504 



REPORT — 189 J. 



Let, now, T be moved round a closed curve whilst the pole is outside 

 its area (fig. 10). The point Q moves to and fro along the circle and 

 returns to its original position when T does so. The motion of the rod 

 will therefore be cyclical. An exception can only occur if during the 

 motion rod and arm should become stretched out to their fullest extent, 

 and on moving on, Q should be allowed to move the wrong way. This 

 can always be easily avoided. 



The corollary to Theorem 1 is therefore applicable. It says that 

 the area is equal to the area generated by the rod QT. But this area, 

 according to Theorem 2, equals the area generated by the motions of 

 translation alone, and this can be recorded by a simple wheel. Let there 

 be mounted anywhere on the rod a wheel W, at a distance c from Q. 



If a small motion of tiie rod be decomposed as before (fig. 8), the 

 wheel will move along the line "SVqW during the translation and along the 

 arc WW during the turning. The first may again be decomposed in the 

 motion from W^W", whereby the rod describes the area pi whilst the 

 wheel rolls, the amount of its 'roll' measuring 7*, and the motion W W, 



Fig. 11. 



Fig. 12. 



Q C 



dui-ing which the rod slides along itself and the wheel slips along the 

 paper wdthout rolling. 



During the turning the wheel will roll along the arc W'\Y=Ci9. The 

 whole roll during an elementary step, wliich will be denoted by tv, is 

 therefore 



'if=^9 + cfl, .•.;; = ?(• — c9 



We found the area swept over 



=iio+{^ r--cJ)3 



For the whole area A we have, therefore, 



In the case under consideration Sfi^O. At the same time Ziv is the 

 whole roll of the wheel. If we denote this simply by v; we get 



A^=hv 

 In the case where the pole lies within the area, as in fig. 11, we have 



