July 19, 1894] 



NATURE 



285 



Physical Society, June 22. — Prof. W. E. Ayrton, F.R.S., 

 PaU-President, in the chair. — Captain Abney, before exhibiting 

 his photographs of flames, demonstrated that a candle flame 

 contains solid particles by passing a beam of polarised light 

 though it, the track of the beam through the flame being clearly 

 seen in one direction, whilst in a direction at right angles it was 

 practically invisible. The same thing was also shown by pass- 

 ing the light through a turbid liquid. Photographs of argand 

 and candle flames with pencils of sunlight and electric light 

 passing through, were then exhibited showing similar pheno- 

 mena. Several series of photographs of flames of candles and 

 various forms of gas-burner taken with diminishing exposures 

 were then shown m order to illustrate the different luminosities 

 at different parts of the llame. Those taken with long exposures 

 showed the bright parts nearly equally white, but as the time of 

 exposure diminished only the most luminous portions were 

 recorded on the plate. From the photographs the author con- 

 cluded that when used with a slit as a photometric standard the 

 argand burner was unsuitable, for portions of different luminosity 

 come into view when the slit is approached or receded from. 

 The ordinary fi^h■tail burner was better in this respect. 

 Questions were asked and remarks made by Prof. S. P. Thomp- 

 son, Prof. Perry, and Mr. Trotter, in reply to which Captain 

 Abney said dropped shutters with slits from one inch to one- 

 sixteenth inch wide had been employed, and some of the 

 exposures were only a few thousandths of a second. The dis- 

 placement caused when the ob'ect was n it stationary could 

 easily be allowed for wtien the velocity of the shutter was 

 known. — Prof. O. Ilenrici read a paper on an elementary 

 theory of planimeters. Considering the generation of areas by 

 the motion of straight lines, the author defined the s;nse in which 

 such areas are to be taken. Choosing the positive sense of a 

 line O T of variable length as outwards from the centre O 

 about which it turns, and the positive direction of rotation as 

 counter-clockwise, the following rule for determining the sense 

 of an area was given. Imagine yourself standing at a point P, 

 and looking along the positive sense of O T, whilst it passes 

 over P. then the area near P will be swept out in a positive sense 

 if T crosses you from ri^ht to left, otherwise it will be 

 negative. Applying this rule to closed curves of any shape, it 

 was shown that if T goes once round the boundary, any area 

 outside the curve was necessarily swept over as many times in 

 •'■e negative sense as in the positive sense, therefore these areas 

 iicelled, and also that the sense of any part of an area depends 

 the sense of its boundary. Passing on to the consideration 

 (.1 areas generated by a line (or rod) of fixed length, which 

 iiioves anyhow in a plane and returns to its initial position. 

 Prof. Ilenrici showed by taking instantaneous centres, that the 

 same rule regarding the sense of the areas holds, and that the 

 area generated by the rod is equal to the difference between the 

 areas of the two closed curves traced by its ends. In the parti- 

 cular case where one end of the rod moves forwards 

 and backwards along the same path the area swept out by the 

 rod is equal to that of the closed curve traversed by the other end. 

 This is the theory of Amsler's planimeter, for the area of the 

 curve whose boundary is traversed by the tracer is the same as 

 that swept out by the rod carrying the tracer when the pole is 

 outside the close<i curve. By resolving small motions of the 

 rod in two component parts, a translation parallel to itself, and 

 a rotation about the point in which the plane of the registering 

 wheel cuts the rod, the author showed- that the areas swept out 

 by the translations were registered by the wheel, whilst the sum 

 of those generated during the rotations cancel. Cases where 

 the pole is inside the curve were next considered, an<l the con- 

 stant then to be added to the wheel reading determined. 

 Instead of registering the transLation by a wheel whose axis is 

 parallel to the rod, a knife-edged wheel which slides and turns 

 freely on an arm perpendicular to the rod would serve the same 

 purpose. This is the principle of Iline and Robertson's plani- 

 meter. In the actual instrument, however, the arm is inclined 

 at about 10' to the rod, and is therefore inaccurate. In the 

 "hatchet" planimeter a bent rod terminates at one end in a 

 tracing point, and at the other in a convex knife-edge or 

 "keel," whose plane contains the point. The area of the curve 

 whose boundary is traversed by the point is approximately equal 

 to twice that of the sector included between the initial and final 

 positions of the rod. . The approximation results from the fact 

 that the area of the curve traced by the keel is not zero. At 

 the meeting the question of reducing the area of the keel curve 

 was discussed at some length, the author showing that in the 



NO. 1290, VOL, 50] 



case of a curve symmetrical about a line, it was possible to 

 reduce this area practically to zero. Even for unsymmetrical 

 curves one couUI obtain a symmetrical one of double the area 

 by drawing a line, cutting the curve, and supposing the area 

 turned over about this line. Prof. Perry inquired if the author's 

 conclusion was that the "hatchet"' planimeter and the Hine and 

 Robertson instrument were inaccur.aie? If so, he was at a loss 

 to understand why the latter gave results more nearly correct 

 than .-Vmsler's. Mr. P.lakesley pointed out that if both arms of 

 a jointed planimeter be simultaneously moved over curves the 

 total reading should give the sum of the two areas traced out if 

 taken in the proper senses. Mr. .\. P. Trotter directed atten- 

 tion to an article by the inventor of the "hatchet" in the 

 current number of Engineering. Dr. Macfarlane Gray said he 

 had examined the proof given in Engineering, and found no 

 error. He then showed how the Amsler planimeter could be 

 explained in a simple geometrical manner by drawing radial 

 lines through the pole and intersecting the curve. Moving the 

 tracer along these radii added nothing to the area ; motion along 

 the arcs was the important component. Mr. O. G. Jones and 

 Prof. Thompson also took part in the discussion. — Mr. F. W. 

 Hill made a communication on the "hatchet" plani- 

 meter. In this paper the author takes a point within 

 the area to be measured, and divides the area into 

 elementary triangles with this point as apex. The tracing 

 point of the planimeter is then supposed to start from the apex 

 and trace out one of the triangles. The inclination between 

 the initial and final positions of the " hatchet " is then expressed 

 in terms of the angle at the apex, the radius vector, and the 

 length of the planimeter. By expanding and integrating the 

 expression, it is shown that twice the area between the initial 

 and final position of the planimeter, after tracing all the tri- 

 angles, is represented by an infinite series of terms, the first 

 of which is the area of the curve, the second is proportional to 

 the moment of inertia of the area about the point, the third pro- 

 portional to its first moment about the same point. The higher 

 terms are usually small enough to be neglected. Starting the 

 tracer at the centroid of the area causes the third term to dis- 

 appear, and the second has its minimum value, so that this is 

 the starting-point recommended. The magnitude of the errors 

 caused by neglecting the various terms are discussed in some 

 detail. In the author's opinion, the instrument can never be 

 strictly accurate ; but usually the errors are within the limits of 

 observation. Prof. Henrici did not agree with the statement 

 that the instrument was necessarily inaccurate, and thought 

 geometry might aid analysis to find the proper starting-point. 

 For a symmetrical curve he had shown that a point existed, 

 starling from which the area traced by the hatchet end was 

 zero. Dr. Macfarlane Gray thought Prof. Henrici's latterjargu- 

 ment was vitiated by his figure not being correctly drawn ; but 

 this Prof. Henrici disputed. Mr. O. G. Jones said Prof. 

 Henrici's construction was not obvious, for the cusp curves 

 traced by the "hatchet" end depended on the starting-point. 

 Mr. Yule sugsjested that by shortening the planimeter it might be 

 possible to bring the third and second terms of Mr. Hill's 

 lormula into greater prominence, and, by going round the 

 curve more than once, determine the first and second moments. 

 — .\ paper on a new integrating apparatus, by Mr. A. Sharp, 

 was taken as read. The paper describes an improved 

 form of harmonic analyser, giving the amplitude and epoch of 

 each constituent term, the mechanism of which is an inversion 

 of that described in a communication made to the Society on 

 April 13. Numerous drawings accompany the paper, showing 

 the various parts in detail. The mechanism is also shown to 

 be applicable for integraphs, and l)y suitable modification may 

 be employed for mechanically integrating differential equations 

 of various forms. A paper on magnetic shielding by a hollow 

 cylinder, by Prof. Perry, and another on " Clark's cells," by Mr. 

 S. Skinner, were postponed, 



Geological Society, June 20.— Dr. Henry Woodward, 

 F.R..S_., President, in the chair. — On deep borings at Culford 

 and Winkfield, with notes on those at Ware and Cheshunt, by 

 W. Whitaker, F. K. S., an 1 A. J. Jukes-Browne. Four borings 

 at Cullord, Winklield, Ware, and Cheshunt were described in 

 detail, so far as the specimens examined would permit ; these 

 were few in the case of Culford, but many from the other 

 borings. The interest of the Culford boring centred in its 

 striking the Palxozoic floor at the small depth of 637J feet ; 

 but the age of the slaty rocks cannot be determined. Although 



