MECHANICAL 
 
 INTEGRATORS 
 
 INCLUDING 
 
 THE VARIOUS FORMS 
 
 OF 
 
 PLAN I M ETERS. 
 
 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 aids to mathematical compu- 
 tation have always deservedly been regarded 
 with interest. 
 
 Aside from the labor-saving quality which 
 most of them possess, they have a value aris- 
 ing from the fact that they represent thoughts 
 of more or less complexity expressed in mech- 
 anism. 
 
 They are of many kinds, and serve widely 
 different purposes. The reader will find in 
 this essay descriptions of many that are use- 
 ful directly or indirectly to engineers. 
 
 k
 
 Mechanical Integrators. 
 
 All measurements are made in terms 
 of 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 em- 
 ployed. 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 ; as, for instance, when the length 
 of a rod or wire is estimated by its 
 weight. 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 gen- 
 erally consists in measuring in terms 
 of the simple units from w^hich the
 
 6 
 
 former is derived, and performing, 
 with the results, the necessary calcula- 
 tion. An example of this latter method 
 is given in obtaining the contents of an 
 area, by taking its length and breadth 
 and multiiDlying them together, instead 
 of adopting the tedious process of ascer- 
 taining, by direct comparison, bow many 
 times the unit of area would be con- 
 tained 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 me- 
 chanical 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 multi- 
 plication 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. The object of the present pa- 
 per is to deal with the theory, design 
 and practical applications to engineering 
 problems of the former class alone. It may- 
 be 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 
 of Harlacher, Henneberg and Smreker, 
 for gauging the velocity of a river cur- 
 rent, 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: 
 
 (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 calcu- 
 lation. The reason of this is that the 
 data of calculation, which are taken di- 
 rectly 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 great- 
 er the number the greater is the labor 
 of determining the result. The instru- 
 ments discussed in this jDaper perform 
 such work mechanically, with the great 
 advantages of rapidity of operation, ac- 
 curacy of results, and without requiring 
 mental effort on the part of the manipu- 
 lator; and all this, moreover, to a great
 
 extent independently of the complexity 
 of the calculation required. All the re- 
 sults the measurement of which will be 
 considered, can be measured graphically. 
 If the observations have been made sep- 
 arately, tliey can be plotted, whether in 
 the form of a diagr im of energy, or on 
 the plan or elevation of an area or sec- 
 tion, and the boundary can be filled in 
 with a tolerably close approximation to 
 accuracy. In the other case the graphic 
 record is, 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 here- 
 after described. A plane surface may, 
 however, be employed upon which to re- 
 present 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 reg- 
 ular or an irregular plane of figure? 
 
 Such measurements are of three 
 kinds : 
 
 (1) The length of its perimeter or 
 boundary. 
 
 (2) The area of its superficial contents. 
 
 (3) Its relation to some point, line, or 
 other figure on the surface, e. g.^ its mo- 
 ment of area or moment of inertia about 
 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 ob- 
 tained 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- 
 tion becomes that of integration or sum- 
 ming up of an infinite series of indefin- 
 itely small quantities. Instruments for 
 performing this operation are therefore 
 called '• mechanical integrators.'' In all 
 such instruments the rolling action of 
 two surfaces in frictional contact is em- 
 ployed, for this, as will be hereafter 
 seen, enables 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 calculating machines. 
 This fact necessitates something more 
 than a mere discussion of the mathe- 
 matical principles upon which the calcu- 
 lations are performed, for though the 
 action of an integrator may be absolute- 
 ly correct as far as its theory of the per- 
 formance of the calculation is concerned, 
 yet there is always some instrumental 
 error depending upon the rolling, and 
 also, as will be seen, of the slipping of 
 the two surfaces in frictional contact.
 
 12 
 
 This error may be exceedingly small. Init 
 it is a matter of great importance to 
 ascertain its exact amount, and the sub- 
 ject will therefore be investigated at 
 length, under the heading "Limits of 
 Accuracy of Integrators," where an ac- 
 count will be also given of the experi- 
 mental results of Professor Lorber, of 
 Leoben; Dr. William Tinter, of Vienna, 
 and Dr. A. Amsler, of Schaflfhausen. In 
 this investigation it will be showu that 
 when integrators are examined upon the 
 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 clussi- 
 fication is that it not only leads to a cl&ir 
 understanding of the nature of the re- 
 sults to be expected from any particular 
 instrument, and teaches the best method 
 of manipulating it, with regard to its po- 
 sition relatively to the figure to be meas. 
 ured, but it also brings out prominently
 
 13 
 
 tlie meolianiral principle uptui wliicb tlie 
 inventor has relied sometimes, as it 
 NVouM appear, unoonsciouHly, for the ac- 
 ruracv of tlie r«->»tiltH »\jnMt« d to be ob- 
 UiintMl. 
 
 It may be here remarked that the wime 
 priiJcipU', by which an intej^'rator is em- 
 j)h)yo(l to determine a result from an au- 
 to^'ni]>!iic record, may Ih) apphed directly 
 to «»btain a continuous result from the 
 machine or bo<ly in motion, such, for in- 
 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 de- 
 tnils of the instruments for direct appli- 
 cation to a machine, and this will be 
 done under the head of *' Continuous 
 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 recti- 
 fying a curve, as it is called, that is. ob- 
 taining its length as a right line, is by 
 rolling a wheel along it. This wheel is 
 connected with a suitable train of wheels 
 for recording the total number of revolu- 
 tions, and as the rolling circle of the 
 first wheel is either a unit in length or 
 contains a known relation to this unit, 
 the length of the curve 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 men- 
 tioned 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 " opisom- 
 eter," in another form the chartometer, 
 or Wealemefna ; it is also employed upon 
 a large scale as a road or route meas- 
 urer. The same principle has been em-
 
 15 
 
 ployed in one of the latest forms of ane- 
 mometer, in which the plane of the wheel 
 is always kept coincident with the direc- 
 tion 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 
 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 re- 
 ferred 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 jDrin- 
 ciple is, therefore, that of direct unit 
 measurement or comparison. The fore- 
 going instrument, w^hich evidently be- 
 longs 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 in- 
 stead of being the addition of an infinite 
 number of infinitely short lines, an in- 
 finite number of continually changing 
 magnitudes has to be added. This is the 
 same both for instruments required in the 
 second kind of measurement, or *'Area 
 Planimeters," and those required iu 
 the third kind, or " Moment Planim- 
 eters." From the fact that the chang- 
 ing magnitudes referred to are sim- 
 pler to deal with in the case of cal- 
 culating the contents of areas than in 
 finding theii' 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. Take any straight 
 lines, OX and OY, at right angles to each
 
 17 
 
 otber, and parallel to OY ; draw a series 
 of straight lines equidistant from each 
 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 adopt- 
 ed 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 rect- 
 angle. 
 
 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 the area of the figure AP>DE. When 
 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 
 will be correctly measured by an instru- 
 ment in which the recording wheel or 
 measuring roller always turns at a rate 
 proportional to the ordinate y of the 
 curve, while the body from which it de- 
 rives its motion moves at a uniform rate 
 along the axis OX.
 
 19 
 
 Area plauimeters have been classified 
 accordinjjf to apparently different modes 
 by which the operation 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 be- 
 long, 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- 
 fore, the names which have been given 
 them will be used ; but this will not de- 
 note any difference of principle, and the 
 classification which will be adopted is 
 that already explained, and depends on 
 mechanical conditions of action. In what 
 follows, one mode of viewing the mathe- 
 matical operation is adhered to through- 
 out, and it may be stated that the object 
 of the author has been to make clear the 
 principles of action of integrators, rather
 
 20 
 
 than to obtain rigid and exhaustive dem- 
 onstrations of their theory. 
 
 Planimeters in which Slipping of the 
 Measuring Roller Takes Place. 
 
 From a brief account of the subject by- 
 Professor Lorber, it appears that the first 
 recorded idea of a planimeter is attributed 
 to Hermann, of Munich, who worked it 
 out with Lammle. This idea of Her- 
 mann's, which was pubhshed in 1814, 
 seems to have fallen into oblivion, for in 
 1827 Oppikofer, of Berne, constructed a 
 planimeter upon similar principles, and 
 it was thenceforth called after his name. 
 On the other hand, Favora gives the 
 priority to Professor T. Gonella, of Flor- 
 ence, who, in 1828, without any knowl- 
 edge 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 won a prize, in Paris, in 
 1836. 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 
 planimeter. Later on, in England, other 
 inventors (Sang, Moseley) worked at the 
 subject, but all these instruments de- 
 pended for their action on the same prin- 
 ciple, which is as follows : 
 
 Let M (Fig. 2) be the plan of a disk 
 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- 
 lution. The plan of the point of contact 
 (B) of the roller with the disk is always 
 made to coincide with that particular 
 point on the curve which is in the line 
 drawn at right angles to OX, through 
 the center C of the disk. The plane of 
 otation of (m) which may be called the
 
 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 
 PQ for a distance :_. .r, eipial to the width 
 of the element AB. 
 
 Then if //, = distance of B from OX. 
 li = radius of disk M. 
 ?•=: radius of measuring roller 
 
 m. 
 ;ij = consequent reading of 
 measuring roller for this 
 travel of disk. 
 
 Then ^ a .'' = linear distance turned 
 through by a point on the 
 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 M these 
 distances are equal. 
 
 ?/ 
 Therefore 2 nrx ^ = ~Ax, 
 
 or, ". = .'AA^X2;i^;
 
 24 
 
 but K — =^ is a constant, which, by taking . 
 
 27rrll 
 r and R in suitable ratio may be made 
 unity. 
 
 Then n^—y^Ax, 
 
 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 AB 
 below OX is being 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 be- 
 fore, since the point of contact is now 
 below C. The final result of these two 
 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 nega- 
 tive, 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 fig- 
 ure, as it practically is for the integra_ 
 tion of the portion DHE. Then it will 
 be found that DKH is subtracted, and 
 DKHE is added, so as to give the re- 
 quired actual area DHE. 
 
 Inasmuch as this reasoning is inde- 
 pendent 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 the 
 series of infinitely narrow strips which 
 compose the figure x4.BDE is given by 
 the final reading of the roller when the 
 traverse of the boundary is completed. 
 
 The Wettli-Starke planimeter (Fig. 3)
 
 26 
 
 acts directly upon this principle, with 
 the exception that it is the disk that 
 is 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 M can be 
 moved to and fro. The disk, which is
 
 27 
 
 made of glass and covered with paper, 
 has two motions, 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 
 roller (m) rests upon the surface of the 
 disk, being carried in another frame (B), 
 which is hinged to the base-plate. The 
 action is as follows : 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 
 of the arm is made to pass round the 
 boundary of the figure, the disk wall be 
 turned through a distance proportional
 
 28 
 
 to the travel along OX, while at any in- 
 stant the roller {m) is at the same dis- 
 tance 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 
 the pointer was " linear ; '' that is, the 
 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 planim- 
 eters have been suggested, but the only 
 modification of the disk and roller which 
 it will be worth while to notice is the 
 cone and roller. 
 
 Let MM', Fig. 4, be the cone cor- 
 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 of ele- 
 ment SB. 
 where the element 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 in- 
 stead 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 plan- 
 imeters." 
 
 In the year 1856, Professor Amsler- 
 Laflbn invented and brought before the 
 world the now well-known polar planim- 
 eter bearing his name, and, since then, 
 no less than twelve thousand four hun- 
 dred 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 Bounia- 
 kovsky, of St. Petersburg (1855), had 
 improved upon previously-existing forms 
 of polar planimeter, though it is well to 
 note that the planimeter s already men- 
 tioned 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 is shown in 
 Fig. 5. It consists of 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 
 figure ; and by means of the graduated
 
 33 
 
 m 
 
 V 

 
 34 
 
 dial h, and the vernier y, in connection 
 with the roller m, the result is given cor- 
 rectly in four figures. The sleeve H can 
 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 disposi- 
 tion of the disk M with regard to the 
 figure ABDE, as in Fig. 2. but now let 
 the roller m move round the edge of the 
 disk instead of across it, its distance 
 from 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 rota- 
 tion 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-= '- = ^m^qlk; 
 
 Iq A £C ^ 1 ' 
 
 but in Fig. 6 ?^=.^::.sin OS^. 
 
 But by similar triangles 
 
 leretore 
 or n^=y^/\xx 
 
 Therefore '=—, 
 
 AX E' 
 
 27rrK 
 
 This is the same result as previously ob- 
 tained, 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 m is 
 exactly the same as if it were in contact 
 at the point B, no matter in what posi- 
 
 I 
 
 tion it may be along the line through B 
 parallel to OX. The turning of m thus 
 measures the area of the element as long 
 as y does not change. If, however, the 
 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 
 are equally distant from OX. The rea-
 
 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 
 final positions of the pointer when the 
 complete travel of the closed curve has 
 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 
 be removed altogether. The roller is 
 then moved in contact with the surface 
 of the figure and with identically the 
 same amount of turning as before, pro- 
 vided that its plane of rotation is turned 
 so as to make the same angle with OX 
 (which is now its direction of transla- 
 tion), as it did before with the direction 
 of motion of the edge of the disk in 
 contact with it. This is the case when 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 
 the axis OX. By employing the bar qSq^ 
 itself as the axis of rotation on which ??i 
 turns, the simple planimeter shown in 
 Pig. 7 is obtained, in which the point 
 of contact of the measuring roller is 
 made to pass around the diagram. 
 The turning of the roller 771 correctly 
 gives the superficial contents. The roller 
 can be moved to any position on the rod, 
 such as shown in dotted lines, Fig. 7, 
 without in any way affecting its result- 
 ant turning, and the former point of con- 
 tact of the roller is replaced by a pointer, 
 which is made to follow the curve instead 
 of the roller. Professor Burkett Webb 
 has described to the author a planimeter 
 of this form used in the United States, 
 known as " Coffin's'' planimeter. 
 
 If, finally, the point S, instead of mov- 
 ing 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 ra- 
 dius TC, the instrument becomes the or- 
 dinary Arasler planimeter (shown in dot- 
 ted Unes). This explanation, so far, is 
 based upon that given by Sir Frederick 
 Bramwell, who has further shown that 
 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 re- 
 turned to initial position. The following 
 demonstration of the truth of this ap- 
 pears, 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), at the ex- 
 tremity of the radius rod, move along 
 the broken line STUVWXYZ, and from 
 these points draw arcs with a common 
 radius =:K, cutting the curve in points 
 t^t.^, ii^u^, t\u„, &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 
 roller ; so also are given the areas of the 
 various other portions, t^u^y^t^, &c. If 
 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 inte- 
 grated 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 continu- 
 ous curve TUVXYZ (Fig. 8a) for the 
 broken line. This curve may be sup- 
 posed 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 ac- 
 tion of the simple polar planimeter in 
 ways more or less different. One of 
 these ways, recently given by Mr. F. 
 Brooks, of Lowell, U. S., may be alluded 
 to. He shows that the area 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) be- 
 ing moved, the measuring roller is not in 
 consequence turned. This is true, both 
 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 other portion of pole-arm 
 
 kxeo. 2)q = \io {a^ + lr -\-1ab cos 6) —^ro 
 {cr + b'-{-2cb)=2ob {a cos O-c)^
 
 46 
 
 and, by a geometrical construction, the 
 travel of the measuring roller is easily 
 shown to be equal to the same expres- 
 sion. Mr. Brooks also explains why the 
 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 still simpler ex- 
 planation. 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 OX when only the portion above OX 
 is measured. In this latter case the 
 curve could never be completely trav- 
 ersed as long as the pointer nioves in 
 one direction. Suppose, however, that 
 the ends of OX are bent round and 
 brought withiu the figure, then the mo- 
 tion in one direction will enable the com- 
 plete circuit to be made ; but only the 
 portion outside the line, i. e., correspond- 
 ing to that originally above OX, will be 
 measured by the roller, and that within 
 must consequently be added to the re- 
 corded 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 of the radius- arm is 
 within the diagram to be measured. 
 
 As the Amsler planimeter alone, so 
 far as the author is aware, has been 
 modified to measure the area of any non- 
 developable surface, this modification 
 may be here noticed. 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 
 radius-bar («), and the other upon the 
 pole-arm h, so as to allow the employ- 
 ment of the integrator for surfaces of va- 
 rying curvature. The joints are equi- 
 distant from the end of each bar, and 
 exactly opposite to each other— the ra- 
 dius-bar and pole -arm being now of equal 
 length, and a pin / is placed on («), 
 which fits into a corresponding recess in 
 [h), so that when the two arms are closed, 
 they can be together bent at the joints
 
 49 
 
 to the required amount, and, the joints 
 being purposely made stiff, they will re- 
 
 i 
 
 main at the proper angle when the in- 
 strument 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 ac- 
 tion of this instrument has been fully 
 exj^lained by Professor Amsler, in an ar- 
 ticle 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 are well 
 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 notic- 
 ing. This is illustrated at Fig. 11, and 
 is a device for readmg at once the mean 
 pressure given from an indicator dia- 
 gram. 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 by the screw 
 S. 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 
 diagram in fortieths of an inch. The 
 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 
 different principles. The former may be 
 considered as measuring the variation in 
 the ordinate (y) by a change of effective 
 radius of the circle on which the measur- 
 ing-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- 
 tion over the surface on which it rolls. 
 They have, in fact, been classified in this 
 way as radius machines, and sine or 
 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 
 changes in value : in the latter, the 
 change is supposed to be effected by 
 turning the pole-arm about its center, 
 without any slipping at all. This dis- 
 tinction 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 
 roller is precisely the same. Recently, a 
 large number of what are called " Pre- 
 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 
 it across the disk in an oblique direction. 
 Thus, the advantages of a uniform and 
 invariable surface of contact for the roller, 
 and the convenience of the polar plan- 
 imeter 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 un- 
 derstand 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
 
 55 
 
 princijDle upon which the correctness 
 of action of j^recision planimeters de- 
 pends. 
 
 c 
 
 
 
 I 2 |0 
 
 &\ 
 
 al 
 
 
 1°^ 
 
 It will be well to approach the matter 
 from the same point of view as in explain- 
 ing the lineal- planimeter. Let the disk 
 M, Fig. 12, rotate about an axis C as it
 
 56 
 
 rolls along the PQ, line parallel to OX, 
 the pivot on the axle at C being attached 
 to a frame which also carries another 
 pivot S. This latter pivot always lies 
 upon OX, and about it rotates a pole-arm 
 b, carrying a pointer j) at one end, and 
 the measuring-roller m at the other end. 
 The plane of rotation of the measuring- 
 roller coincides with the .direction of the 
 pole-arm, and is carried over the disk in 
 contact with it, along th^ arc rmn' . Then 
 from what was proved, p. 402, the motion 
 of the roller in is exactly the same as if 
 it were moved, so that its axis {ihvays co- 
 incides with Qj, the perpendicular upon 
 the pole-arm from the center of the disk 
 — provided only that its axis is always 
 parallel to this line. Thus, adopting the 
 previous notation, and taking 
 
 8^:*= length of upper portion of pole- 
 arm =R. 
 Then when the disk rolled through a 
 distance ax, 
 n^^reading of roller m 
 2/j = ordinate SB.
 
 57 
 
 turning of roller _Jl7trn^ 
 distance turned by edge" i\x 
 
 
 therefore ^ — ^i 
 
 AX K 
 
 or 7? 
 
 wliich is the same result as in the case of 
 both the linear and polar planimeters. In 
 practice, the portion of the pole-arm 
 which carries the poii>ter is usually per- 
 pendicular to the other portion, as shown 
 by the dotted lines, Fig. 12. In this 
 case, the direction of motion of the disk 
 and frame carrying the center of the pole- 
 arm 8 must be taken parallel to the guide 
 P'Q', that is, perpendicular to the former 
 direction. It has already been shown in 
 the case of the Amsler planimeter, that it 
 does not matter in what path the center 
 S of the pole-arm is carried, so long as 
 the foregoing conditions are observed, 
 and thus there are several forms of pre-
 
 58 
 
 cision polar planimeters in which the 
 point S is carried in the arc of a circle 
 instead of along a straight line. It may 
 now be made clear, from Fig. 13, that 
 the first two kinds of planimeters are 
 special cases of the last. 
 
 (A) Fig. 13. Let R be the radius of 
 the disk, Rj the radius about which the 
 roller in is carried. Then the area of 
 the diagram as already explained can be 
 measured by either pole-arm S/) or S^:)'. 
 
 (B) Fig. 13. Let the radius R, of the 
 pole-arm become infinitely great, while R 
 remains finite; thence on moves across 
 the disk M in a straight line usually, but 
 not necessarily, through the center, and 
 the linear planimeter is the result. 
 
 (C) Fig. 13. Let the radius of the 
 disk become infinitely great, and any 
 motion of such an imaginary disk under 
 these conditions would make the result 
 equivalent to moving the roller over the 
 surface of the paper. Therefore the disk 
 may be removed, and the elementary 
 form of polar planimeter is obtained, the 
 roller being placed in either position
 
 59 
 
 Fis. 13 
 
 ^(C)
 
 60 
 
 as shown at m or m, without affectmg 
 the result. 
 
 ^Ar 
 
 7-. 
 
 I / 
 
 ' '/ 
 \ 1/ / 
 
 ^"^-J 
 
 The following is a simple explanation 
 of the action of the precision polar plan- 
 imeter
 
 61 
 
 Let M (Fig. 14) be the disk, which can 
 be turned by any suitable means through 
 a distance corresponding to the hnear 
 travel of its center about C. 
 
 Let r„ = radius of zero circle (E'ES')- 
 
 7'= radius of any circle FF'. 
 
 a=:/ turned through by pole- 
 arm, when the pointer moves 
 from the zero circle to the 
 circle FF'. 
 
 9''= /turned through by radius 
 arm, CS, when an element 
 EE' F'F is being described. 
 
 a = radius arm = CS'. 
 
 i = pole- arm = FS'. 
 
 d=SS\ 
 
 Then from the figure — 
 
 and r; = a' + ^r 
 
 r' = a' + h'-2ab cos (90 + a) 
 = a' -{-b"^ -{-2((b sin a. 
 Therefore 7-^ = 9\'^ + 2ab ^n a, 
 
 or sm a=—-—-^ 
 
 2ab 
 
 Now the turning of the plate is pro- 
 portional to c'', and may, for the arc FF', 
 be taken as equal to r^(J'c. 
 
 y=SK = SS' sin ZSS'K=d nin a.
 
 62 
 
 where c and d are constant quantities ; 
 also if ft equal the radius of the roller. 
 , y _ linear distance by ed^e of roller 
 K distance traveled by edge of disk 
 
 ■'■"=^^' 
 
 7\(,''C lab 
 
 ^'' — ^\\/ ^^^^ \ 
 
 7* ccl 
 ^ut?r— 4^ — ris a constant quantity, and 
 
 anay be made equal to unity. 
 
 r' — r ' 
 Therefore oi^^ — -^-^^^area of ele- 
 ment EETF'. 
 
 Thus n, is a measure of an element of 
 area, and as the motion of m due to the 
 turning of the pole- arm in moving to a 
 larger or smaller cu'cle does not affect 
 the reading when the pointer at F has 
 passed round a closed curve, the final 
 reading of the roller gives the area of 
 any figure.
 
 
 
 63 
 
 
 
 
 The 
 
 actual construction 
 
 of 
 
 the 
 
 pre- 
 
 cision 
 
 polar 
 
 l^lanimeter appears 
 
 to 
 
 have 
 
 
 \M 
 
 ITuy.. lo 
 
 
 
 
 V 
 
 ^ 
 
 S 
 
 a 
 
 
 c 
 
 
 I- ^ 
 
 
 
 
 ^— t-^Li 
 
 & 
 
 
 
 
 
 
 Fi-. IG 
 
 been first carried out by Mr. Hohmann, 
 Bauamtmann of Bamberg, in 1882, in 
 conjunction with the well-known mechan-
 
 64 
 
 ician Mr. Coradi, of Zurich. A plan of 
 the first instrument is given in Fig. 16 ; 
 but it will be more easily understood by 
 reference to the diagram, Fig. 15, which 
 shows a frame (a) pivoted at one end (c) 
 to a weight (G), about which it turns. 
 This frame carries a small disk (ic), which 
 rolls. in contact with the surface of the 
 diagram, and gives motion to the disk 
 M. The roller 771 is moved across the 
 disk in the horizontal direction by a 
 pole-arm centered on S as axis. Re- 
 ferring to Fig. 16, which is lettered in 
 a similar way to Fig. 15, it will be seen 
 how the pole-arm, in turning about the 
 center S, effects this motion. A plan and 
 elevation of the frame F, which carries 
 the roller m, is shown on a larger scale, 
 and this frame is moved backwards or 
 forwards through a slot in the support- 
 ing frame. The roller m has the ar- 
 rangement of the screw and worm for 
 obtaining the readings of the dial //, as 
 in the Amsler planimeter, and also the 
 vernier in conjunction with the measur- 
 ing roller. Two rollers, j j\ serve to
 
 65 
 
 Fi-. IT 
 
 Uii^J — p[L^ 
 
 M '"(©^ 
 
 i 
 
 E 
 
 %^^^€PI 
 
 iM 
 
 Q SECT! 
 
 ONAL ELEVATION 
 
 FiiX. lO 
 
 i)>~tt@ 
 
 PLAN 
 
 fw:TH Disk removed)
 
 66 
 
 balance the instrument. The details of 
 the arrangement by which the length of 
 the pole-arm b is adjusted are also shown 
 on a larger scale. 
 
 In this instrument, the fact that the 
 disk is inclined at an angle makes no 
 diffei'ence in the theory of its action, and 
 as the roller W obviously drives the disk 
 so that the angular motion corresponds 
 with the angular motion of the radius bar 
 a, the explanation already' given makes 
 its mode of operation clear. The case is 
 rather simplified by the fact that the 
 roller m is moved radially across the 
 disk. 
 
 An instrument of similar kind has been 
 designed and recently described by Pro- 
 fessor Amsler-Laffou. This is shown in 
 Figs. 17, 18 and 19, where it will be seen 
 that this disk M, which is now horizon- 
 .tal, is turned by means of bevel-wheels 
 Sj z^, the back of one of which forms a 
 portion of a frustum of a cone rolling 
 about the center C of the radius-arm a a. 
 The center is itself a sphere, which al- 
 lows any side motion of the instrument
 
 67 
 
 due to the inequality of the surface to 
 take place without affecting the accuracy 
 of the result. The necessary pressure cf 
 the roller upon the disk is obtained by 
 allowing the weight of the portion of the 
 frame b which carries the roller in to rest 
 upon the disk by being pivoted by the 
 centers KK (Fig. 18). A peculiar fea- 
 ture of the instrument is that the pole- 
 arm frame can be centered either within 
 or without the frame. If placed in the 
 former position, the reading is twice as 
 great as in the latter, the positions of 
 the centers being purposely adjusted to 
 effect this. The frame can be taken off 
 one center, s' (Fig. 17), by unscrewing a 
 set screw at x, and at once placed upon 
 the other. The weight v^ can be adjusted 
 in any position by means of the nut and 
 screw t (Fig. 17), and so the pressure of 
 the pointer p upon the surface of thQ 
 diagram may be regulated. 
 
 In both the above instruments the 
 disks derive their motion from a roller 
 in contact with the surface of the dia- 
 gram, but in the next two instruments to
 
 68 
 
 be described, Messrs. Holimann and Co- 
 radi have caused the disk to be turned in 
 a manner which prevents any such error 
 as from the possible sHpping of the above 
 roller. The first instrument of the kind 
 is shown (Figs. 20, 21) in plan and ele- 
 vation. The disk M is carried by a frame 
 (aa) as before, but the frame now swings 
 about a circular stand, the edge of which 
 is toothed, so that the pinion (^), which 
 is upon the axis of the disk, is turned, 
 and therefore the disk itself, with the 
 same angular velocity. The weight of 
 the frame and disk is, to a great extent, 
 taken off by means of the light rod [l), 
 which swings about a central pillar P. 
 A side view of the pole-arm is shown, 
 and the mode of adjusting it and sup- 
 porting the portion which carries the 
 roller (m), so that by means of centers 
 KK the weight of that portion of the 
 frame is allowed to rest upon the disk. 
 
 It is evident that this instrument 
 works upon identically the same princi- 
 ples as the foregoing ones. This "Freely 
 swinging " precision planimeter was fol-
 
 69 
 
 Fi^f^. i20 aiid 21 
 
 SECTIONAL ElE.'ATiON
 
 70 
 
 lowed by another, called the '' Plate " 
 planimeter (Fig. 22), which is of still 
 simpler construction. In this form ad- 
 vantage is taken of the fact that the 
 measuring roller need not have its path 
 through the center of the disk, and a sup- 
 port {v) is obtained above the disk, so 
 that its pivot [q) can work between cen- 
 ters, the weight of the frame being sup- 
 ported by rollers {j). The portion of 
 the pole-arm which carries the roller {711) 
 is (as in the last case) pivoted between 
 the centers KK. The dial for higher 
 readings is as in the case of the previous 
 instruments denoted by A. 
 
 The last and most recent modification 
 is the " Rolling planimeter," of Coradi. 
 This approaches nearest to the diagram 
 (Fig. 12), which completely explains its 
 action. Here the center of the radius- 
 arm is removed to an infinite distance, 
 and the center of the disk and that of the 
 pole arm are carried along straight lines 
 parallel to the axis OY in that figure. 
 The way in which this is efi'ected is seen 
 from Figs. 23 and 24, which show Co-
 
 71 
 
 radi's rolling- planimeter in plan and ele- 
 vation. Two rollers {ec/) are in contact 
 with the surface of the diap^ram, i n 
 
 ITiiZ. 
 
 SECTIONAL ELEVATION 
 
 their axis is a bevel wheel (z^) (Fig. 24), 
 which gears with another bevel-wheel (2,), 
 which is upon the axis of the disk. Thus 
 the wheels 2, 2, are turned as the frame 
 is rolled along, and, consequently, the
 
 73 
 
 disk itself. The axis of the rollers cc 
 works upon the centers ee^ which are set- 
 screws in the frame (aa). The disk M 
 is also carried between centers {qq)-, as in 
 the instrument last described, and, also, 
 as in that case, the path of the roller does 
 not pass throuf^h the center of the disk. 
 This instrument, which has many advan- 
 taj^es, and, notwithstanding:^ that it rolls 
 on the diagram surface, <]fives results of 
 great accuracy, has been examined with 
 great care by Professor Lorlier, who has 
 given a lengthy description of it and a 
 full account of its theory. 
 
 The last planimeter of this kind to be 
 examined is one by Professor xVmsler. 
 This instrument, shown in Figs. 25, 26, 
 27, (lifters from the last in that the tooth- 
 wheel z,^ works in gear with a rack 2, 2,, 
 which is cut upon a fixed frame DD. 
 Thus, although it is supported by the 
 rollers cr, there is no possibility of slip- 
 ping as far as the turning of the disk is 
 concerned. The rollers run in a groove 
 cut in the frame DD, and the action of 
 the instrument is easy and smooth. The
 
 74 
 
 Fio-. 24 
 
 ELEVATION 
 
 Kiss. ;ao,a6 and '^T
 
 75 
 
 theory of its action is identical with that 
 of the foregoinj^ one, as explained by- 
 
 means of the diagram (Fig. 12). The 
 various parts are lettered in the figures 
 to correspond with the explanations of 
 thatjnstniment previously given.
 
 76 
 
 In the instruments hitherto described 
 the surfaces of revohition are Hmited to 
 the disk and cone, but various other sur- 
 faces may be made to replace these. The 
 only one that has been so employed is 
 that of the sphere; and in the present 
 class of instruments, in which slipping 
 takes place, the following property of the 
 sphere is made use of : Let a sphere M 
 (Fig. 28), which replaces the disk (Fig. 
 2), roll along the axis OX. Then sup- 
 pose the roller m can, by suitable means, 
 be moved round the surface so that its 
 plane of rotation shall always contain the 
 center of the sphere and be perpendicular 
 to the arm CB, which corresponds to the 
 pole-arm of the former instruments ; it is 
 evident that if the perpendicular be 
 drawn from q, the jjoint of contact of m 
 with the sphere, to CZ the axis of rota- 
 tion of the sphere, meeting it in the point 
 u, the line qu is the radius of the rolling 
 circle of contact of the measuring roller. 
 _,, » motion of measuring roller _ 
 motion of sphere along OX ~ 
 qu qu
 
 77 
 
 But from the figure ^^3^=^ — sin a. 
 
 Therefore, adopting the same notation as 
 hitherto used, 
 
 motion of m _27rr)i ^ __y 
 motion of .\l A x K 
 
 which proves that the area of the curve 
 may be measured by any device, on the 
 principle of Fig. 28. It may be shown 
 in the same way as on p. 18, that the re- 
 sult is similar if the sphere rolls upon the 
 arc of a circle, about any center as T, 
 instead of along the straight line OX. 
 Planimeters of this kind have been con- 
 structed by Mr. Hohmann and Professor 
 Amsler. In both cases only portions of 
 the whole surface of a sphere have been 
 employed, and the motion is given by 
 means of an axis, instead of by rolling the 
 spherical surface upon the diagram. In 
 Mr. Hohmann 's planimeter, shown in 
 plan and elevation, Fig. 29, the concave 
 surface M is used. Rotation is given to
 
 SECTIONAL ELEVATION
 
 
 79 
 
 this by means of an axis gr/, an enlarged 
 portion of which (r) rolls npon a cii'cular 
 metal path R. The pole-arm P turns 
 about a center r, and so causes the rolling 
 circle of the measuring roller iit to ^'ary 
 according to the foregoing principles. 
 This instrument has not come into use. 
 Professor Amsler has employed the con- 
 vex surface in an instrument somewhat 
 similar to the one described, except that 
 better provision is made for obtaining 
 the required pressure between the sur- 
 face of the roller and sphere, and for 
 giving rotation from the roller-path.
 
 80 
 
 Planimeters in which only Pore Eolling 
 Motion is Assumed to take Place. 
 There have been many efforts to de- 
 sign instruments in which no sHpping 
 shall take place. These efforts have re- 
 sulted in the production of various in- 
 struments which, though they differ in 
 external form and mechanical action, yet 
 rely upon the same mathematical prin- 
 ciple of action as the planimeters al- 
 ready dealt with, the particular form of 
 disk and roller, or sphere and roller, be- 
 ing taken. Thus, in every case there is 
 a measuring roller, or its equivalent, the 
 rate of motion of which has to l^e varied 
 by some means or other. It is in the 
 method by which this is done that this 
 class of planimeters differs from the 
 other. Instead of obtaining the varia- 
 tion of the measuring roller in bringing 
 it into contact with circles of different 
 linear velocity by sliding it over the sur- 
 face of the disk or sphere, one or other
 
 81 
 
 of the two following principles are em- 
 ployed. A device equivalent either to 
 (1) bringing in succession a series of 
 measuring rollers into contact with the 
 different imaginary circles ; or (2) bring- 
 
 ing circles of different linear velocity in- 
 to contact with a single fixed measuring 
 roller. 
 
 The disk-globe and cylinder-integrator 
 of Professor James Thomson belongs to 
 the former class. In this a sphere G 
 (Fig. 30) rolls over the surface of the disk
 
 82 
 
 M, but is also in contact with a cylinder 
 mm'. The motion of G in direction OY is 
 that in which the roller would shp in the 
 ordinary disk and roller, and does not 
 
 31 
 
 affect the motion of rotation of mm' . On 
 the other hand, the motion in direction 
 OX, which is due to the turning of the 
 disk, is entirely imparted to nuti. Thus, 
 as G rolls along m7n\ the same effect is, 
 in theory, produced as if a series of roll-
 
 83 
 
 ers ?7i, mj, m^, etc., upon the same axis as 
 the cylinder, were successively applied to 
 the surface of the disk, and all slipping, 
 at any rate from this cause, is avoided. 
 The actual mechanism which has not 
 been employed for a planimeter takes a 
 slightly difterent external form in the 
 harmonic analyzer of Sir W. Thomson's 
 tide-calculating machine. 
 
 The devices which have now to be con- 
 sidered as solutions of the problem un- 
 der consideration by the first method, 
 are used in connection with the geomet- 
 rical property of the sphere already dis- 
 cussed, and upon one similar to it. 
 
 Let M (Fig. 31) be the plan of a sphere 
 rolling along the line OX, carrying with 
 it, by a frame not shown, a cylinder {in) 
 which can roll about it so as to come into 
 contact at anj point q upon its horizon- 
 tal great circle. Then the rotation of 
 the cylinder may be employed exactly in 
 the same way as the rotation of the roller 
 on the integrator described on page 416? 
 and shown by Fig. 28 ; but in the pres- 
 ent case, instead of causing the roller to
 
 84 
 
 slip over the surface, the rolHng of the 
 cyhnder is practically equivalent to 
 briuging in succession a series of roll- 
 ers m, ni^, rii^, etc., upon one axis in con- 
 tact with it. 
 
 This principle has been employed both 
 by Professor Mitchelson of Cleveland, 
 XJ. S., and Professor Amsler. The mech- 
 anism of Professor Mitchelson's instru- 
 ment is shown in Fig. 32. In this a flex- 
 ible steel band or chain F, passing round 
 a semi-circular arc D, forces the cylinder 
 C to roll on the sphere G. The cylinder 
 is carried by a frame E, which slides 
 along the bar A, by which it is supported. 
 The mode in which it is proposed to ap- 
 ply it to the ordinary Amsler planimeter 
 is shown on a smaller scale, Fig. 32a, 
 where b is the pole-arm, a the radius 
 bar, and t the center of rotation of the 
 latter. 
 
 Professor Amsler's planimeter on this 
 principle is similar to the foregoing, ex- 
 cept that instead of being carried by two 
 guides as sleeves by a bar, the cylinder 
 frame is supported on rollers from a
 
 85 
 
 Fig. 33
 
 86 
 
 frame above, the rolling friction on the 
 latter being less than that of the cylinder 
 on the sphere. Thus, the cylinder always 
 moves to its required position. The mo- 
 tion of the spherical surface is obtained 
 from a bevel- wheel upon its axis, which 
 gears with a larger one formed upon the 
 edge of a circular stand or suj^port of 
 the instrument. 
 
 A similar principle of the geometry of 
 the sphere has also been employed in an 
 instrument suggested in a p^iper in 1855 
 by the late Professor Clerk Maxwell, 
 when an undergraduate at Cambridge. 
 Instead of the cylinder in Fig. 31, let a 
 sphere m' roll around on the sphere (M), 
 as shown in Fig. 33. Then, from the 
 property of the sphere, which is proved 
 at length in the above paper, the tiu'ning 
 of the sphere m about its axis of rota- 
 tion xx^ relatively to the turning of M 
 along OX, is proportional to the tangent 
 of the angle a in the figure. In the other 
 case it will be remembered that the turn- 
 ing of the cylinder or disk was propor- 
 tional to the sine of the same angle. By
 
 87 
 
 suitable means the "principle can be em- 
 ployed in the constiTiction of a planim- 
 eter. Two forms of such planimeter 
 are shown in the paper, and though they 
 are both in the form of the linear plan- 
 imeter, and are scarcely suitable for prac- 
 tical application, yet the matter is dealt 
 with in a way worthy of the inventor. 
 It is evident that this is another case of 
 bringing the eqiiivalent of a series of 
 rollers ;//, m ^^ ?//,, into contact with the 
 sphere, thougli these are no longer of one 
 size, but vary from a diameter zero to a 
 diameter of the size of that of the sphere 
 
 Coming now to the instruments in 
 which the alternative device adopted for 
 the avoidance of slipping is by bringing 
 into .contact with one roller different 
 circles of the disk M, or of its equivalent. 
 
 This may be done in the following 
 way : Instead of allowing the cylinder 
 (m) to roll on the sphere M (Fig. 31), 
 and so to change the radius of the imag- 
 inary rolling circle (whose diameter is 
 qq') on which it rolls, suppose that the
 
 88 
 
 cylinder is kept in contact as shown by 
 the dotted lines, and the axis of rotation 
 zz' of the sphere is tui'ned, as, for in- 
 
 (Z) 
 
 stance, would happen if a sphere in com- 
 bination with rollers were used as sug- 
 gested for an anemometer by Mr. Ven- 
 tosa, through an equivalent amount, i. e., 
 through the angle a. This will give the
 
 89 
 
 same result as far as the rotation of the 
 cylinder is concerned, but with an im- 
 portant difference. The cyhnder (in Fig. 
 31) or sphere (in Fig. 33) is no longer 
 needed, and may be replaced by the orig- 
 inal measuring roller, whose axis has a 
 fixed position parallel to OX. It will be 
 seen that this device practically amounts- 
 to bringing different circles on the sphere 
 M into contact with the measuring roller 
 (ni), with the great advantage that ex- 
 actly the same circle on the sphere M is 
 scarcely likely to again roll in contact 
 with the roller {)/i), tliough of course the 
 radius maj' be the same. This method 
 has been recently proposed by the author^ 
 and the mode of carrying it out without 
 involving slipping, by what is called the 
 " sphere and roller mechanism," which 
 mechanism has been explained and de- 
 veloped at length in a paper before the 
 Royal Society. It need here be only re- 
 marked that the planimeter there de- 
 scribed, and afterwards exhibited to the 
 British Association at Montreal, was of 
 the linear form, and of little practical
 
 90 
 
 use ; but the author has since completed 
 a polar planimeter and exhibited it be- 
 fore the Koyal Society. 
 
 But one more area planimeter re- 
 mains to be mentioned, aod this is the 
 one invented and brought before the 
 Physical Society by ]\Ir. C. V. Boys. The 
 principle of action is briefly this : A 
 wheel or roller, which is not supposed 
 to slip sideways on the diagram, has its 
 plane of rotation kept always at an angle 
 a to the axis OX of the figiu-e to be inte- 
 grated, such that 
 
 2/= ordinate of the curve 
 =rtan axK 
 where K is a constant, and y is the ordi- 
 nate with resj^ect to OX of that point on 
 the curve which the pointer of the instru- 
 ment is at the same instant tracing. If 
 the component of a small motion of the 
 wheel parallel to OX is a a, and the com- 
 ponent of the same movement parallel 
 to OY is i\t. 
 
 Then 
 
 A^ 
 
 A.!' 
 
 = tan a — 
 
 y 
 
 K 
 
 A^ = 
 
 -y Axx 
 
 1 
 K'
 
 91 
 
 or the distance moved by the wheel par- 
 allel to the axis OY becomes the measure 
 of an element of area. It is easy to see 
 that the height moved by the wheel be- 
 comes a direct measure of the area of the 
 figure. Various examples of the action 
 of this planimeter, called by the inventor 
 the tangent integrator, are given by Mr. 
 Boys ; but the action is obviously limited, 
 and an investigation of the theory reveals 
 the fact that it is only a special case of 
 the general problem, not only of the 
 method of applying circles of varying 
 diameter to one roller, but of the sphere 
 and roller mechanism itself. This will 
 be rendered clearer by stating that, in 
 order to employ the component parallel 
 to OY, the roller was made to work 
 against a cylinder, which, by its turning, 
 acted as the measuring roller. Evidently 
 the length of the cylinder limited the 
 travel in that direction. The cylinder 
 was carried bodily along in the direction 
 of its axis (corresponding to OX), and 
 made to effect its own turning, the 
 amount of turning varying with the tan-
 
 92 
 
 gent of inclination of the wheel, and this 
 was sufficient in the application to the 
 steam-engine integrator to be hereafter 
 described, where the longitudinal motion 
 of the cylinder could be made propor- 
 tional to the stroke. Mr. Boys endeav- 
 ored, by various means, to obtain con- 
 tinuous motion in both directions, one 
 being equivalent to bending the ends of 
 the cylinder round, and so attempting to 
 solve the difficulty by what he has termed 
 a "mechanical smoke ring." The author, 
 however, by approaching the matter from 
 a different point of view, designed the 
 sphere and roller integrator, which is 
 nothing more or less than the inversion 
 of the mechanism of Mr. Boys. In this 
 the roller of Mr. Boys is replaced by the 
 sphere, and instead of the two motions, 
 one of the cylinder about its axis, and 
 one of the cylinder longitudinally, the 
 two rollers are used. It may be easily 
 shown that the turning of the plane of 
 rotation of the roller of the tangent in- 
 tegrator is equivalent to changing the 
 axis of the sphere in the sphere and 
 roller integrator.
 
 93 
 
 Moment Planimeters. 
 The moment of an area, and its mo- 
 ment of inertia about a given line, may 
 be obtained mechanically upon similar 
 principles to those by which a simple 
 area was obtained. If ABCDE, Fig. 1, 
 be the figure whose moment of area and 
 moment of inertia are required about any 
 line OX ; then, taking any element of 
 area AB, if // = height of upper portion 
 SB, then the moment of area of the 
 element SB about OX 
 
 is mz=area of SBx~- 
 
 Similarly, the moment of inertia of the 
 element is 
 
 1=^1/; AX. 
 
 The sum of an infinite number of 
 such expressions as these, when A x be- 
 come infinitely small, gives respectively 
 the moment of area and the moment of
 
 94 
 
 inertia of the whole figure according to 
 the expressions. 
 
 Moment of area =M = ^ /'yVZa;, 
 Moment of inertia = I ^=z^f (fdx. 
 
 Now, there are two possible ways of 
 obtaining these results mechanically. 
 One of these ways is by applying for the 
 purpose the suggestion made by Sir Wil- 
 liam Thomson in connection with the 
 disk globe and cylinder integrator of 
 Professor James Thomson, of using a 
 train of such mechanisms to obtain the 
 integration of a simple linear differen- 
 tial equation. By certain simple arrange- 
 ments. 
 
 The first mechanism would gi\efydx, 
 " second " " "- fy^dx. 
 
 " third " " " fy'dx. 
 
 This method need not be further con- 
 sidered here, since, so far as the author 
 is aware, it has never been carried into 
 actual practice. It maybe, however, said 
 that the mechanical difficulties in the way 
 of causing the measuring wheel or roller 
 of the first mechanism to actuate the sec- 
 ond, and the roller of the second to actu-
 
 95 
 
 ate the third, without introducing serious 
 error, are not easy to overcome, and re- 
 quire a very easily working piece of ap- 
 paratus. The author has discussed the 
 apphcations of the sphere and integrator 
 for tlie purpose, in a paper to the Koyal 
 Society. 
 
 The other principle is to cause the 
 measuring roller to be directly turned 
 at a rate which is made to vary, not as 
 in the simple planimeter with the value 
 of the ordinate (y), but with its second 
 or third power. Though no method of 
 directly doing this has apparently yet 
 been suggested, yet the same result is 
 practically effected by the beautiful appli- 
 cation of a mathematical principle in the 
 "moment integrator" of Professor Ams- 
 ler. 
 
 Let the pole-arm CB (Fig. 34) be at- 
 tached to a toothed segment {z^), one 
 portion of which gears with a toothed 
 wheel ^2, the radius being as 2 to 1. Let 
 the center C of s be carried along OX, 
 while the center of C„ of s^ is carried 
 along a line ox parallel to OX. Let m^
 
 96
 
 97 
 
 be a roller acting in every way as the 
 measuring roller of the Amsler planim- 
 eter, whose axis is carried in the plane of 
 the wheel z.^, its direction passing through 
 the center C^. When the pole-arm coin- 
 cides with OX, let the plane of rotation 
 of the roller m^ be parallel to OX, and 
 its axis parallel to OY. When the pole- 
 arm is turned through an angle SCB=:a, 
 the angular motion of the wheel z^ is 
 twice that of the arm ; thus the roller m^ 
 takes the position shown in the figure. 
 
 This is so because ^ — 
 
 / ^ motion of :c^ 
 
 radius 2, _2 
 radius z^ !> 
 
 .•./Kc-/=2a. 
 
 Suppose the pointer p to move through 
 the width of the element SB at a height 
 =:y, and with it z, and 2.^, the roller m^ 
 being in contact with the diagram sur- 
 face. Then, by what was proved in the 
 case of the Amsler planimeter, and adopt- 
 ing the same notation.
 
 98 
 
 Turning of yn^ _Jl7trn^_ lc„, 
 
 Motion of translation of t)}^ Ax c^K 
 
 =cos 2a 
 = 1-2 sin' a, 
 
 but — ^ = =(^r=sin a (where CB=R ), 
 
 27r7vi, -,0-2 -, 2 , 
 AX R/-^^' 
 
 or 
 
 ."' = (2Fr)^*^-(2^'y^^- 
 
 When the complete travel of the curve 
 has been made, the sum of a series of 
 quantities similar to the first becomes 
 zero ; so that, by making the constant 
 
 I — rr-gl equal ^, the reading of the roller 
 
 gives the value — 
 
 or the moment of area of the figure 
 BDEA. 
 
 For the moment of inertia the segment 
 of 2, is used, the radius of which is three 
 times that of another wheel z^, with 
 which it gears. The action of a roller 771^, 
 carried by the wheel z^, is exactly the 
 same as that of m^, except that its angu-
 
 99 
 
 lar motion is three times as great as the 
 pole-arm CB, instead of twice as great, 
 as in the case of the other roller. 
 
 By reasoning similar to that already 
 adoj)ted, and taking the plane of rotation 
 of rn^ perpendicular to OX in its initial 
 position, instead of, as in the former case, 
 parallel tolt — 
 
 travel of rti^ Inrn^ l'c„ 
 
 motion of translation 
 
 of 
 
 m.^ A X 
 
 -c,k' 
 
 
 
 
 ' = sin 3a 
 
 
 = 3 sin a- 
 
 -4 
 
 sin^ a. 
 
 
 SB 
 CB" 
 
 "K" 
 
 = sin «, 
 
 
 Therefore ? = 
 
 :3 sin 
 
 [ — 4 sin'a = 
 
 
 
 
 
 -B^- 
 
 
 n. .-/ ^ U 
 
 A, 0« 
 
 
 (. * \„. 
 
 A --.• 
 
 which, when the pointer is taken around 
 the curve, gives, wuth suitable values of 
 the constants, 
 
 = area of BDE A.— moment of in- 
 ertia of BDEA 
 =^ A -I. 
 
 or 1= A — ^<,.
 
 100 
 
 The instrument, Figs. 35, 36, has an 
 area planimeter attached to it, so that, by 
 reading the rollers m^ and m^^ and sub- 
 tracting the results, the moment of iner- 
 tia is obtained. 
 
 The details of the moment planimeter 
 shown (Figs. 35 and 36) are easily ex- 
 plained. A guide PQ of steel has a 
 groove gg, which is placed parallel to the 
 axis OX by means of the gauges G, one, 
 as shown, being at each end, which are 
 adjusted with their points q upon the 
 line OX. The rollers KR run in the 
 grooves gg, and support a frame FF, 
 which carries, by means of an axle J J, 
 the frame EE. This frame supports the 
 toothed segment 2,, and the two toothed 
 wheels z^, 2^, upon vertical axis. The 
 former between centers, one of which is 
 shown, Fig. 36, ^, the latter by steel axles 
 within the column ij,^. The pole-arm 
 carries, in addition to the pointer p at 
 the end, the roller m^ with its dial A,, 
 forming an ordinary planimeter, and is 
 itself carried on the centers s^s^. The 
 two other rollers and dials are shown as
 
 101
 
 102 
 
 rn^rn^ and hji^ respectively. The weight 
 W serves to balance the instrument, so 
 as to avoid undue pressure on the paper, 
 and the motion is so smooth as to en- 
 able a curve to be traced with the great- 
 est ease and accuracy. 
 
 Attention has been called by the late 
 Dr. Merrifield and others to the valuable 
 applications of this instrument for pur- 
 poses of naval architecture, but so far as 
 the author is aware, no account has been 
 given in this country of its applications 
 in civil engineering, as proposed by Pro- 
 fessor Amsler. The following brief ac- 
 count of the methods in the case of cal- 
 culating the contents of embankments, 
 cuttings, etc., is therefore given from an 
 abstract for which the autlior is indebted 
 to the kindness of Dr. A. Amsler : 
 
 Let Fig. 37 be the plan of a portion of 
 an embankment or cutting, the character 
 of which is supposed to be the same 
 throughout, viz., of uniform width of 
 roadway, and uniform side-slopes, the 
 surface of the ground, the gradient, and 
 the horizontal curvature of the roadway,
 
 103 
 
 being restricted in no way. AA' repre- 
 sents the center line of the railway ; 
 B^B/ and B^B/ its two borders ; C^C/ 
 and C^C/ the intersections of the side- 
 slopes with the surface of the ground. 
 Suppose now the embankment or cutting 
 
 Fig. sr 
 
 to be divided into thin layers by vertical 
 planes, perpendicular to the center line 
 AA' of the roadway ; PQ and PjQj may 
 be the intersections of two adjacent 
 planes with the plane of the diagram. 
 
 Then if p=area of section PQ, 
 
 As=interval between PQ and 
 and P,Q, measured upon 
 the center of gravity of 
 the section.
 
 104 
 
 Total volume of embankment is (from 
 one of the properties of Gnldinus) — 
 
 Y=fpds, 
 the integral extending over the whole 
 length of the embankment under con- 
 sideration. 
 
 There are three cases dealt with in the 
 
 Paper of Professor Amsler, correspond- 
 ing to the three forms of sections, I, II, 
 or III, Fig. 38. 
 
 The first of these, I, is simple enough, 
 since the center of gravity of the section 
 always coincides in plan with the center 
 line of the roadway, and the plan of oper- 
 ation is as follows : 
 
 Let Fig. 39 represent a longitudinal 
 section of a portion of the embankment 
 of uniform gradient, developed into a
 
 105
 
 106 
 
 23lane; the straight hne E'EE" represents 
 the top of the embankment ; G'GG" the 
 profile of the ground ; the straight Hne 
 A'AA'^ which is parallel to E'EE", is the 
 locus of the imaginary vertex of the 
 trapezoidal cross-sections. The level 
 line MN is the line to which the offsets 
 of the profile of the surface of the ground 
 refer. BG shows the intersection of a 
 vertical cross-section with the figure, and 
 AG the intersection of a plane perpen- 
 dicular to the top of the embankment 
 (and also to the line A'AA") with the 
 figure. 
 
 Let i= Z AGB=: / MNA=:gradient ; 
 
 y=:AG =di stance of 
 vertex to bot- 
 tom of em- 
 bankment ; 
 2/o=AE ^distance of 
 vertex to top 
 of embank- 
 ment ; 
 2 /i= angle at vertex at A. 
 
 It may be easily proved the area of 
 the section made by the plane AEG is — 
 p^{kG^'--kW) ian/i={y'-^;) tan/i
 
 107 
 
 but since Y=fpdx 
 
 Therefore volume = tan ftf {y'^—y^) <^^^- 
 
 And thus, if (i is known, the volume of 
 
 the portion E'E" G"G' (Fig. 39) is easily 
 
 found with the mechanical integrator, 
 
 thus: 
 
 Take A'AA" as the axis of moments, 
 and adjust the rail of the instrument so 
 as to be parallel to it. Start the pointer 
 anywhere on the shaded figure, and trace 
 round it ; the travel of the roller m^ being 
 denoted by M, the scale of the drawing 
 longitudinally being : 
 
 l"=m feet, 
 and vertically l"=7i feet; 
 
 then volume =:Y = 20 mn^ tan 
 
 /ixM cubic feet. 
 
 It only remains to insert a known 
 value for tan fi, which is easily done, 
 thus : 
 
 Let Fig. 40 be a perspective view of 
 the sections AEG and BG (Fig. 39), 
 where : 
 
 C G 
 
 Then from the diagram ^-J^=tan ;
 
 108 
 
 also 
 
 BG 
 
 = tan a 
 
 Therefore 
 
 or 
 
 ^ AG 
 
 and — — =cos i. 
 
 ±>(jr 
 
 tan a 
 
 tan /?=■ 
 \ cos ^/ 
 
 cos I 
 ,tan a> 
 
 ns.40 
 
 where Z a and Z ^' are known constants. 
 To complete the 
 - calculations for the 
 whole route separate 
 portions are taken, 
 with the various pro- 
 posed gradients. 
 
 The above formula 
 is exact for the integ- 
 rator shown in Fig. 
 36, as arranged for 
 English measures, a 
 complete revolution of 
 the measuring roller 
 being taken as a unit 
 of reading. 
 It is to be noted that nothing is 
 supposed as to the curvature of the 
 center line of the roadway horizontally, 
 as it is supposed to be developed in
 
 109 
 
 the figure. Also, that the aggregate 
 error arising from the assumptions that 
 the cross-sections are exact trapezoids 
 will in most cases be verj^ slight, on ac- 
 count of the errors in cuttings and those 
 in embankments partly compensating for 
 each other, in addition to the cutting and 
 filling in each section, as shown in Fig. 
 41, where the small triangular portion in 
 dotted lines C'DH represents the amount 
 taken off the former, and added to the 
 latter. 
 
 Alterations of the proposed roadway, 
 otherwise involving tedious calculations, 
 simply necessitate an alteration in the 
 line A'AA", and a repetition of the me- 
 chanical work of the integrator, but need 
 no fresh diagram. In preparing the 
 drawing, allowance should be made for 
 ditches along the roadway in cuttings, 
 which is easily done, as shown in Fig. 42, 
 where B^B,, which equalizes the amounts 
 taken and left, must be considered as the 
 roadway line. In the case shown in Fig. 
 43, the excess of the embanking over 
 the cutting is approximately equal to the
 
 110
 
 Ill
 
 112 
 
 layer above the dotted line CjC^. The 
 contents of this layer could be measured 
 either by considering it as an embank- 
 ment, and treating it as such, or by the 
 simpler — and for a first estimate suf- 
 ficiently accurate method — of assuming 
 its section to be a parallelogram. The 
 area of the shaded portion (Fig. 43) is 
 then simply to be measured, and the re- 
 sult, multiplied by the length of the road, 
 gives the required contents. The sup- 
 position that the slopes CD and CD' are 
 the same is also sufficiently accurate. 
 
 The foregoing is the first method de- 
 scribed by Professor Amsler, and is 
 extremely simple, but obviously only 
 approximately accurate. The two other 
 methods are capable of giving very 
 accurate results, and are dealt with by 
 him at considerable length. Only a short 
 account of them will be given here. 
 
 The first thing to be noted is that, as a 
 rule, the center of gravity of tbe section 
 will not really coincide in plan with the 
 center line of the roadway, but will curve 
 at the line SPP'S', Fig. 44, AA' being the
 
 I' 
 i'M
 
 114 
 
 true center'line. Thus, in the expression 
 fpds, the value of ds does not coincide 
 with dx^ as hitherto assumed. From the 
 figure it is seen that : 
 
 A s_K + e 
 
 where R is the radius of curvature of the 
 center Hue. 
 
 Therefore As=Aa3l + :^j 
 
 
 'pedx 
 
 or V=: J pds = J pdx, + J -^5—1 
 
 and this expression must be used. 
 
 The first of the two methods assumes 
 the base of section to be inclined, but 
 not broken (Fig. 38, II), and the side- 
 slopes, gradient, and radius of a given 
 portion to be constant. A diagram is 
 prepared, as shown in Fig. 45, in which 
 the dotted lines now represent intersec- 
 tion of the sides of embankment with the 
 surface of the ground, which do not, as 
 before, coincide with the contour of the 
 center line.
 
 115 
 From this figure y^=AI!> 
 
 2/?= Z at vertex A. 
 
 P^ig. 45 
 
 Then, by similar reasoning to that 
 previously employed, it may be proved 
 that : 
 Area of element section 
 
 and 
 
 ^p=y^i/, (2/,-?/J 
 
 tanVi
 
 116 
 .♦. V^tan ft / {y,y-y:) cU^^^^ 
 
 By a simple transformation this ex- 
 pression is brought into such a form as 
 to allow of mechanical integration. The 
 final formula being : 
 
 where 
 
 ^ ={/(}/' -y:)<i^-hr(y :-y:) dx- 
 
 hf{y-y,)'dx- 
 
 Another simple diagram has to be pre- 
 pared, and by means of three operations 
 of the integrator, the values of U and Y 
 are given thus : 
 
 U = 20x??i?i' (^1 + ^0 — ^3) 
 -^=mn' [320 {u^-u^-u;)-100 {lo- 
 
 where m and n have the significations 
 formerly explained, and u, v, and lo are 
 the respective readings of the area, mo-
 
 117 • 
 
 ment of area, and moment of inertia 
 rollers in each of the three operations. 
 Considering the great amount of calcula- 
 tion thus saved, and the accurate nature 
 of the results, this second method, al- 
 though involving rather more labor than 
 the first, is a very important one. 
 
 The third method, which deals with the 
 broken base, is much the same in princi- 
 ple as the second, but the expressions 
 become more complicated, whilst six 
 readings of the measuring rollers are in- 
 volved. The case of an embankment, 
 consisting partly of a cutting, is com- 
 pletely and accurately worked out by this 
 method. 
 
 The Paper of Professor Amsler con- 
 cludes with an example of the application 
 of his integrator to the problem of the 
 strength of a girder.
 
 118 
 
 Continuous Integrators. 
 
 Any piece of mechanism which contin- 
 uously adds np results may be regarded 
 as a continuous integrator. Of such in- 
 struments, revolution counters as em- 
 ployed in meters of various kinds, form 
 the simplest example, and correspond in 
 action to the devices already described, 
 by which the linear measurement of a 
 boundary is performed. These will not 
 be further referred to, and it is only nec- 
 essary to consider those computing 
 mechanisms which, dealing with the re- 
 sult of two simple imit measurements, 
 correspond in principle to area planim- 
 eters. 
 
 It appears that Poncelet, before the 
 year 1838, suggested the employment of 
 a continuous integrator for computing 
 the two factors in dynamometrical meas- 
 urements. This was described by Morin 
 in 1838, as applied in his " compteur " 
 for registering the work done by a team 
 of horses, dragging a loaded carriage at
 
 119 
 
 any given velocity over any length of 
 road. The princii^le employed was that 
 of the disk and roller, the use of which 
 as already shown, had been suggested 
 for application in a planimeter more than 
 twenty years before. In the case in 
 question, the disk was turned by an end- 
 less cord or band from one of the wheels 
 of the carriage, while the position of the 
 roller on the disk was caused to vary 
 with the tractive force, and its reading 
 thus gave the product of force and space, 
 or the actual work done. 
 
 In 1840, a Committee of the British 
 Association, consisting of Professor 
 Mosely, Mr. Enys, and Mr. Hodgkinson, 
 was appointed to procure the dynamo- 
 metrical apparatus of Mr. Poncelet, and 
 to obtain a series of experiments on the 
 duty of steam-engines by means of that 
 apparatus, the sum of £100 being placed 
 at their disposal for the purpose. The 
 report of this committee, in 1841, de- 
 scribes at length the "constant indi- 
 cator " of Professor Mosely, which was 
 in reality a continuous steam-engine in-
 
 120 
 
 tegrator. It was entirely a new instru- 
 ment, except that the principle of the 
 disk and roller was employed, and also 
 the traction springs of General Morin. 
 The pressure of steam was allowed to 
 act upon a piston so as to vary the posi- 
 tion of the measuring roller, while motion 
 was given to the surface of revolution by 
 means of the stroke of the engine. The 
 surface of revolution was a cone, which 
 was substituted for a disk, as by this ar- 
 rangement the rapidity of the changes of 
 velocity due to corresponding changes in 
 the position of the integrating wheel is 
 diminished in the same proportion in 
 which the sine of one-half the angle of 
 the cone is less than unity. The force 
 necessary to drive the integrating wheel 
 is diminished in the same proportion, and 
 therefore the chance of an error arising 
 from the slipping of the edge of the in- 
 tegrating wheel on the surface which 
 gives it motion. The reports of the 
 committee in 1842, 1843, and 1844 (which 
 was joined in the first of these years by 
 Dr. Pole), show that the action of the
 
 121 
 
 above instrument was, as far as could be 
 determined in its application to a single- 
 acting Cornish engine, very satisfactory, 
 but apparently no mention has been made 
 of it, or results obtained from it, in any 
 succeeding report. 
 
 Various other steam-engine integrators 
 have been brought forward since then, 
 most of them acting upon the same prin- 
 ciples ; amongst these is the recent power- 
 meter of Messrs. Ashton and Story, which 
 is described at considerable length in the 
 American edition of Weisbach's Mechan- 
 ics as apparently something new. It is, 
 however, the same instrument as Mose- 
 ley's integrator, except that it employs a 
 spiral instead of a straight spring, and 
 returns to the use of the flat disk. 
 
 The disk and roller has also been since 
 the time of Morin applied in many dy- 
 namometers, which thus become really 
 " ergometers " or " work " measurers. In 
 a series of articles which recently ap- 
 peared in La Lumiere Electrique those 
 of Hirn, Megy, Bourry, and Darwin, are 
 described as having a " totalizer '' or in-
 
 122 
 
 tegrator of this kind attached to them. 
 The position of the measuring roller 
 varies with the force exerted or trans- 
 mitted, and the motion of the disk with 
 the revolutions of the motor or machine. 
 Thus by suitable counting apparatus the 
 continuous product is given of force and 
 space, or work done. 
 
 A cone has sometimes been used in- 
 stead of the disk with dynamometers, as 
 in that of Baldwin and Eickemeyer, used 
 at the Centennial Exhibition in 1876 for 
 testing mowing machines, in which, in- 
 stead of a measuring roller, a cylinder is 
 placed with its axis parallel to one side of 
 the cone, while an endless band of round 
 cord is rolled along between the two sur- 
 faces, so as to transmit the varying mo- 
 tion of circles of different radius on the 
 cone. Fig. 46 shows the apparatus of 
 Mr. Koury applied to a transmission dy- 
 namometer. In this the force is trans- 
 mitted through a differential train of 
 three bevel wheels, the middle one of 
 which is attached to the spindle (S)» 
 which supports the weight (W), and is
 
 123 
 
 suspended from the main shaft by the 
 joint at (k), about which the whole can 
 turn. Thus, the deviation of the weight 
 
 from the vertical (as shown by the dotted 
 lines) changes with the force. The 
 change of position of the spindle (S) 
 causes a band {l>) to move along the sur-
 
 124 
 
 faces of revolution RR,, the upper one, R, 
 being turned from the shaft by the spin- 
 dle (ZZ). It is to be noted that the dis- 
 tance of the band (b) from its zero posi- 
 tion is not directly proportional to the 
 force represented by the change of posi- 
 tion of the weight, and, therefore, the 
 surfaces must be formed with a certain 
 curve, found by construction, in order 
 that the dial and counting apparatus at 
 D may correctly give the product of the 
 two variables, force and space, and so 
 the work transmitted through the dy- 
 namometer. 
 
 It cannot be said that continuous in- 
 tegrators of this kind are at present prac- 
 tically employed to any great extent. 
 There are probably two reasons for this. 
 One is the want of durable and reliable 
 instruments. The other, the question as 
 to how much, and to what degree they 
 are really needed. 
 
 With regard to the first of these, it is 
 evident that in all the arrangements 
 hitherto considered (with the exception 
 of Baldwin and Eickemeyer device) there
 
 125 
 
 Fia-. 47 
 
 is that slipping of surfaces in contact, 
 
 which, though of little effect as far as 
 wear goes in the 
 limited operations 
 of a planimeter, 
 becomes a very se- 
 rious considera- 
 tion when contin- 
 uous action is re- 
 quired to be main- 
 tained. The only 
 integrator of the 
 second, or non- 
 slipping class, 
 which, as far as 
 the author is 
 aware, has yet 
 been practically 
 applied, is the 
 " power- meter " 
 of Mr. Vernon 
 Boys. This instru- 
 ment is shown in 
 Figs. 47 and 48, 
 and acts upon the 
 
 same principle as Mr. Boys' integrator. 
 
 The piston C, subject to the varying
 
 126 
 
 pressure in the engine-cylinders, with 
 which the barrel A is connected by 
 the connections at B and B', is moved 
 up and down against or with the ten- 
 sion of the spring D ; its rod acting on 
 the arm g causes the plane of rotation 
 of the roller G to take positions more or 
 less inclined to the axis of the cylinder 
 H. This cylinder H is moved to and 
 fro with the stroke of the engine by 
 means of the cord L, Fig. 48, and the 
 roller G bemg in frictional contact with 
 it causes it to turn round to a greater or 
 less extent, according as the plane of G 
 is more or less inclined to the axis of H. 
 The amount of its revolution is registered 
 by the counting apparatus in I (Fig. 48), 
 to which the axis of H is geared, and is 
 thus a measure of the power of the en- 
 gine, for it gives the product of the tan- 
 gent of the angle to which G is inclined 
 and the distance moved through by H, 
 that is the product of pressure of steam 
 into the stroke of the engine. The steam 
 being (as originally in Moseley's and also 
 in subsequent integrators) supplied both
 
 127 
 
 ff 
 
 Fig. 48 
 
 ^ty= 
 
 frS 
 
 I I 
 
 
 B
 
 - 128 
 
 above and below the small piston, the ab- 
 solute pressure is given. Thus, in the 
 present case, as the change of pressure 
 on C at the beginning and end of each 
 stroke causes the rod of g to be alter- 
 nately above or below the axis of H, so 
 the motion of the cylinder to and fro will 
 always cause the cylinder H to turn in 
 one direction, and thus to continuously 
 integrate the work done. This device 
 only enables a reciprocating movement 
 of the cylinder H to be made, and the 
 author has already mentioned the device 
 of the sphere and rollers, which by the 
 inversion of the higher pair of Mr. Boys, 
 enables continuous motion to be obtained, 
 and is suitable for application in dyna- 
 mometers, electric-motors, and other jDur- 
 poses. 
 
 With regard to the want of such instru- 
 ments, a very strong case was made out 
 by the committee, already mentioned, in 
 their report in 1841, where the applica- 
 tion of a continuous integrator to steam- 
 engines was alone discussed. The appli- 
 cation has been made to electric-motors,
 
 129 
 
 and in trials of motors and machines 
 generally, and there is little doubt if con- 
 tinuous integrators combining the three 
 qualities of durability, accuracy and 
 cheapness could be produced, that in 
 these days of increased regard for meas- 
 urement of all kinds, there would be a 
 much larger and increasing application 
 of them. 
 
 Limits of Accuracy of Integrators. 
 
 In all calculating machines, accuracy of 
 the result must be the question of first 
 importance. Assuming the theory relied 
 on in the various instruments for the 
 mathematical operation to be correct, the 
 accuracy depends primarily upon the me- 
 chanical arrangements, though in the 
 case of planimeters it also depends upon 
 the skill and care of the manipulator, and 
 involves the question of a personal er- 
 ror. This latter point need not be con- 
 sidered, partly because this occurs more 
 or less in all results obtained by observers, 
 but also because it is less than might be
 
 130 
 
 at first anticipated, from the fact that in 
 tracing the pointer around the curve 
 there is no reason why the error due to 
 moving it on one side should exceed that 
 due to moving it on the other side, that 
 is, why equal errors of opposite effect 
 upon the final reading should not be made. 
 
 It has been seen that the action of all in- 
 tegrators, except mere revolution count- 
 ers, depends upon the motion of the meas- 
 uring roller, or its equivalent, over sur- 
 faces of various forms, therefore the 
 above-mentioned mechanical arrange- 
 ments resolve themselves into an exam- 
 ination of the nature of the frictional 
 contact of two surfaces. It was for this 
 reason that integrators have been classi- 
 fied according to the nature of this fric- 
 tional contact, and it now remains to in- 
 vestigate the nature of this, to show to 
 what the classification leads, to give the 
 direct results of experiments upon the 
 subject, and also the indirect results 
 obtained from the instruments them- 
 selves. 
 
 Planimeters and integrators generally 
 have been divided into —
 
 131 
 
 I, Those in whicli the frictional sur- 
 faces shp as well as roll over each 
 other. 
 
 II. Those in which slipping of the 
 
 Fis. 49 
 
 surfaces is supposed not to take place. 
 
 The order of this arrangement was 
 adopted upon historical grounds, and 
 also because the former class is at pres- 
 ent by far the most important ; but it 
 would be more convenient, upon mere
 
 132 
 
 grounds of mechanical simplicity, to in- 
 vert the order. 
 
 Let AB (Fig. 49) be the plan of the 
 measuring roller. Suppose a force ap- 
 plied in the direction OX, making an 
 angle (a) with the plan of the axis of 
 AB. 
 
 Let <:?=! OX = distance through which 
 the force acts. 
 
 1st. Suppose that frame which carries 
 the measuring roller is free to move in 
 any direction horizontally, but maintains 
 the plane of rotation of the roller verti- 
 cal, then the application of a force along 
 OX, at the center of AB, will cause it to 
 roll along the line coinciding in direction 
 with the plan of the center line AB of 
 the roller, that is, along the line OY. 
 This will always be the case, except 
 when this force is applied in the limiting 
 case in the direction perpendicular to 
 the plane of AB {i. e., when a = 0). 
 
 Thus, the distance in this case traveled 
 by the center of AB, which is the same 
 as the path rolled by it, is 
 
 sm a
 
 133 
 
 and the distance moved through by the 
 center at right angles to OX is 
 XY=^cot a. 
 
 The latter value is the one usually 
 taken or recorded by the instruments at 
 present in use, but depends directly upon 
 the former. 
 
 Next, suppose the frame carrying the 
 roller is constrained either by guides, as 
 in the linear planimeter, or by the radius 
 bar of the polar planimeter, or other- 
 wise, to move in the direction of OX, 
 that is, in the direction in which the force 
 acts. When the center of the roller has 
 reached the point X, that is, when the 
 force has been exerted through a dis- 
 tance OX, 
 
 Then 0Z = distance slipped by AB = 
 d cos a. 
 XZ==: distance slipped by 0B= 
 d sin «. 
 
 Upon the degree of accuracy with 
 which the above conditions are fulfilled 
 depends the correctness of the working 
 of all integrators ; for not only do these 
 two cases entirely" cover the action of
 
 134 
 
 the two classes of planimeters, and 
 the corresponding continuous integra- 
 tors, but one of the limiting cases in 
 each, viz., that in which the force acts in 
 the plane of rotation of the wheel (when 
 a=90°), represents the conditions under 
 which the wheel of the boundary meas- 
 urer or opisometer is employed. It may 
 be therefore said that the theory of me- 
 chanical action of integrators is based 
 upon one or other of the following as- 
 sumptions, in which the limiting case 
 (namely, when a=r90), is included. 
 
 Class I. — That the rollmg of the 
 planimiter, when slipping is allowed, is 
 
 Nj^X^jfZsin a. 
 
 Glass II. — That no slipping takes 
 place, which amounts to the assertion 
 that 
 
 ■'sm a 
 
 Nj and N^ being the readings in each 
 case, and k^ and h^ suitable constants for 
 the instruments. 
 
 It is easy to see that the first of these 
 
 J
 
 135 
 
 is really the assumption made for all in- 
 struments in Class I. ; but in the various 
 instruments in Class II., it is only with 
 the i^lanimeter of Mr. Boys that it be- 
 comes directly obvious that the above as- 
 sumption is made. With the others, 
 though it is less evident, nevertheless, it 
 will be found, on examination, to be 
 equally true that the second supposition 
 is really made, and that upon its truth 
 the correct action of all instruments in 
 the second class depends. The forces 
 acting in each of the two cases must 
 therefore be taken into consideration 
 and the mechanics of the problem exam- 
 ined. 
 
 Proceeding in order of simplicity, Class 
 II. will be examined first. 
 
 Let AB in both cases (Fig. 50) be the 
 plan of the measuring roller. 
 
 Let S= reaction of surface upon which 
 AB rolls, that is, the force 
 with which it is kept in con- 
 tact with it ; 
 
 /^= coefficient of friction between 
 roller and surface ;
 
 136 
 
 P=OC==reaction of surface, which 
 must be brought mto action 
 in a horizontal direction to 
 cause the roller to turn on 
 its axis. 
 
 Class 11. {Fig. 50).— Suppose the 
 frame in which the roller is carried to be 
 free to move in any direction horizontal- 
 ly, let a force be gradually applied at the 
 center of the roller AB in the direction 
 perpendicular to the plane of rotation. 
 This will produce no effect as long as it 
 is less than the maximum resistance, 
 which can be opposed by friction be- 
 tween the edge of the roller and the sur- 
 face upon which it rests, that is, as long 
 as 
 
 K=:OD (Fig. 50) <S//. 
 
 When the force K is equal to S//, and 
 acts within the angle 6, the roller AB 
 will move with uniform motion along the 
 line of action of the force without turn- 
 ing. The same thing will hold if, instead 
 of the force acting perpendicularly to 
 the plane of rotation, it acts at some 
 oblique angle to it not exceeding a cer-
 
 187 
 
 CLASS 2.
 
 138 
 
 tain value measured from the normal. 
 The limiting* value of this angle depends 
 on the resistance of the friction of the axle 
 of the roller AB to turning. Let this 
 angle be (0), and draw OE perpendicular 
 to OC, meeting the circle draw^ with O 
 as center and radius OD in E, and let 
 
 6^=angleE0D. 
 When the line of action of the force falls 
 without the angle 6, as, for instance, 
 when it takes place in direction OF, the 
 roller will still slip along the line of the 
 force, but the roller will now also turn. 
 The component in the plane of rotation 
 will now, however, be of a magnitude 
 such that the motion of rotation of the 
 roller is no longer uniform. Since only- 
 uniform motion is being considered to 
 take place, the conclusion is, that when 
 the force acts at an angle greater than 6 
 to the axis of rotation (^. e., when a>6) 
 it must never be so great as to cause 
 the roller AB to slip, and therefore only 
 a motion of pure rolling can take place. 
 By proper mechanical devices the roller 
 can be made to turn very easily, and angle 
 be kept very small. 
 
 i
 
 139 
 
 The magnitude of the force which 
 must be appHed in any position of the 
 roller to effect this motion, is 
 
 sin a 
 =Y cosec a, 
 
 and is at once given by the intercepts 
 drawn from O to OE in the construction, 
 shown in Fig. 50, for any other value 
 of a. 
 
 Class I. (Fig. 51) — Suppose that the 
 frame does oppose restraint, and that 
 this restraint is such as to always cause 
 the center of AB to move in the direc- 
 tion in which the force acts. Let OF 
 (Fig. 51) lie in this direction, making the 
 angle a with the axis of AB, draw EF 
 perpendicular to OF from the point E, 
 then by the triangle of forces. The force 
 required to move AB is 
 
 P^ = OF = Syucos {a-6). 
 
 The reaction which is supplied by the 
 frame is 
 
 QrrEF = S/<sin(a-0). 
 
 By describing a semicircle upon OE a
 
 140 
 
 Fig. 51 
 
 CLASS 1,
 
 141 
 
 construction is at once given, as shown 
 in Fig. 51, in which, by drawing radial 
 lines from O, the value of P^ for any 
 angle is at once given by the intercept. 
 
 The above investigation is by no means 
 a complete one, for this would require 
 a discussion of the moments acting, but 
 having obtained the above result by the 
 more complex method, the author con- 
 sidered it unnecessary to introduce a 
 more detailed jDroof than has been 
 given. 
 
 Comparing the two foregoing cases by 
 means of the diagrams in Figs. 50 and 
 51, Diagram AB, it is clear that the 
 forces acting always diifer, except in the 
 limiting cases; then 
 
 If a = dO° P,=P,=F=OC; 
 
 a= 0" P^ is <S//; 
 
 P, is ^S//. 
 
 In order to examine to what extent the 
 assumptions hold good upon which the 
 accuracy of integrators rests, the exjDeri- 
 mental determination of the following 
 points is needed:
 
 142 
 
 I. Case of boundary measures and lim- 
 iting case of botli classes of area and 
 moment integrators. — The conditions of 
 rolling of two surfaces when the force 
 acts in dii'ection of the plane of rotation. 
 
 II. Class I. of area and moment plan- 
 
 'F^i<^.CM2 
 
 imeters. — Whether the rolling is propor- 
 tional to the slipping, as assumed. 
 
 III. Class II. of area and moment plan- 
 imeters. — Whether there is any slipping 
 in this case, and if so, to what extent. 
 
 Case J. There are two separate prob- 
 lems under this head; one, where the 
 planes of revolution of the two rolling
 
 143 
 
 surfaces coincide ; and the other, when 
 they do not coincide. 
 
 Dr. A. Amsler has experimentally in- 
 vestigated the former thus : Two accu- 
 rately turned disks, SS', (Fig. 52), each 
 200 millimeters in diameter, were placed 
 with their edges in frictional contact, so 
 that the mark shown on each coincided 
 at the point of contact. By means of the 
 apparatus (Fig. 53) the lower one S w^as 
 turned by means of a band (BB') through 
 1700 revolutions one way (thus turning 
 the upper one S,), and then 1700 revolu- 
 tions in the opposite direction. The 
 marks which should have coincided if no 
 slij^ping had taken place, were now found 
 to be as in Fig. 52, the marks being a 
 distance = f„ = 0.05 millimeter apart. 
 
 The relative error is thus = ^ ^tt^, = 
 
 ('^'^""^^ 40,000,000 - This experiment is 
 only of value to show the error when the 
 w^heel travels back by turning in the op- 
 posite direction, and at the most, shows 
 that the error is nearly the same in both
 
 145 
 
 directions, aud does not prove anything 
 with regard to the action of the measur- 
 ing roller of the boundary measurer; 
 concerning this point, observations were 
 wanted upon the results of the first 1700 
 revolutions. A very good way of exam- 
 ining the point would be to note the diver- 
 gence of the marks in x revolutions when 
 S drives S,, then cause S^ to drive S back 
 through the same number of revolutions, 
 and it would be seen whether the diver- 
 gence was due to a difference in the per- 
 iphery of the wheels, or to a slipping of 
 the surface of contact. During Dr. Ams- 
 ler's experiments no one was allowed to 
 enter the room, in order to avoid altera- 
 tion of temperature. He found that the 
 heat radiated from the human body, or 
 even from a lighted candle placed at 
 some distance, had a perceptible influence 
 on the result. 
 
 Fig. 54 shows a most important case, 
 in which the plane of rotation of the sur- 
 faces in contact do not coincide with that 
 of the measuring roller {m) being actu- 
 ated by the disk M. Fig. 55 shows Dr.
 
 146 
 
 Amsler's apparatus for exaroining this 
 case, in which, if the edge of the roller 
 has any width, there must be slipping 
 action, even, though the force alwa^-s acts 
 
 
 
 Fig. 
 
 o4 
 
 
 
 
 \ 
 
 7Jl i 
 
 
 
 
 
 \ 
 
 /:>>-' — r~- ---. 
 
 
 
 ^ 
 
 ><^ 
 
 
 
 
 
 v-^ 
 
 y^ 1 
 
 
 
 
 
 ^^ 
 
 \ 
 
 
 ^x 
 
 
 / 
 
 
 \ ! 
 
 
 \ 
 
 
 
 
 \ 
 
 
 
 \ 
 
 / 
 
 
 \ ! 
 
 
 
 \ 
 
 1 
 
 
 \ 1 
 \ 
 
 
 M 
 
 \ 
 \ 
 
 1 
 
 
 
 
 \ 
 
 
 1 
 
 
 
 
 \ 
 
 
 1 
 
 \ 
 
 
 
 \ 
 
 
 1 
 
 \ 
 
 
 
 \ 
 
 
 1 
 1 
 
 \ 
 
 
 
 \ 
 
 
 / 
 
 \ 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 \ / 
 
 / 
 
 
 N 
 
 
 
 \.^ 
 
 
 
 
 "^■"■-.. j 
 
 ^^'' 
 
 
 in the direction of the plane of rotation. 
 The roller m which rests on the upper 
 surface of the disk, which latter has its 
 edge divided, and is in juxtaposition with 
 a vernier (y). The axis of the roller is 
 fixed, and its edge is thus kept always 
 vertically under a microscope (K). The
 
 147 
 
 :Fis.55
 
 148 
 
 position of the disk is noted, and it is 
 then moved forward about 8 revolutions 
 (or exactly 2,900°), which gives the roller 
 about 130 revolutions, and a mark is ob- 
 served on the latter. Then in theory the 
 result of giving 8 more revolutions to the 
 disk in the same direction should be to 
 bring the same mark of the roller under 
 the microscope, Practically the succes- 
 sive motions of the disk will be a little 
 different, so that the second advance of 
 the disk will not be exactly the same as 
 in the first case. The same mark on the 
 roller is, however, always brought under 
 the microscope, and. the difference in 
 turning of the disk is what is noted. 
 In the following table — 
 
 ^=: number of experiment, 
 (p= angle by which the disk differs 
 from last reading, 
 
 so that the second column gives the po- 
 sitions of the disk at the end of succes- 
 sive advances in which the roller is made 
 to take 130 complete revolutions, the 
 third column shows the travel of disk in 
 minutes (2,900° having, of course, to be 
 added to the readings). The fourth
 
 149 
 
 gives the difference between these and a 
 mean value. The last gives the ratio of 
 these differences to the travel. 
 
 T 
 -1: 
 1 
 
 s 
 
 1 
 
 05 
 
 of 
 
 COt> OOCOi>OOJ>0 
 
 oooooooooo 
 oooooooooo 
 
 1 1 1 1 1 + 1 ++ 1 
 
 7 
 -a- 
 1 
 
 1 
 
 OO^OOQOCOWOOOOOGO 
 
 •- oooooooooo 
 
 1 1 1 i 1 ++ 1 + 1 
 
 of 
 
 T 
 
 1 
 
 -SI 
 
 ^^^ssss^g^ 
 
 ^^^^^^^^^^^ 
 
 o * 
 
 •.s> 
 1 
 
 §§g55^gi^g^S 
 
 ^ 00 O^ rH O -t CO ^ O o 
 
 •^ 
 
 OTH05C0Tj<oi:Di>X)OO 
 1—1
 
 150 
 
 
 y/m 
 
 
 a. 
 
 
 
 
 
 CC 
 
 
 \A 
 
 £
 
 151 
 
 Case II. To test the results when a 
 roller partly rolls and partly slips, Dr. 
 Amsler used the apparatus shown (Fig. 
 56) in plan and elevation. In this C is a 
 carriage, running upon four wheels, on 
 the base (B), which has parallel grooves 
 planed in it ; the travel of the carriage 
 being limited by two stops at the ends. 
 Upon the surface of C the measuring 
 roller {iii) rests, being attached to a plate 
 A. By means of the graduations on A 
 the axis of {iii) can be set at any required 
 angle with reference to the direction of 
 motion of the carriage. The frame sup- 
 porting the roller is carried on the disk 
 by means of pivots, so as to allow (m) to 
 rest on the surface of C with the constant 
 pressure of its weight. 
 
 If a wangle of axis of {jii) in the direc- 
 tion of the motion of the car- 
 riage ; 
 
 s= motion of the carriage ; 
 ^< = turning of the roller; 
 
 Then w=s sin a = s cos( ^ — aj=s cos /i.
 
 152 
 
 If <7>= actual reading of vernier v 
 and (p=q)^ when = 90 or /i = 0; 
 then u=zs cos (^ — ^o)' 
 
 iL=s COS {(p„ — cp^), etc. ; 
 Then tan ", '^"^ ■^,-". '^»^ ■?,, 
 
 COS {(p,-Cp^) COS (^,- ^)o 
 
 In the following Table, which repre- 
 sents the results of experiments when 
 the disk was covered with a surface of 
 pear-tree wood, carefully polished (paper 
 being, however, found to afiord almost 
 as good results) : 
 
 z=as before, the number of the experi- 
 ment ; 
 
 Bi= angle of inclination of roller for ex- 
 periment (/) ; 
 
 «/^= motion of roller as observed for ex- 
 periment (e) ; 
 
 ?^i= motion of roller as calculated.
 
 153 
 
 
 
 1 
 
 OrHC0Ti<.rHO-.---rHOTHO 
 
 1 + + + + + + 1 1 
 
 ^ 
 
 
 -d 
 o 
 
 
 =3 . 
 3 ""^ 
 
 a ^ ^ T-> r^ CI y^ '^ a> a <^ 
 
 "^ lO r^ CO IC C^i O O O "^ tH 
 CO IC *-■? > O C5 t> L- C> w ^ 
 
 1 ^' 
 
 o 
 
 ociCia:)OOOco-<*rt^THr-i 
 
 
 , 1 
 
 
 perim 
 u'. 
 
 ci c-> 1-1 1- o o o t- o o L-r 
 ■"^ o 1-H o »o T— 1 <o o c; o th 
 O ic CT c5 O C: I- i?5 0> O rt^ 
 
 c:ooxoo-,oOTi^'*TH,-H 
 
 
 
 i 
 1 
 
 ; O O O O O tS O CO O CO O 
 CO CO CO JC 
 
 ■^ 
 
 ^ oocoooo^^cgoo 
 
 II 
 
 o OOOC:COt--^OCQCOOi-H 
 (MC^l-^TfCOCOOOOO 
 
 ^ 
 
 
 
 Oi-KNCOTtOCOt'XCSO 
 
 Class III. — The actual conditions of 
 motion when a force smaller than the 
 component of S/^, acts obliquely to the 
 plane of rotation of the measuring roller, 
 do not appear to have been made the 
 subject of direct experiment. It is ap-
 
 154 
 
 parently always tacitly assumed that no 
 slipping takes place. But this crucial 
 point cannot be thus left to mere con- 
 jecture, and the author has designed a 
 method of carefully testing this, which 
 he has not yet been able to properly 
 carry out. From a few rough observa- 
 tions, there seems little doubt, howevei, 
 that some slipping always does take 
 place, and that its amount is, in the 
 limiting cases, by no means inconsider- 
 able. 
 
 Lastly, a few words may be said con- 
 cerning the work of Professor W. Tinter 
 and of Professor Lorber. The former 
 has examined most carefully no less than 
 nine different planimeters, from which he 
 concludes that the different angles at 
 which the measuring roller of the jDolar 
 planimeters has little effect upon the re- 
 sult, and that, taking one turn of the 
 measuring roller as fz=100 square cm., 
 the average error in the reading was 
 only from ^0.00075 to 0.0013, according 
 as the center of rotation was without or 
 within the area to be measured. The
 
 155 
 
 work of Professor Lorber is so exten- 
 sive and elaborate that it is impossible 
 to do more than give in the most brief 
 form the results at which he has arrived 
 after many thousands of experiments. 
 He concludes that error in the reading is 
 always represented by an equation of 
 the form — 
 
 dn=K ' -\-M\/ny 
 (172= the error in the reading, 
 K and // being con- 
 stants, 
 where ??=:the reading of the meas- 
 
 uring roller; 
 
 the above equation gives rise to one of 
 the following form : 
 
 where r= actual area to be meas- 
 
 ured, 
 
 and ^/F,i = tl]e error in the result ex- 
 
 pressed in terms of the 
 area. 
 
 The following are the results given in 
 his latest paper : 
 Linear planimeter dF 
 
 = 0.00081/+ 0.00087^F7
 
 156 
 
 Polar planimeter 
 
 = 0.00126/ + 0.00022 VF/ 
 Precision polar planimeter 
 
 = 0.00069/+ 0.00018 ^jy' 
 Freely swinging planimeter 
 
 = 0.00060/"+ 0.00026y'jy 
 Simple plate planimeter 
 
 = 0.00056/+ 0.00084^1/ 
 Rolling (Coradi) planimeter 
 
 = 0.0009/+ 0.0006 ^jy 
 
 The degree of accuracy represented 
 by these results may be inferred from 
 the fact that in one case of the last plan- 
 imeter, when 
 
 d¥ 1 
 /=100 the relative ^^'I'^^'^-p ^fg-ggQ"
 
 157 
 
 Discussion. 
 
 Sir Frederick Bramwel], President, 
 said that the paper having been read only 
 in abstract, there had been no mention 
 whatever of what the author himself had 
 done. The members would no doubt, 
 under the circumstances, allow him con- 
 siderable latitude in personally explain- 
 ing the apparatus on the table. He would 
 not, however, ask them to defer the ex- 
 pression of their thanks to Professor 
 Shaw for his valuable paper until this 
 explanation was given ; but he felt sure 
 that after this was done, it would be still 
 more clear that those thanks were well 
 deserved. 
 
 Professor H. S. Hele Shaw said that 
 the paper had been only read in the form 
 of a brief abstract because from the nature 
 of the subject, and its method of treat- 
 ment, it appeared advisable that he should 
 personally give a short accouut of its 
 contents. The engravings, of which
 
 158 
 
 there were a good many, could not be 
 prepared in time to be sent to those who 
 were Kkely to take part in the discussion, 
 and, therefore, he would explain the 
 principal points which he believed to be 
 original, and which he hoped would be 
 thoroughly discussed. He would take 
 this opportunity of thanking Professor 
 Amsler for kindly lending him several 
 instruments, some of which were now 
 shown for the first time in this country, 
 having been sent from Switzerland for 
 the pur^DOse, at Professor Amsler's own 
 expense. He also wished to thank Mr. 
 C. V. Boys for lending him models of his 
 tangent integrators, and also two of the 
 actual instruments to exhibit ; and his 
 friend and colleague, Mr. C. D. Selman, 
 for several valuable suggestions and as- 
 sistance in the preparation of some of the 
 diagrams. The author then proceeded 
 to explain, by means of the diagrams on 
 the walls, and by models which had been 
 constructed on a large scale, the princi- 
 ples of the classification adopted in the 
 paper and the various instruments exhib- 
 ited.
 
 159 
 
 Mr. William Anderson (of Erith) ob- 
 served that he had had considerable ex- 
 perience with continuous integrators in 
 measuring work done by agricultural 
 implements. There was a good deal to 
 be said about the use of those instru- 
 ments and the defects to which they were 
 liable, about the personal error, which 
 was an important point, and errors from 
 imperfect adjustment. The conclusion 
 at which he had arrived with regard to 
 continuous integrators, in which the space 
 passed over and the strain were multi- 
 plied together and registered continu- 
 ously, was that they were exceedingly 
 good for comparative results, but were 
 not altogether to be trusted for positive 
 indications. In comparing, for example, 
 a number of machines working in a field 
 under similar circumstances as to weather 
 and everything else, with the same oper- 
 ator, the comparative results would be 
 trustworthy ; but if there were any vari- 
 ation in any of the conditions, they would 
 not. There was always a good deal of 
 doubt about the positive results. The
 
 160 
 
 causes of error were these. In the con- 
 tinuous integrators the integrating wheel 
 was attached to a train of wheel -work 
 which possessed a considerable amount 
 of inertia and friction. In planimeters, 
 and integrators of that class, where the 
 observations could be made slowly, at a 
 steady speed, and where the conditions 
 did not vary, inertia did not count for 
 much ; but in the steam-engine, and in 
 agricultural implements or in traction in- 
 dicators, there were great and rapid vari- 
 ations of speed. The sudden strains 
 which were put on hj the tractive force 
 shifted the integrating wheel along the 
 disk suddenly, and the speed changed in 
 a similar manner. The force necessary 
 to accelerate the movement of the inte- 
 grating wheel and its mechanism tended 
 to cause a slip, which was partly counter- 
 acted by a slip produced by the force 
 necessary to arrest its motion when the 
 speed changed, but the friction of the 
 mechanism always acted in the same di- 
 rection, tending to augment the error. 
 "With one implement, for example, if there
 
 161 
 
 were a tolerably steady pull with no sud- 
 den variation in the velocity, there would 
 be one amount of error, whereas if an- 
 other implement were worked in a jerky 
 fashion there would be a totally different 
 amount of error. With regard to per- 
 sonal error, to which the author did not 
 attach much importance, he had reason 
 to think that it was of great consequence. 
 The degree of care and skill exhibited 
 in adjusting the instrument and taking 
 the measurements had an important in- 
 fluence ; he was therefore always careful, 
 in a series of experiments with agricultu- 
 ral implements, to have the same ob- 
 server throughout, if possible, because 
 the results aimed at were rather compar- 
 ative than positive. Still, he was bound 
 to say that, with care and experience, 
 satisfactory positive results could be at- 
 tained. The author did not appear to be 
 aware of the extensive use which the 
 Royal Agricultural Society had made of 
 the integrator which had always been 
 known as Morin's, but it appeared that 
 honor had been ascribed where it was
 
 162 
 
 not strictly due ; probably the reputation 
 of the great French mechanic, who had 
 done so much to introduce it, had ob- 
 scured the claims of the real inventor. 
 For the last thirty-five years the Royal 
 Agricultural Society had used continuous 
 integrators, and he thought that there 
 was no one more competent to speak of 
 their action than Sir Frederick Bram- 
 well, who had himself conducted many 
 of the experiments. The issues had often 
 been very important, involving the for- 
 tunes of manufacturers of agricultural 
 implements, which had, in a great meas- 
 ure, hung upon the indications of the 
 dynamometers. Most of the apparatus 
 used by the Eoyal Agricultural Society 
 had been designed and constructed by 
 the late Mr. C. E. Amos, M. Inst. C. E., 
 and by his son Mr. J. C. Amos, who for 
 many years filled the office of Consulting 
 Engineers. With reference to continu- 
 ous indicators for steam-engines, he had 
 little or no experience. He had tried 
 them, but the results had not been, so 
 far, satisfactory. One of the chief defects 
 
 1
 
 163 
 
 was the difficulty of keeping the little in- 
 tegratiug wheel perfectly free from flats. 
 It was not easy to find any metal per- 
 fectly uniform throughout ; and if it were 
 not uniform, a fiat would soon form, and 
 then all the results would be utterly un- 
 trustworthy, because the wheel tended to 
 hesitate at the fiat place. Formerly in- 
 tegrating wheels were made of gun-metal, 
 cast with great care, and under a great 
 deal of head. Latterly they had been 
 made of steel, and a better result had 
 been obtained. But even when the wheel 
 wore uniformly, if the width of the sur- 
 face in contact with the disk varied, then 
 again there was a source of error, because 
 the surface of the integrating wheel lying 
 upon the disk became greater, and then 
 there was uncertainty as to the true di- 
 ameter of the periphery of the disk on 
 which the integrating wheel was Avorking. 
 The only way of eliminating these errors 
 was by repeated testing of the appar- 
 atus. 
 
 Mr. C. Vernon Boys said the first part 
 of the paper on which he desired to say
 
 164 
 
 a few words was the division of the sub- 
 ject into different classes. The author 
 had referred to one system of classifica- 
 tion which Mr. Boys had adopted in a 
 jDaper published in the Philosophical 
 Magazine, a division into three classes. 
 The author had rightly shown that the 
 two classes which Mr. Boys had called 
 the " radius class," and the " Amsler 
 class.'' were in a mechanical sense, that 
 was, so far as the connection between 
 the surface of the integrating wheel or 
 roller and the surface on which it worked 
 was concerned, absolutely identical. But 
 though that was undoubtedly the case, he 
 thought the division might still hold 
 good, for an inventor could not have con- 
 trived machines in one class or in the 
 other without having had some such sys- 
 tem of division in his mind at the time 
 of the invention. There was one con- 
 siderable omission in the paper, and that 
 was the only point on which he felt it 
 necessary to find fault. There was no 
 mention of a very large series of most 
 beautiful machines, designed, and he be-
 
 165 
 
 lieved partly constructed, by the author 
 himself. As those mstruments had been 
 fully described in a paper by Professor 
 Shaw before the Royal Society, possibly 
 he thought that they were so well known 
 that it was unnecessary to describe them 
 again ; but it would certainly have ren- 
 dered the paper far more complete and 
 valuable if that large amount of work had 
 been incorporated in it. Of all the in- 
 struments brought before them, he 
 thought that the new precision phmim- 
 eters and that extraordinary spider- look- 
 ing instrument criuvling on the sphere, 
 were those which called for the utmost 
 admiration. It was impossible to look 
 at them or to use them without being 
 impressed with the extreme mechanical 
 beauty of their construction and design. 
 But though that was undoubtedly the 
 case, he thought that no one could see 
 some of the combinations of a sphere 
 with three, eight, or six little w^heels 
 round it, inventions of the author, with- 
 out classing them even above those in- 
 struments in point of beauty.
 
 166 
 
 The author had spoken of the tangent 
 principle as being a particular case of 
 machines which worked without slipping. 
 This was perfectly true ; but there was a 
 very great distinction between integrators 
 dej^ending upon the tangent action and 
 integrators depending upon a rolling ac- 
 tion of the radius class, to which alone 
 the other machines belonged. There was 
 one accidental mistake in the wall dia- 
 grams representing two spheres which he 
 desired to point out, as it was a little 
 puzzling. In the instrument of Clerk 
 Maxwell, one of the non-slipping radius 
 class, the axes of the two sp'ieres were 
 shown in such a position that rolling 
 would ultimately make them coincide. In 
 reality, the equator of one rolled over the 
 pole of the other, as was obvious to those 
 who had anything to do with integrators. 
 The distinction between instruments of 
 the tangent class and those of the radius 
 class might be represented by the little 
 wheel of the instrument. The accuracy 
 of integrators of the radius class depended 
 upon the exact size of the wheel, and the 
 
 I
 
 167 
 
 exact size of the surface upon which it 
 rolled; and, as Mr. Anderson had re- 
 marked, it was very much impaired when 
 flats formed upon any of the surfaces in 
 contact, for there was then a little hesita- 
 tion. The mere action of integrating 
 machines in which there was slipping was 
 sure to produce those flats at some time 
 or other, so that the time they were likely 
 to last and the amount of work they were 
 capable of doing were limited. The actual 
 size of the roller was of importance, for 
 as it wore and became gradually smaller, 
 the number of rotations were affected, 
 and therefore the recorded result gradu- 
 ally increased. These were the objections 
 to instruments of what he had called 
 the radius type. In the class of instru- 
 ments at which he had worked exclusively, 
 and which he believed he had originated 
 until he found that to a certain extent 
 Mr. B. Abdank- Abakan owicz had pre- 
 ceded him, the wheel was allowed to roll 
 along just in the direction in which it 
 was pointing. Anyone who had been in 
 the streets of London must have noticed
 
 168 
 
 with wliat extraordinary persistence and 
 power the wheels of all vehicles went in 
 the direction in which they were point- 
 ing. A butcher's cart driven round a 
 corner with tremendous fury would often, 
 in sjDite of its jumping on the ground, 
 still continue its course with very little 
 side slip. If the springs were sufficiently 
 good, or the road sufficiently smooth, 
 so that the wheel kept in contact with 
 the ground it would apparently not slip 
 at all. In the case of the first instrument 
 which he had made, which he had called 
 the cart integrator, he was astonished to 
 find with what extraordinary accuracy 
 results were obtained by its use, such, for 
 example, as the area of a circle and other 
 things that were known. In that instru- 
 ment he had depended entirely upon the 
 fact that the steering wheel of a tricycle 
 would go along in the direction in which 
 it pointed. The instrument had to pull 
 a heavy brass cart after it and slide in a 
 comparatively roughly-made groove, but 
 even so he found the value of ;r, on squar- 
 ing the circle, came out 3.14 when using
 
 169 
 
 a very rough home-made instrument. It 
 seemed, then, that if the sources of side 
 slip which were undoubtedly present, 
 due to the great friction that had to be 
 overcome in dragging the cart, could be 
 removed — that was, if the ground under 
 the cart could be made easily movable, 
 which was the case when it was converted 
 into a cylinder, the cylinder would follow 
 exactl}^ in the direction in which it should 
 go. For that reason an integrator of 
 that class was free from the objections 
 which had been raised, that in time inte- 
 grators wore out, the little wheel got 
 flats upon it, and as the size varied the 
 record varied. In the case in point there 
 was no slii)ping and no tendency to make 
 flats, and if the wheel was made half or 
 twice the size the record was the same, 
 for it only depended on the direction in 
 which it wanted to go, not upon the 
 amount that the wheel turned in going 
 along in that direction. He desired to 
 say a word or two with regard to an in- 
 strument, his engine-power meter, which 
 was not so well known as he had hoped
 
 170 
 
 it might be. There were certain sources 
 of error apparently present in it, but 
 which were really imaginary. In the 
 first place, it would seem that if the posi- 
 tion of the piston-rod, and, therefore, the 
 angular position of the little tangent 
 wheel G varied in the least, then as the 
 cylinder H traveled along there would be 
 an error due to the angular want of true 
 precision. But that was not so, for sup- 
 posing the spring to be too long or too 
 short, and the tangent wheel to be per- 
 manently deflected, when there was no 
 steam pressure, at an angle, say, of 2° or 
 3°, then as the cylinder moved in one di- 
 rection, the wheel would run up a certain 
 slope, and when the cylinder moved in the 
 other direction, it would run back along 
 the same slope, and so far as that was 
 concerned, no error would be introduced. 
 The wheel might be set at a permanent 
 angle, and the roller work backwards and 
 forwards, and nothing would be recorded 
 at the end of the operation. He could 
 show it with a rough paper and wood 
 model. If he permanently compelled
 
 171 
 
 the tangent wheel to assume a certain 
 angle by holdmg it with his thumb, as 
 the cylinder traveled in one direction, it 
 would rotate a certain amount, and when 
 it traveled in the other direction it would 
 rotate back again, and on the dial there 
 would be no permanent record. The 
 disk-cylinder integrator had one serious 
 defect ; it was very difficult to apply it to 
 cases in which growth of time or of mo- 
 tion was continuous. If it was desired 
 to have a time-integral and a motion-in- 
 tegral, when the motion was continuous, 
 it could not be easily applied, because 
 when the cylinder had got to one end of 
 the stroke, there was nothing for it but 
 to stop and come back again. It was, 
 therefore, necessary to apply a mangle 
 motion, so that as the motion was con- 
 tinuous the cylinder went backwards and 
 forwards ; then by causing the cylinder 
 to work between two tangent wheels al- 
 ternately, or by letting the mangle-motion 
 work ordinary reversing gear between 
 the cylinder and the recording mechan- 
 ism, continuous integration could be ef-
 
 172 
 
 fected. But of course the instrument, 
 where continuous integration was con- 
 cerned, would not compare with the 
 spherical integrators designed by the 
 author. On the other hand he did not 
 think that any instrument, in the peculiar 
 case presented by the automatic integra- 
 tion of an engine diagram, could compare 
 with it, for those peculiarities of motion 
 which interfered with the ordinary form 
 of radius machines, by causing a perpet- 
 ual scrubbing, to which reference had 
 been made, produced no trouble at all, 
 for there was no side-slipping, and be- 
 cause the very large moment of the radius 
 integrator was replaced by the extremely 
 small moment of a little cup containing 
 a wheel not the size of a three-penny bit, 
 and by a little bar of steel. In fact, the 
 moment of the piston and its attachments 
 was nothing like so great as that of an 
 ordinary Richards Indicator, because 
 there was no multii^lication of motion. 
 
 Mr. W. Anderson wished to state that 
 he had been able to bring to the meeting 
 the integrating part of one of the old
 
 173 
 
 dynamometers of the Royal Agricultural 
 Society. The date of its construction 
 was unknown, but he believed it w^as 
 about the year 1848. 
 
 ■ Mr. J. G. Mair observed that the au- 
 thor had not given himself sujB&cient 
 credit for the machine he had invented, 
 and he would like to ask him if the read- 
 ing of the counter gave such accurate 
 results as the vernier on Amsler's plan- 
 imeter. To read by means of a counter 
 was of great advantage, and especially so 
 where large numbers of indicator dia- 
 grams had to be taken out ; on some of 
 the engine trials he had made, three 
 hundred diagrams had to be averaged, and 
 constantly reading the vernier on Profes- 
 sor Amsler's instrument was trying to 
 the eyes. The application of the inte- 
 grating machine as a power meter was a 
 most useful one, and he had made several 
 trials with the one invented by Mr. 0. 
 Vernon Boys ; on three trials the read- 
 ings were 68.5, 68.2, and 73 by the me- 
 ter, against 67, 67.5, and 72 in the pump. 
 The pump-power was taken from the dis-
 
 174 
 
 placement of the pump piston, and the 
 head as shown on a mercury gauge. He 
 thought those results were a proof of the 
 correctness of the iustrumeut. He did 
 not think that reading a counter was 
 more difficult than reading the dial of a 
 watch. One other measurement had to 
 be made, namely, the distance between 
 the stops, and as that could be measured 
 with an ordinary rule, a child could 
 almost make the simple calculation, so 
 that he did not think personal error very 
 much affected such an instrument. There 
 was naturally a good deal of diffidence in 
 using a machine, the details of which 
 were not thoroughly grasped. With an 
 indicator diagram, there was something 
 to see which was readily understood ; but 
 where compound measurement was shown 
 on a counter, it was at first difficult to 
 realize the reading as correct. As soon, 
 however, as the instruments were better 
 known, he had no hesitation in saying 
 that, where the absolute shape of an in- 
 dicator diagram was of no consequence, 
 the continuous integrator would entirely
 
 175 
 
 supersede all other means of measuring 
 power. 
 
 Mr. Druitt Halpin said that the author 
 had referred to one of the minor im- 
 provements by putting a locking-spring 
 to the frame. Mr. Mair had noticed the 
 great difficulty there was in reading the 
 instrument, on account of the small scale 
 to which it was graduated, l)ut the author 
 had not completely followed the idea of 
 adding the locking arrangement. The 
 convenience of reading the instrument 
 was doubtless very great, for it could be 
 taken up and put in a good light, but the 
 real object of the locking gear was to 
 make the instrument do its own addition, 
 and also carry over the decimal places 
 now lost. Instead of taking each dia- 
 gram by itself, and writing down the 
 result of each separately, and adding 
 them up, the instrument w^as set at and 
 and locked ; the instrument was run over 
 the diagram, and it was locked, and so on. 
 So that by taking ten diagrams succes- 
 sively, and putting the decimal point one 
 point back, the instrument was made to do
 
 176 
 
 its own addition It bad the further ad- 
 vantage that, whatever the last decimal 
 might be, it was carried on by the instru- 
 ment. If it was 2.38, it could not be 
 said whether it w^as 2.389 or 2.381. With 
 regard to Mr. Boys' application of these 
 instruments, he had had an opportunity 
 of testing it on an engine of 120 or 130 
 I. H. P., and he found the results coin- 
 cided within from 1 to 2 per cent, of the 
 indicated power which was obtained with 
 standard indicators. With Mr. Webb's 
 permission, he also put one of his power- 
 meters on the large rail mill at Crewe, 
 and took a series of observations there 
 of the exact power required to roll a rail, 
 from the moment the ingot touched the 
 rolls to the moment the next ingot 
 touched them, which gave the true pow- 
 er, correction being made for any differ- 
 ence of velocity in the fly-wheel. It gave 
 the power both while the engine was 
 running with the bloom in the mill and 
 when it was running empty. Whether 
 the machine in its j)resent form would be 
 suitable for locomotives, he was hardly
 
 177 
 
 prepared to say, because he feared that 
 the attachment which was provided for 
 taking diagrams was, perhaps, so heavy 
 that its inertia would interfere with the 
 correct action of the instrument. But, if 
 that was Hghtened, or the attachment 
 left out, he was sure the best results 
 would be obtained from it. 
 
 Mr. H. Cunynghame said that his at- 
 tention had been called a great deal to 
 Mr. Boys' machines, in the development 
 of which he had assisted, as they were 
 first designed for practical application to 
 steam-engines. He had made many at- 
 tempts to apply continuous integrators 
 for the purpose of integrating electric 
 power, and he certainly could safely bear 
 out all that Mr. Anderson had said about 
 the imperfections of integrators of the 
 slipping type. If a wheel was running 
 in any direction in which it was likely to 
 slip upon the surface, it would be rubbed 
 into facets, and in like manner, a sphere 
 would be rubbed into a polyhedron. In 
 this state both wheel and sphere would 
 be worse than useless for integration.
 
 178 
 
 But if these machines had an integrator 
 of the roller type, even if it had facets 
 already, the rolling wonld take the facets 
 out of it just as an apothecary rolled his 
 pills. Instead of an integrating wheel 
 of hard steel, as was necessary in all in- 
 tegrating machines of the slipping type, 
 in those of the rolling type, soft metal 
 might be used, and the more it was used 
 the better it would get. Moreover, in 
 such machines, since what was counted 
 was not the revolutions of the wheel, but 
 the revolutions of something caused to 
 roll by means of the wheel, the accuracy 
 of shape of the rolling wheel became of 
 minor importance, and even if it " skated" 
 over the surface instead of rolling, a fair 
 result would be obtained. He thought 
 Mr. Anderson's remarks were somewhat 
 unjust to planimeters of the rolling type, 
 which he did not think had been brought 
 before the public so as to be within the 
 experience of anyone. That led him to 
 say a few words on Mr. Boys' steam- 
 power meter. He believed that Mr. Boys 
 and he were the first in this country to
 
 179 
 
 make a trial of those machines. They 
 made a trial at the works of Messrs. Ran- 
 some & Jocelyn at Battersea, and for 
 that purpose they had an extremely good 
 indicator made by Messrs. Elliot. It 
 would be quite understood that Mr. 
 Boys' indicator was not in any way cali- 
 brated by mere trial. They took the 
 measurements of the engine and the di- 
 ameter of the cylinder. Then thej^ took 
 the amount by which a certain pressure 
 of steam would cause the spring to rise, 
 just as was done before taking the H. P., 
 by means of a Richards Indicator. They 
 thought that if those two machines were 
 used upon the same engine, and if the re- 
 sults given by the two machines, namely, 
 Richards and Boys, were independently 
 calculated from the constants of the two 
 machines, then, if these results nearly 
 corresponded, a very remarkable coinci- 
 dence of testimony would be obtained. 
 They therefore put Boys' machine on the 
 engine alternately with Richards', and he 
 thought they must have alternated twenty 
 times, and the results of Boys' machine
 
 180 
 
 were found to be uniformly, during the 
 first series of experiments, about 23 per 
 cent, too high. That puzzled them ex- 
 tremely, but upon examination they found 
 that the manufacturer, instead of making 
 the piston 1 inch in diameter, had made 
 it 1 square inch in area, and when they 
 made the proper correction they found 
 the results to agree to about Ih per cent., 
 and with those stated by Mr. Hal^Din and 
 Mr. Mair. Mr. Mair's tests were made 
 in this way: he raised, by means of a 
 pump, a given weight of water through a 
 number of feet, and then he estimated 
 the foot pounds and compared them with 
 one another, and the result showed a re- 
 markable degree of accuracy, the discrep- 
 ancy of one or two per cent, being account- 
 ed for by the loss of work owing to the 
 raising of the pump valves, which were 
 large and heavy. 
 
 Dr. William Pole, after testifying to 
 the high character of the paper, offered 
 a few remarks on the very early example 
 of mechanical integration with which the 
 author had connected his name. It had
 
 181 
 
 come about in the following manner : 
 Some half century ago the engineers of 
 the center and north of England became 
 aware of the reports published from 
 time to time of the extraordinary econ- 
 omy of the pumping engines of the mines 
 of Cornwall. These reports at first ob- 
 tained no credence, and even when they 
 were found to have some foundation, the 
 most singular attempts were made to 
 explain them away. In the midst of the 
 controversy, the late Mr. Thomas Wick- 
 steed, M. Inst. C. E., the Engineer to the 
 East London Water Works Company, 
 determined to throw light on the ques- 
 tion by buying an engine in Cornwall, 
 and setting it up to pump water on his 
 own premises at Old Ford, where it 
 could be thoroughly tested and exam- 
 ined. 
 
 The subject had previouslj^ been 
 brought before the notice of the British 
 Association for the Advancement of Sci- 
 ence, and had attracted the attention of 
 Professor Henry Moseley. He, in writ- 
 insr his excellent work on " The Me-
 
 182 
 
 chanical Principles of Engineering and 
 Architecture," had become acquainted 
 with a principle of dynamometrical ad- 
 measurement proposed by Mr. Ponce- 
 lot, and carried out in 1883 by General 
 Morin ; and it occurred to him that a 
 machine might be contrived on a similar 
 principle, applied to record the work 
 done by a steam-engine, l)y a species of 
 mechanical integration, combining the 
 pressure exerted on the piston with the 
 space moved through ; and it was seen 
 that such a machine would be most use- 
 fully applied in testing the performance 
 of Mr. VVicksteed's Cornish engine. 
 
 At the meeting of the Britisli Associa- 
 tion in ] 840, a grant was made for the 
 purpose, and a committee, consisting of 
 Professor Moseley, Mr. Eaton Hodgkin- 
 son, and Mr. J. Enys, was a2)pointed to 
 carry it out. The machine was con- 
 structed under Professor Moseley's di- 
 rection, and a full account of it was given 
 in the report for the following year. 
 The trials were then made on the engine, 
 and the results were exceedingly satis-
 
 183 
 
 factory. The integrator worked for a 
 month without intermission, and its in- 
 dications, when calculated out, were 
 found to agree closely with the results 
 obtained, as accurately as they could be, 
 by other means. 
 
 The fixed data of the engine had all 
 been well ascertained, but, to render the 
 comparison complete, it was found de- 
 sirable to get, if possible, an accurate 
 measurement of the velocity of the pis- 
 ton at various parts of its stroke ; this 
 velocity was very variable, depending not 
 only on the mass in motion, but also on 
 the ever-varying force of steam acting 
 on the piston, and on means which had 
 hitherto been devised for ascertaining 
 the velocity experimentally. The atten- 
 tion of the committee had, however, 
 been directed to an admirable chrono- 
 metrical instrument contrived by Messrs. 
 Poncelot and Morin, and Professor 
 Moseley undertook to adapt a machine 
 on this principle to the Old Ford en- 
 gine. 
 
 At this time Dr. Pole (who had been
 
 IS I 
 
 oooupioil iuiioptMultutlv it\ invostijjfrtting 
 tho rtoiion of the OiU'uish eiij^ino) \\i\i\ 
 the honor of bein*j' itwittnl to join the 
 eoiniuittee» ivnd the two siu'cotnlino- re- 
 ports, \\\ IS4;> ixud \SAA, wore writttMi hv 
 hiiu. 
 
 By luertus of these two iustnimeiits, 
 and by t\\\ ordinary indieator, aided by 
 eai^eful observiitions ns to the consump- 
 tion of ooa\ i\\\<\ water, and other vari- 
 able factors, the workin*^ of the OKI Ford 
 Cornish engine was investigated, both 
 theoretieally and practically, i!\ a most 
 con^plete and accunUe maniier, and the 
 peculiarities of this fornt of engine, as 
 comjvared with the oi\linary engines iu 
 use at that time, weit) thoi\^ughly 
 brought out, so a^ to clear up all the 
 doubts that had been so long entertained 
 by eugineei*s. This application of a me- 
 chanical integnUor to i-eal work of mag- 
 nitude and importance was probably the 
 earliest made. It was intendevl to fol- 
 low it \ip by adapting the machine to 
 other purposes, especially to oi»ean-goiug 
 steamers, and it was actually fitted to
 
 the cTjf^iueH of the (Ireat Wehtern htearu- 
 ship un her firnt voyaj^e, but by an acci- 
 dent which it waH difficult to repair at 
 sea, the experiment wan rendered UKe- 
 lesB, and the attempt wan never re- 
 newed. 
 
 Mr, \V. li. iJouHlield naid ho liad nut 
 had the advantage of reading the proof 
 of ProfeHBor Shaw'n paper, but Jjc had 
 heard it read in ith nhort form at the 
 previous meeting, and he thought po»- 
 ftibly it miglit be of inferent, and uneful 
 in connection with a paper whicij dealt, 
 more or lesB, with the different typen of 
 mer.'hitnical integratorn, if he mentione<l 
 a principle which, bo far as he knew, had 
 not been alluded to. It occurred to him 
 8ome years ago that, Vy the use of vari- 
 ous curves in connection with planim- 
 eters, various products and squares 
 might be reaxlily integrated. A simple 
 form of area plani meter of this kind 
 might be made Vjy combining a T-square 
 and a set-square, or two set-squares. 
 Referring to the illustration^ Fig. 57, the 
 lower T-square or set-sqtiare ABC would
 
 186 
 
 have in it a slot KL, and upon it a rib 
 GH, and would be capable of motion 
 about a point R in line with the slot. 
 The upper set-square would have a slot 
 MN capable of sliding upon the rib GK, 
 and also a parabolic groove OP. A 
 pointer — in practice a ring with cross 
 wires— would slide in the slots KL and 
 OP. In using this planimeter a needle 
 point would serve to pin the lower set- 
 square through the point R, at a conve- 
 nient pole in relation to the area to be 
 integrated. The cross-wires or pointer 
 would be carried round the curve enclos- 
 ing the area to be integrated, the two 
 set-squares at the same time sliding 
 on one another by means of the rib 
 GH and slot MN. The measurement 
 would be made by the roller Q placed 
 at the vertex of the parabolic curve. 
 The principle of the apparatus was very 
 simple. Taking polar co-ordinates, with 
 R as the pole, the area to be integrated 
 could be expressed as 
 
 i
 
 187 
 
 Fiir. oT 
 
 -Kv. - 
 
 AREA PLANIMETER
 
 188 
 
 If now the equation of the parabolic 
 slot were 
 
 and the parabola moved, as in the appar- 
 atus, so that y was always =r, the ex- 
 pression for the area would become 
 
 f2axd e. 
 
 This showed that the area was meas- 
 ured by the roller Q at the vertex of the 
 parabola. This proof rather suggested 
 that possibly the use of polar co ordi- 
 nates in some of Professor Shaw's proofs 
 would shorten them. By the use of 
 other curves, instead of a parabola, the 
 principle could be applied to the me- 
 chanical solution of various problems in 
 integration. 
 
 Professor W. H. H. Hudson remarked 
 that the classification adopted by the au- 
 thor coincided, if not entirely, very con- 
 siderably with this, that in the instru- 
 ments in which slipping was bound to 
 take place, when it was attempted to ex- 
 press the element of area, in mathemati- 
 cal language y dz, dz corresponded to the
 
 189 
 
 angle turned through and y to the radius, 
 so that the area corresponded to the 
 length. That was certainly so in some 
 of the earlier instruments ; but those in 
 which no slipping w^as allowed to take 
 place seemed to correspond rather to a 
 different mode of translating the mathe- 
 matical expression. There was a certain 
 amount of turning which was propor- 
 tionate to dz^ and then by a suitable 
 mechanism that was turned into some- 
 thing else proportional to y dz by using 
 a multiplier proportional to y. One 
 wheel was made to turn so many times 
 faster than another, the number of times 
 corresponded to y instead of, as in the 
 first class of instruments, the radius be- 
 ing proportional to y. That multiplier 
 occurred in the various forms of the in- 
 strument, sometimes as a sine, and some- 
 times as a tangent. Those of the sine 
 class w^ould have this disadvantage, that 
 the multiplier must, of necessity, be a 
 proper fraction. That would limit the 
 range, whereas those dependent on the 
 tansfent seemed to have no limitation
 
 190 
 
 whatever, and the multiplier might be 
 anything. This seemed to be a great ad- 
 vantage of the sphere instrument of Pro- 
 fessor Shaw, that there was, apparently, 
 no limitation at all, that which represent- 
 ed y being a multiplier which represent- 
 ed a tangent. He had always taken 
 great interest in mechanical illustrations 
 of mathematics. In most cnses the 
 mathematics had been incidental, but in 
 this case it was of the very essence of 
 the whole business. These were instru- 
 ments designed to illustrate not only 
 mathematical results, but even mathe- 
 matical processes, and he was very much 
 struck by the way in which the author 
 had developed Amsler's planimeter out 
 of the simple form of the disk and roller. 
 It illustrated a mathematical conception 
 which was sometimes received with in- 
 credulity, by those who met with it for 
 the first time, namely, treating parallel 
 straight lines as meeting in a point at in- 
 finity. This was done in pure mathe- 
 matics, and he found that the author had 
 been obliged to do it in order to bring 
 
 I
 
 191 
 
 his instruments all under one head. It 
 seemed to him that this was a striking 
 illustration of the inter-action of mathe- 
 matics and mechanics, and the benefit 
 which each gave to the other. 
 
 Mr. W. E. Rich said the integra- 
 tion of the areas of plane surfaces, which 
 these instruments were ordinarily de- 
 signed to do, was much less difficult than 
 the integration of work done on a dyna- 
 mometer in the manner spoken of by 
 ^Ir. Andersnn. One of the great desid- 
 erata in dynamometrical measurements 
 was to get a spring which followed 
 Hooke's law ; but, unfortunately, in prac- 
 tice they never got one that followed it 
 precisely. The uext thing was to adjust 
 the instrument for zero. If the instru- 
 ment were not accurately so adjusted, 
 complex formulas had to be used for re- 
 ducing the results, and as in many in- 
 stances zero was not accurately deter- 
 mined, and these formulas were at the 
 same time overlooked, the results record- 
 ed were in such cases very erroneous. 
 One point of practical difficulty with in-
 
 192 
 
 tegration for a dynamometer ^was the 
 slipping of the small disk upon the 
 large one. In the field, when it was 
 raining, the observers were much ham- 
 pered in that way, and on a stony road, 
 when traveling at a high speed with a 
 small integrating disk, the contact be- 
 tween the disks was frequently broken by 
 the shake and jar of the whole instru- 
 ment. These observations were perhaps 
 not directly pertinent to the question of 
 integration ; but one of the most useful 
 objects of integration was for dynamomet- 
 rical purposes, and therefore he thought 
 it worth while to mention them. 
 
 Professor H. S. Hele Sbaw, in reply, 
 said that the question of mechanical in- 
 tegrators was a very much wider one 
 than many would at first think ; but the 
 discussion upon the paper had embraced 
 two principal points: Planimeters such 
 as were shown upon the table, and the 
 subject of Continuous Integrators. The 
 shpping class of plaiiimeter bad been 
 brought to a great state of perfection, 
 and their accuracy was, as he had shown.
 
 193 
 
 remarkable. It seemed to him that the 
 only objection to them — an objection 
 which several speakers had adverted to 
 — was the diflSculty of reading them. 
 Most of the remarks had been devoted 
 to continuous integrators. Perhaps that 
 was not altogether unnatural, because 
 they had, as Mr. Rich had pointed out, 
 such a directly practical application. 
 Mr. Anderson had given valuable testi- 
 mony on the subject of the defects of 
 such instruments as were in use for dy- 
 namometrical purposes, and his experi- 
 ence agreed with that of Mr. Rich and of 
 all who had much occasion to use them. 
 Professor Shaw thought the general con- 
 clusion was that none of the continuous 
 integrators in use for this purpose were 
 really satisfactory in their action. To 
 say of an instrument of measurement 
 that it was not an accurate instrument, 
 that it was merely capable of giving com- 
 parative results, was. after all, to give it 
 very poor praise, for mere comparative 
 results simply meant that no constant 
 for the instrument could be obtained.
 
 194 
 
 He bad no doubt that, for temporary 
 purposes, the difficulties might be over- 
 come, but it was the general opinion 
 that an instrument like the disk and 
 roller, which, by the way, in spite of 
 what Mr. Anderson had said, ica?^ first 
 applied by Morin after its invention by 
 Poncelot before 1840, was not reliable 
 when in continuous use. An instrument 
 that worked one day differently from an- 
 other day, because the conditions were 
 slightly changed, must certainly be pro- 
 nounced to be very imperfect. It was 
 because of this very difficulty that he 
 was led, as he had no doubt Mr. Boys 
 and others had been, to work at the sub- 
 ject, and to endeavor to overcome the 
 difficulty of the slipping action. Mr. 
 Mair, Mr. Druitt Halpin, and Mr. Cun- 
 ynghame had all testified to the satisfac- 
 tory action of the tangent integrator of 
 Mr. Boys, which was one solution of the 
 problem of avoiding the slipping action 
 in question, but had been employed in a 
 far more trying apphcation than with a 
 dynamometer, viz., a steam-engine integ-
 
 lt)5 
 
 rator. This he was very glad to hear, 
 because the instruments of his own, ex- 
 hibited on the table, depended upon the 
 same principle for their successful action, 
 and it was just in this particular that 
 they differed from the integrator referred 
 to by Dr. Pole. His instruments had 
 been alluded to in complimentary terms 
 by one or two of the speakers, and per- 
 haps he might be allowed briefly to de- 
 scribe the principle upon whicli they 
 worked, and also to call attention to two 
 new instruments wliich were sliown for 
 the tirst time on the previous Thursday 
 at the Royal Society. The difficulty 
 which he endeavored to overcome was to 
 make use of those mathematical proper- 
 ties of the sphere, which he had j^revi- 
 ously discovered and pubhshed, and 
 which Professor Hudson had alhided to 
 as to sine and tangent forms. As was 
 not surprising, he hsid found, when he 
 came to collect information on the sub- 
 ject, that the same properties had been 
 employed by others, amongst whom were 
 Professor Mitchels(m and Professor Ams-
 
 196 
 
 ler. But what the author had endeav- 
 ored to do was to avoid the sHpping of 
 the measurmg roller over the integrating 
 sphere. That was his first thought. 
 There was generally a germ in all inven- 
 tions, and that was the germ of his. If, 
 instead of moving the measuring rollers 
 round the sphere, they could keep them 
 in contact with the sphere, and move the 
 axis of the sphere, or, as it at first oc- 
 curred to him, have the two rollers in 
 contact and roll them around the spheres, 
 they would hold it in such a way that 
 they would allow it to rotate around the 
 different axes in the solid. At first he 
 did not think it would work, but he 
 tried it with a rough boxwood ball, and 
 two disks with india-rubber rings arouDd 
 them, and he was surj^rised to see that 
 the rollers did move round, and to see 
 the ball rolling round on a different axis. 
 He thought at first it was merely a ques- 
 tion of rolling centers, but after some 
 time he saw that it was really a question 
 of geometrical i^rinciple, and having cer- 
 tain rollers in contact around one equa-
 
 197 
 
 tor it did not matter where they were- 
 placed, so long as the others were per- 
 pendicular to the first, acting, in fact, on 
 the same principle as that by which two 
 beveled wheels would work together- 
 The axis of rotation around which the 
 sphere must turn, because of the con- 
 tact of the other set of rollers, was in 
 the vertical plane ; there was but one in- 
 tersection of the two plates, and that in- 
 tersection was the axis of rotation of the 
 sphere. That principle once understood 
 enabled him to construct various other 
 models based upon it. He was indebt- 
 ed almost entirely to his brother, Mr. 
 Edward Shaw, Stud. Inst. C. E., and to 
 Mr. W. E. Kerslake, for working out this 
 in a practical form, and constructing the 
 various instruments exhibited. One of 
 these instruments was a continuous integ- 
 rator, which 'had been designed for ap- 
 plication to the dynamometer of the 
 Rev. F. J. Smith, of Taunton, who in- 
 tended to have been present and take 
 part in the discussion, but from whom he 
 had just received a telegram stating that
 
 198 
 
 he had tried it most carefully and ex- 
 haustively, and bad found that it had 
 given absolutely uniform results. He 
 had tried it himself, but it was a very 
 satisfactory thing that it went to Mr. 
 Smith straight from the hands of the 
 maker, not adjusted in any way. It was 
 remarkably simple. According as the 
 arm was moved into a more or less in- 
 clined position, so there resulted a larger 
 or smaller travel of the indicator. No 
 appreciable effort was required to turn it ; 
 it was done with a minimum of power, 
 and it entirely got over the difficulty of 
 unequal wearing. He had every reason 
 for thinking that, if anything, it would 
 rather tend to wear the ball round than 
 otherwise. Another instrument that he 
 had on the table was the moment integ- 
 rator. Professor Amsler's moment in- 
 tegrator was an expensive one, and at 
 the time of writing the paper he had not 
 any idea of constructing one : but di 
 rectly he had made a small area planim- 
 eter, he saw that a much simpler instru- 
 ment could be devised by employing the
 
 199 
 
 wheels in a very small compass, and he 
 should be haj^py to explain after the 
 meeting, to anyone who might be inter- 
 ested, the performance of the operation 
 by the application of these small wheels. 
 Mr. Mair, he believed, asked whether the 
 readings of the instrument were accu- 
 rate. It had not been tried so exhaust- 
 ively as Amsler's instruments, and there- 
 fore he could not compare it with them ; 
 but he saw no reason why they should 
 not expect the same accuracy from it as 
 from those instruments in the non-slip- 
 ping class, concerning the accuracy of 
 which they had already heard testimony. 
 He greatly regretted that it had not 
 been possible to prepare the engravings 
 in time to be sent around with the paper, 
 or some points he had been anxious to 
 have discussed would, no doubt, have 
 been dealt with more fully by the vari- 
 ous speakers. For instance, the points 
 which Professor Hudson had referred to 
 were actually treated in the paper, and, 
 he ventured to think, were, by means of 
 diagrams, made perfectly clear. He
 
 200 
 
 would remark that, with reference to the 
 range of multiplier alluded to, although 
 it was quite true that this was much 
 greater in the case of the tangent form 
 of integrator, in fact was infinitely great, 
 yet for all practical purposes the sine 
 form enabled quite sufficient range to be 
 obtained while the latter had the great 
 advantage of allowing of the polar in- 
 stead of the linear form of eonstructiou. 
 He wished to say in conclusion, that the 
 point about which he felt extremely 
 gratified was that none of the speakers, 
 although many of them had had consid- 
 erable exjDerience concerning integrators, 
 had taken any objection to tlie division 
 of the subject he had made. When he 
 began to write the paper, his only ob- 
 ject was to endeavor to classify and put 
 in more concise form the details of the 
 subject, but he found the utmost diffi- 
 culty in doing so. He found out that 
 the other classifications in use would not 
 embrace all the instruments, and were 
 very unsatisfactory ; directly, however, 
 the mode of treating the whole question
 
 201 
 
 which he had employed in the paper, oc- 
 curred to him, the matter began to as- 
 sume a clearer aspect, and he had found 
 little trouble in grouping and explaining 
 all known integrators by its means ; and 
 inasmuch as the classification had led 
 him to certain modifications of his own 
 instruments, he hoped that it would not 
 be altogether without results to other 
 workers on the subject.
 
 202 
 
 Correspondence. 
 
 Mr. B. AbdEink-Abakanowiez submitted 
 a linear planimeter of precision of his 
 construction (Fig. 58). This instrument 
 was based on the employment of a cylin- 
 der C turning freely round its axis, and 
 of a roller r, the plane of which could be 
 inclined as wished in respect to the axis 
 of the cylinder, and which could be moved 
 along the generatrix of the cylinder. The 
 displacement of any point of the cylinder 
 was in proportion to the tangent of the 
 angle that the plane of the roller formed 
 with the axis of the cylinder. On a bar 
 DD, fixed on the surface of a drawing, 
 were mounted the pedestals S, S' carry- 
 ing between their points the cylinder C. 
 The roller r was mounted on a carrier B, 
 movable along the rod XX. Another 
 carriage, A, supported a bar YY, perpen- 
 dicular to XX. The two carriers were 
 joined together by the rod g, and their 
 distance could be varied to change the 
 value of the constant. On the bar DD
 
 203
 
 204 
 
 as well as on YY racks were cut, in which 
 worked the pinions^; and p' . The rod I, 
 which made the plane of the roller devi- 
 ate, was placed on the plane of r by 
 means of regulating screws mounted on 
 the arm R. In turning the two pinions 
 p and p', it was easy to follow exactly 
 with the pointer P, the curve bb of which 
 it was desired to find the superficies. 
 There was then imparted to the triangle 
 ABP a movement of translation, and at 
 the same time its height AP was made to 
 vary according to the ordinates of the 
 given curve. The roller was alw^ays 
 pressed by a spring against the surface 
 of the cylinder. The number of turns of 
 the cylinder was read on a counter placed 
 on an endless screw v (this counter was 
 not shown on Fig. 68), and the fractions 
 on the drum T supplied with a vernier, 
 V. The integrators of Mr. C. V. Boys 
 were founded on the same principle, and 
 he took the opportunity of stating that 
 Mr. Boj'S had found this principle of in- 
 tegration independently of him, and that 
 Mr. Boys had recognized the priority of
 
 205 
 
 his invention. In other respects the 
 form of the apparatus of Mr. Boys differed 
 from that he had constructed. 
 
 The other apparatus he had made on 
 the same principle, for a special object 
 he was now engaged in pursuing. He 
 sought to make apparatus capable of 
 tracing the integral curve, and it was to 
 this special end he had directed his 
 labors. This was briefly the problem : 
 given the curve OLD (Fig. 59), of which 
 the equation, was y=f [x), to trace me- 
 chanically another curve EGF, of which 
 the equation should heY=J\f {x) dx + C. 
 The constant=AE. Since 1878 he had 
 constructed machines to solve this prob- 
 lem, and he had brought them forward 
 on various occasions. It would occupy 
 too long a time now to explain these in- 
 struments. They were made with the 
 object of rendering service, particularly 
 in the art of engineering, and in limits 
 much more extensive than could be ac- 
 complished by planimeters. In reality 
 the integral curve could be met with in 
 almost all problems of statics. Those
 
 206 
 
 who were familiar with the elegant pro- 
 cesses of graphic statics knew how im- 
 portant to the calculation of bridges, 
 arches, statical moments, and of inertia, 
 was the outline traces by funicular 
 curves. The machines that he had called 
 
 briefly " Integraphs '' traced these curves 
 mechanically, and to obtain them it was 
 sufficient to have the same data as for the 
 ordinary methods of graphic statics. It 
 would be possible to arrive at the same 
 result with linear planimeters, liut it 
 would be necessary to operate by suc- 
 cessive additions, whilst the machines 
 would register at each moment the incre- 
 ment of the integral, tracing a curve on
 
 207 
 
 wbich could be made all the necessary 
 geometric operations. 
 
 Major- General H. P. Babbage remarked 
 that what most interested him was the 
 contrast between arithmetical calculating- 
 machines and these integrators. In the 
 first there was absolute accuracy of re- 
 sult, and the same with all operators ; 
 and there were mechanical means for cor- 
 recting, to a certain extent, slackness of 
 the machinery. Friction too had to be 
 avoided. In the other instruments nearly 
 all this was reversed, and it would seem 
 that with the multiplication of reliable 
 calculating machines, all except the 
 simplest planimeters would become ob- 
 solete. 
 
 Professor A. G. Greenhill said, suppose 
 O to be the fixed center, GAP the planim- 
 eter, A the joint containing the sphere 
 (Fig. 60). 
 
 1. Fix the joint A, and move P to P, 
 on the arc of a circle, center O ; then if 
 OA turned through an angle 0, the dial 
 at A would register an angle M d cos q), 
 if the angle between AP and OA pro-
 
 208 
 
 duced was q)^ and M was some cod- 
 stant. 
 
 2. Fix the joint O, and move P, to P, 
 
 on the arc of a circle, center A, ; the dial 
 at A would not move. 
 
 3. Fix the joint A, and move A, back 
 to A, and P,^ to P3 ; then the dial at A 
 would move back an angle M d cos cp\ 
 if (p' was the angle between AP, and OA 
 produced .
 
 209 
 
 4. Fix the joint O, and move P3 to P 
 on the arc of a circle, center A ; the dial 
 would not move. 
 
 In completing the circle of the finite 
 area PPjP^P^, the dial would then have 
 registered an angle Md (cos 9 — cos qj'). 
 
 But the area PP^P^P,^ area PP,Q,Q 
 
 =i ff (OP'-OQ^) 
 
 =i 6 («' + 2 ab cos cp-hb'-^r — 
 2 ab cos cp'—b"^) 
 
 =. a b 6 (cos cp — coH qj'), 
 
 if 0A = (/. AP = />. 
 
 Then if "M — ab, the dial would register 
 the area PP,P,P3. 
 
 Any irregular area must be supposed 
 to be made up of infinitesimal elements 
 formed in the same manner as PP,P^ 
 
 Mr. E. Sang observed that in the 
 " Transactions of the Koyal Scottish So- 
 ciety of Arts " there was a description of a 
 plantometer, by Arthur Beverley, of Dun- 
 edin. New Zealand. In it the rubbing wheel 
 was guided in a straight path. A very 
 simple analysis showed that, because the
 
 210 
 
 tracer returns to its first position, the re- 
 sult was true, whatever might be the 
 curve of this guide, so that Beverley's 
 and Amsler-Laffon's were two cases of a 
 general law. They were identical in 
 principle ; it was not likely that, in his 
 out-of-the-way position, Beverley had 
 known of the other. 
 
 Professor Shaw remarked that he had 
 already called attention to the fact that 
 Mr. Abdank-Abakanowicz had anticipated 
 Mr. Boys in the use of the non-slipping 
 principle for integrators, but he was glad 
 that the former gentleman had alluded 
 to the tracing of an integral curve by in- 
 struments acting on this principle. This 
 was a problem of considerable interest, 
 and the author migbt mention that he 
 had, in a j^aper before the Royal Society, 
 called attention to a particular case of the 
 integral curve. In this case if the section 
 of any solid were taken on the plane of 
 the paper, a curve, which he had called 
 the curve of areas, might be drawn, the 
 ordinates at any point of which repre- 
 sented the area of the cross-section of the
 
 2J1 
 
 solid at that point perpendicular to the 
 plane of the paper. Such curves might 
 no doubt be drawn by an adaptation of 
 the instruments of Messrs. Abdank-Aba- 
 kanowicz and Boys, and there seemed to 
 be considerable scope for further investi- 
 gation in that direction. 
 
 The author was obliged to express his 
 disagreement with the opinion of Gen- 
 eral Babbage, that all integrators except 
 the simplest planimeters would become 
 obsolete and give place to arithmetical 
 calculating machines. Continuous and 
 discontinuous calculating machines, as 
 they had respectively been called, had 
 entirely different kinds of operations to 
 perform, and there was a wide field for 
 the employment of both. All efforts to 
 employ mere combinations of trains of 
 wheelwork for such operations as were 
 required in continuous integrators had 
 hitherto entirely failed, and the author 
 did not see how it was possible to deal 
 in this way with the continuously varying 
 quantities which came into the problem. 
 No doubt the mechanical difficulties were
 
 212 
 
 great, but that they av ere not insuperable 
 was proved by the daily use of the disk, ■ 
 globe, and cylinder of Professor James ^ 
 Thomson in connection with tidal calcu- 
 lations and meteorological work, and, in- 
 deed, this of itself was sufficient refuta- 
 tion of General Babbage's view. 
 
 €
 
 i
 
 University of California 
 
 SOUTHERN REGIONAL LIBRARY FACILITY 
 
 )5 Hilgard Avenue, Los Angeles, CA 90024-1388 
 
 Return this material to the library 
 
 from which it was borrowed. 
 
 APR 1 9 199^ 
 SHLF 
 QUARTER L(pAb} 
 
 '^
 
 B 000 014 556 5 
 
 ?A' '. ^ROOL