50. 



PROOF OF THE PRINCIPLE OF AMSLER'S PLANIMETER. 

 [From the Cambridge Philosophical Society's Proceedings. Vol. i. (1857).] 



Let be the fixed point, 

 P the tracer, 

 Q the hinge, 

 W the centre of wheel, 

 M the middle point of PQ, 

 OQ = a, PQ = l, MW=c. 



The area of any closed figure whose boundary is traced out by P, is 

 the algebraical sum of the elementary areas swept out by the broken line 

 OQP in its successive positions. 



Let (f> and \fj be the angles which OQ, QP at any time make respec- 

 tively with their initial positions. 



s the arc which the wheel has turned through at the same time. 



If now OQP take up a consecutive position, and <, \jj, s receive the 

 small increments 8<f>, St|, 8s, we see that 8s = motion of W in direction 

 perpendicular to PQ. 



