406 PROOF OF THE PRINCIPLE OF AMSLER'S PLANIMETER. [50 



Hence motion of M in the same direction =8s + c8\jj, and therefore the 

 elementary area traced out by QP = b (8s + c8\jj). Also elementary area traced 

 out by 



Hence the whole area swept out by OQP in moving from its initial 

 to any other position is 



^d~<f> + bcijj + Its. 



If OQP returns to its initial position without performing a complete 

 revolution about 0, the limits of <f> and t/f are 0, and the area of the 

 figure traced out by P is 6s. 



If OQP has performed a complete revolution, the limits of <f) and x// 

 are -lir, and the area traced out is 



TT (a 2 + 2bc) + bs. 



