80 Mr. C. V. Boys on the Drawing 



reading cannot be directly observed ; but since by tbe time 

 that the rule is parallel to the axis the needle-point must 

 have reached, in the particular case chosen, the division -5, 

 the two or three steps required may be made with more 

 than sufficient accuracy by moving the needle-point a few 

 divisions at a time so as to arrive at this point when 

 parallelism is being reached ; in fact at this stage equal 

 angular changes correspond to equal changes on the scale. 

 As soon as this position is passed the central line of the strip 

 cuts the axis on the other side, and so now these readings 

 have to be added to the total curvature *5 in order to find the 

 remaining readings upon which the needle-point should rest. 

 In fig. 2 one complete loop of this particular nodoid has been 

 drawn, and the second one has been carried to the stage at 

 which *16 is the axial reading, and *34 the reading of the 

 temporary centre of curvature. Unduloids may be drawn in 

 a similar way, but in this case the needle-point has to move 

 off to an infinite distance, when the axial reading is equal to 

 the total curvature. The length of line for which the centre 

 of curvature is beyond the end of the rule is generally very 

 small, and this may be drawn so that no error can be detected 

 by alternately moving the needle from one end to the other, 

 each for a very small arc. The catenary may of course be 

 set out in the same way and with still greater ease and accu- 

 racy, for the infinite values do not come into any finite part 

 of the curve. The axial reading and that on which the needle- 

 point rests must be kept the same, but on opposite sides of the 

 tracing-point. 



It will be evident that if the little tripod is just pressed 

 sufficiently upon the paper to make small impressions of the 

 needle-feet, it may afterwards be taken over the same course 

 and the third needle-point pressed upon the paper, and thus a 

 series of points on the evolute determined. In fig. 2 the evolute 

 is shown dotted. The points of most importance on this are 

 the cusps a, h, corresponding to the places on the curve where 

 the curvature is a minimum or a maximum, the distance a b 

 being the half period of the curve, and the maxima or minima 

 c, g, which are the ordinates of the curve where the abscissa 

 is a minimum or a maximum. By means of these and the 

 curve itself the geometrical constants of any curves drawn in 

 this way may be determined more accurately than from the 

 curve itself alone. Prof. Greenhill has given me the formula 

 by which, with the use of tables of elliptic integrals, the chief 

 constants of the unduloid and nodoid may be calculated. 

 Miss Stevenson, a student of the College of Science, has drawn 

 a large series of these curves, and we have found that while 



