler, in an article in which the theory of the relations between measurement upon a spherical surface and upon a plane surface is dis- cussed. The various applications of the simple planimeter for finding areas well are known, and need not be explained but there are some slight modifications of the instrument for special purposes, and one of these recently applied by Professor Amsler to his planimeter is worth noticing. This is illustrated at Fig. 11, and is a device for readmg at once the mean pressure given from an indicator diagram. Two points, U and V are seen, one (IT) being upon the upper side of the bar A, which slides in the tube H, and one (V) upon the tube H itself. These points can be adjusted to the length of the diagram by inverting the instrument in the way shown in the figure, and the ; sliding-bar is then clamped S. by the screw This setting causes the reading of i 51 : 52 the instrument, when the diagram is traversed by the pointer in the usual way, to give at once the mean height of the The diagram in fortieths of an inch. simple relation is as follows Beading of measuring-rollers = Mean height of dia- gram in inches. Mean pressure = Mean height X vertical scale of diagram. As an instance of the great saving of labor by the use of the Amsler planimeter, the author happens to know a civil en- gineer's office, where a large amount of earthwork quantities had to be taken out, the calculations proceeded slowly and with many repetitions, until one of the draftsmen procured a planimeter, and then the other, with the result of a great expedition of the work, and the almost complete absence of errors and even then only in decimal places— where — previously the divergence had been as much as by units. Although the connection between the 53 disk and roller or linear planimeter and the polar planimeter has been shown, it is possible to regard them as acting upon The former may be different principles. considered as measuring the variation in the ordinate (y) by a change of effective radius of the circle on which the measuring-roller works, the latter measuring the same thing by a coiTesponding change in the sine of the angle which its plane of rotation makes, with its direction of mo- on which it rolls. They have, in fact, been classified in this way as radius machines, and sine or tion over the surface cosine machines, for the slipping, al- though occurring in both, appears in the ordinary way of viewing the subject to affect the result in different ways. In the former, slipping is sui)posed to be entirely due to the variation in the value of (y), and only takes place when the ordinate in the latter, the changes in value change is supposed to be effected by turning the pole-arm about its center, without any slipping at all. This distinction is, however, quite an imaginary : 54 one, for it will be seen that if the curve be obliquely inclined in either case to the axis OX, the action of the measuring Recently, a of what are called " Pre- roller is precisely the same. number large cision Polar Planimeters," have been de- signed and constructed, which combine in an obvious manner the above two prin- ciples of action, the disk giving motion to the roller, while the pole-arm carries the disk in an oblique direction. Thus, the advantages of a uniform and it across invariable surface of contact for the roller, and the convenience of the polar planimeter are combined, with the still more important advantage of a large relative turning of the measuring roller. Before describing a few different forms of the best of these instruments, the general theory upon which they work will be given it will then not be difficult to understand the action of the several instru; ments without repeating the exi)]anation in each case. It will be found that both the linear and polar planimeter are only special cases of application of the general