360 



FOREST AND STREAM. 



[April 14, 1893. 



POLAR PLANIMETER AND INTEGRATOR. 



THE two instruments here illustrated have done much to lighten 

 the labor of the designer and naval architect, all the calcula- 

 tions of a vessel being greatly shortened by them, while the liability 

 to error is lessened as well. They were invented by Dr. J. Amsler- 

 Laffon, a Swiss scientist. The planimeter has been in common use 

 by designers and engiueers for a number of years, the integrator 

 being of more recent invention, having been used in England and on 

 the Continent for about a dozen years, though as yet hardly known 

 in tin's country- The theory upon which the two are based is inti- 

 mately connected with the calculus, and consequently a knowledge 

 of the higher mathematics is necessary to understand it perfectly: 

 but the operation of each in practice is so simple that they are used 

 by the younger apprentices in the large shipyards, several inde 

 pendent readings, of course, being made and checked off. 



The planimeter, the simpler of the two, is shown in Fig. 9, 

 and consists of a graduated roller and dial working freely in a 

 frame composed of two jointed arms, one being of constant length, 

 while the other may be lengthened or shortened to suit the scale of 

 the drawing. At the outer extremity of the fixed arm is a needle- 

 point or an anchor weight, about which the whole instrument re- 

 volves in use. At the outer extremity of the shifting arm is a tracing 

 point which is passed over the entire outline of the figure it is 

 desired to measure, the inclosed area being read off on the dial and 

 rolling wheel. The shifting arm may be set to various scales, as 

 shown by the graduations on its side, or in some cases the entire 

 top of the sliding rod is graduated in milimeters, so that by calcula- 

 tion the arm may be set to any scale. In the common graduation 

 the arm is marked for 0.10 decimeter, O.lOsq ft. and 10. OOin., a sub- 

 sequent calculation being necessary to translate the reading to any 

 special scale of the drawing other than these. In use the arm is first 

 adjusted to the most convenient scale, the instrument is then placed 

 on the drawing in such a position that the tracing point will reach 

 every part, the reading of the dial and wheel is noted, the point being 

 placed on some certain spot, the point is then moved over the entire 

 outline to the same spot from which it started, and a second reading 

 taken, the result being the area of the figure, 



The integrator (Fig. 10) is much more complicated, the two arms 

 carrying three separate dials and rollers; while, in place of revolving 

 about a fixed point, the entire apparatus traverses back and forth on 

 a grooved straight-edge. The dial marked A gives the area of the 

 figure, the dial i gives the moment of inertia, and the third dial, M, 

 gives the statical moment. The instrument is shown as adjusted to 

 the body plan of a vessel, as in calculations of the stability, for which 

 purpose it is largely used. By means of the three sets of readings the 

 displacement, the vertical and longitudinal positions of the center of 

 buoyancy, and the other elements which enter in the calculation of 

 the stability, are rapidly obtained. The instrument is quite expen- 

 sive, the price in Switzerland ranging from $75 to $150, and being just 

 about doubled in the United States through the duty. 



One of the first imported into this country is owned by Mr. Robert 

 Center, of New York; Mr. Burgess employed one in his office for sev- 

 eral years, another is used iu the Navy Department at Washington, 

 one has lately been added to the equipment of the School of Naval 

 Architecture and Marine Engineering at Cornell University, and one 

 was lately on sale at an instrument maker's in New York. Th re are 

 probably others in use, but these are the only ones we have been able 

 to learn of. 



The following graphical demonstration of the operation of the 

 planimeter was written by Mr. Emil Thiers for the American Machin- 

 ist, and is so much clearer than any other that we have yet read that 

 we reproduce it, by the permission of that journal, for the benefit of 

 those of our readers who are familiar with the use of the instrument 

 without clearly comprehending its principle of operation. 



Although the polar planimeter is an instrument in daily use. yet 

 the theory of its action is probably fully understood by few of the 

 many to whom it is indispensable. A clear and elementary exposi- 

 tion of its theory will therefore, it is supposed, be of interest. 



It is presumed that the construction of the instrument is sufficiently 

 well known to make a description unnecessary. 



Ia order to arrive at the relation existing between the arc through 

 which the measuring wheel of the instrument revolves and the part 

 described upon the plane of the paper by the point of tangency, let it 

 be assumed that a straight-edge M A, Fig. 1, is provided with any 

 number of small wheels, all in one plane, and all touching the paper. 

 For any movement of the straight-edge all wheels will revolve 

 through the same arc. In proof of this, in Fig. 2, let M N be moved 

 into the position M' N'. The movement may be considered as made 

 up of a rotation of M N about >S, the point of intersection of the 

 directions M N and M' A", and of a motion of translation along M' 

 A". The rotation of the straight-edge about any point in the plane of 

 the wheels is not taken cognizance of by the latter, while the motion 

 of translation in the plane of the straight-edge causes all wheels to 

 revolve through the same arc. Any movement, however compli- 

 cated, of the straight-edge upon the paper must be considered as 

 made up of rotations and translations as described: therefore, the 

 wheels being set to indicate similarity in the beginning, it is impos- 

 sible to move them out ot these position of coincident readings. 



In Fig. 3 let F be the tracing point; P the fixed point or pole; G the 

 joint; M the measuring wheel. M N is the trace of a plane coin- 

 ciding with that of the wheel, and P A" a perpendicular upon it from 

 P. Now let F be moved to the right length, P F remaining constant. 

 Every point of the system wili then describe a circle about P, and all 

 relative distances. MN among them, will remain constant. The 

 measuring wheel M will therefore, as in the case of the wheels on the 

 straight-edge, revolve similarly to an imaginary one at point N. As 

 the latter remains tangential to its circular path, it follows that the 

 arc s, measured by it, as also by the measuring wheel M, is equal to 

 the circumference of a circle of radius P N = p; i.e., s = 2 it p. 



The measuring wheel will revolve right-handed, as seen from F; 

 for the reverse direction of the motion of F the direction of rotation 

 of the wheel will likewise be reversed. 



The value of p is readily obtained from the right-angled triangles 

 FPQ and G P Q. F G- a, G M = c, G P = r. F P - R. 



P Q- = P 2 — (a + c -p) 2 = r 2 — (p — c, 2 

 it 2 — a 2 + 2 a (p — c) — ?- 2 

 g 3 -r 2 a c -f r 2 R- 

 2 a 



a a 

 For another circle of radius R' described by the tracing point, the 

 ai c registered by the measuring wheel s' is similarly— 

 J _ « 2 + 2ac-f t 2 ^ _ E ' 2 it 

 a a 

 the lengths a, c and r remaining unchanged. If now, in Fig. 4, point 

 F is moved over the outlines of the ring. 1 2 3 4 5 6, the movement of 

 F will be right-handed over the outer circumference, and left-handed 

 for the inner, while the crossing of the ring radially takes place 

 twice, and m opposite directions, and is therefore without effect on 

 the final reading of the wheel. The direction of rotation of the wheel 

 being right-handed for s and left-handed for s', the arc registered 

 will be b = s— s'. Substituting the values of s and s' found above, 

 we shall have— 



R*-it-R'°-Tt- 

 l> = ' or R*Tt - R'*7t = ab. 



Th e left side of this equation is the area of the ring=/,- hence we 

 obtain the simple relation f=ab, i.e.. the area of any ring whose cen- 

 ter is P is equal to the product of the length a of the movable arm 

 by the arc registered by the measuring wheel. 



It is manifestly immaterial at what paint we cross the surface of 

 the ring, the motion of the wheel being right-handed or left-handed, 

 according as the tracing point is moved from the outer circumference 

 inwardly, or from the inner one outwardly. If point F is therefore 

 made to move over the outlines of the ring segment F 1 2 3 4, Fig. 5, 

 the movements of the wheel due to the radial lengths 23 and 41 will be 

 equal and opposite, and the arc registered will be one n'th of that 

 registered before, supposing the subtended angle to bear this ratio to 

 360°. Both sides of the equation / = a.b have therefore been decreased 

 in the same ratio, a remaining unchanged, the equation still holds, 

 and the area of the ring segment likewise is equal to the product of 

 the movable arm of the instrument by the length of the registered 

 arc. 



If now a figure, Fig. 6, composed of any number of annular seg- 

 ments whose common center is P is run over as indicated by the ar- 

 rows, we shall obt3in the area as for the single segment. The result 

 will evidently remain unchanged if we omit all lengths which are 

 traversed twice in opposite directions, leaving only the outlines of 

 the figure. Tbe number and tize of the elementary segments is a 

 matter of indifference, and as any figure may be considered as made 

 up of segments infinitely small, it follows that in general the area 

 may be. found according to tbe formula /= ab. by moving the tracing 

 point Fover the outlines of the figure, the only condition being that 

 *he instrument must be brought back to the position from which it 



As the length P G = r of the guide arm does not enter into the 

 result, it may be made of variable length or even infinite, by guid- 

 ing G iu a straight slot. This form of tbe instrument is identical with 

 what is known as Coffin's averaging instrument. To convert a figure 

 of area/ into a rectangle I X h, e.g , to find the mean pressure of an 

 ndicator diagram, it would be convenient to make the length of arm 



Fiy. 3. 



' e \ 



a 





"""x /it 







Fiy. 8. 



a equal to that of the diagram; then I x h = / = a X b; and as I = a, 

 we have h = b. 



If the area to be measured is too large for the instrument, the pole 

 being outside, we must either divide it up into sections, or else place 

 the pole witnin the area to be measured. Iu this ease it is necessary 

 to omit that portion of the surface immediately surrounding the pole, 

 and take account of it separately. The process is somewhat tedious, 

 as the tracing point must be moved over the outlines of the figure, 

 over those of tbe portion omitted, and twice over the radial length 

 connecting the two (see Fig. 7). However, the process may be sim- 

 plified by making the area omitted such, that the wheel wili not turn 

 while tracing its outline. This is a case for a circle whose center is the 

 pole and whose radius R is such that thep corresponding is zero, the 

 plane of the measuring wheel passing through the pole (Fig. 8). For 

 p = o, we have, as found above: 



a 2 + 2 a c -f r n - — R* = o, 

 and the area of the omitted portion is 



-B 2 jt =(oH2«c + ' ! ) it 

 To find the area, therefore, the pole being inside, proceed as be- 

 fore and add the constant (a s -f2ac-f r 2 ) 



ft was stated above that for measurements with the pole outside 

 the result was independent of the length r: it may be added that it is 

 likewise independent of c. Any motion of the measuring wheel in 

 the direction of its axis, due probably to readjustment of its bear- 

 ings, will therefore have no influence on the readings. For meas- 

 urements with the pole within the area, it is important, however, to 

 check the accuracy of the constant. To this end place the points P 

 and F of the instrument upon a slip of paper, Fig. 8, so that the pole 

 Pis as nearly as possible in the plane of the wheel. JSow rotate the 

 slip, keeping its tension constant about pole P, and notice if the 

 wheel turns. According as it turns in one direction or the other P F 

 is increased or diminished, until no motion is perceptible. The area 

 of the circle of radius P Pis the constant required. 



"The crew in the forecastle were soon on deck (those in the cabin 

 were unable to get out) and one of them rushed forward and let go 

 the head sails. The vessel soon came up. It was a narrow escape, 

 and bad the Bohlin not been an extra good craft and the squall 

 abated somewhat she might have filled and sunk. The vessel had 

 lain flat in the water, her sails half under. One of her crew walked 

 along her side from the wheel box to the fore rigging, so flat did she 

 he. The bait boards were torn off the house and two of the dories 

 floated off by the water." 



The Nannie C. Bohlin is one of the deep shooners and something 

 like the Fredonia designed by the late Mr. Burgess, and has before 

 this occasion demonstrated in the highest degree her special fitness 

 for the business in which she engaged, so far as both seaworthiness 

 and speed are concerned. It is evident to any one at all familiar with 

 naval architecture, and the peculiar peril in which she was placed, 

 that had she been as shallow as the vessels in common use in the 

 New England fisheries a tew years ago, none of her crew would ever 

 have returned to tell of their experience. 



IMPROVEMENTS IN THE FISHING FLEET. 



THE importance of tbe recent improvements in the fishing vessels 

 of New England, clue fo the precept and example of the U. S. 

 Fish Commission, though generally acknowledged, never have been 

 more strongly exemplified" than in a recent occurrence, the partic- 

 ulars of which are thus stated in the Gloucester Times of April 1: 



"Sch. Nannie C. Bohlin, from the Banks Sunday, reports a most 

 thrilling experience. On tbe morning of March 12. at about daylight, 

 while bowling along by the wind under full sail, with the usual watch 

 on deck, a sudden squall arose. Capt. Bohlin was just coming on 

 deck, and was standing in the companionway when a fierce gust 

 from the northwest threw the vessel down. Tne captain managed to 

 reach the deck. The man at the wheel, with great presence of mind, 

 threw the wheel down, although both he and the wheel were sub- 

 merged. He then rushed for the starboard side of the vessel and 

 hung out over the stern, which was almost under water. One other 

 of the crew also hung over the side and escaped being washed over- 

 board. 



YACHT NEWS NOTES. 



The annual general meeting of the St. Lawrence Yacht Club for 

 the election of officers and the reception of reports was held on Sat- 

 urday evening in the club rooms, about fifty members being present. 

 The secretary's statement shows the past season to have been a very 

 successful one in every way. The membership has increased from 

 202 to 314. A comfortable club house at Dorval has been erected and 

 furnished, a boat house and wharf built, and work commenced on 

 the construction of a breakw T ater, the dredging of a basin, and the 

 laying down of ways. The treasurer's statement shows a substantial 

 surplus, and the club commences its season with more enthusiasm 

 and brighter prospects than ever before. The following officers were 

 elected: Com., A. W. Morris, M.L.A.; Vice-com., Chas. H. Levin; 

 Rear-com., Lionel J. Smith; Hou. Secy., A. F. Mitchell; Hon. Treas., 

 V?. A. C. Hamilton; Meas., Messrs. F. P. Shearwood and George 

 Marler; Committee, Messrs. G. Herrick Duggan, David A. Starr, H. 

 Markland Molson, David A. Poe, J. Dudgeon, E. Kirk Greene, J. 

 Simmons and J. D. Miller.— Montreal Witness, April 4. 



It is now reported that Vice-Corn. Morgan will not race Gloriana 

 this season, but will sail the new flu keel 35-footer. Capt. Harry 

 Craven has sailed for England, which makes it probable that he will 

 bring out an English steam yacht for Mr. Morgan. It is also reported 

 that Mr. Ogden Goelet, owner of Norseman, schr., has purchased a 

 steam yacht abroad. 



A new yacht club to be named the Gravesend Yacht Racing Club , 

 has been organized by the racing owners of small boats about 

 Gravesend Bay, New York Harbor. 



Two of the new 21 footers have recently been named, the Bigelow 

 fin keel, building at Bristol being called Vanessa, while the Small 

 boat will be the Exile. 



Constellation, schr., is fitting out at Foster's wharf, Beverly, under 

 the direction of Capt. Watson.' 



