Mr. F. P. Purvis on Amsler's Planimeter. 11 



serve for the determination of the virial can be formed with the 

 employment of any variables ; and with this I think I may on 

 the present occasion be satisfied, without entering upon special 

 applications of the equations — which may be of very various 

 kinds, and, hence, would lead to extended discussions. 



II. On Amsler's Planimeter. By F. P. Purvis, Esq.* 



THE following is a simple and thoroughly general explana- 

 tion of the action of this perplexing little instrument. 

 Suppose, for simplicity and greater generality, that the instru- 

 ment consisted simply of the straight bar A B, of length /, car- 



rying a pencil at each end, A and B ; and suppose any lines A a, 

 B b were traced out by these pencils : we will consider how the 

 area Aab B may be expressed in terms of / and the motion of 

 some point in the line AB. 



Let the motion from AB to a/3 represent an elementary 

 motion of the bar, the centre of it C moving from C to 7, and 

 the bar turning about y through the angle dd ; let dn= the 

 normal distance from y to A B ; this motion may be considered 

 to take place in two parts : — 1st, the motion of A B parallel 

 to itself into the position wy ; 2nd, the motion of AB about its 

 centre into the position a/3; the required area A a j3 B is, in 

 this elementary motion, equal to the area Axy B (*=ldn), since 

 the area 7 a x = the area y /3 y, and the areas A a x and B /3 y 

 are negligible with respect to Idn, being the product of two in- 

 finitesimal quantities, while Idn is the product of one infinitesi- 

 mal quantity (comparable with each of the two just mentioned) 

 and the finite quantity /. 



Integrating for the whole area A a b B, we see that it is ex- 

 pressed by In, where n is the travel of the point C normally to 

 the bar A B. 



Now we may obtain that normal motion n by centring a 

 wheel on the bar at C, free to revolve in the plane at right angles 



* Communicated bv the Author. 



