83 



THE FOLLOWING DEMONSTRATION OF THE WORKING 



OF THE PLANIMETER is FROM THE PEN OF WM. 

 D. GELETTE, CIVIL ENGINEER, FORMERLY OF 

 BOSTON. 



Case I. When the anchor point is outside of the 

 figure to be measured. Let X be the figure to be 

 measured, and let a be the pole or origin, and R the 

 radius of polar co-ordinates of the point t in Fig. 1. 

 And let Rj be the radius of a second point f, on the out- 

 line of the figure X, and let be the angle t a ^ then 



radius R may be taken constant, then area t a t t = 

 By polar co-ordinates the area X = 2 -^ 



the area of the triangle t a t^-. But if 9 be 

 taken so small, that for the small distance t t v the 



Let a be the anchor point, t the tracing point, and w the point of contact of the 

 flange of the wheel of a polar planimeter, and call tp = y, ap = z, andp w =/, and 

 the angle op to = a then after the motion of the tracing point t to , the point p 

 comes to jD t , w to to t and the angle a changes to cq. But when, as in this case, it is 

 supposed that during the small motion t t the radius R is constant, then for the 

 same length of motion, a will be constant and p a p t will = 6. 



Expressing R in terms of y, z, and a, we have R 2 = (t m) s -f (a p-{- pni) *= 

 <z-f ycos) 2 -f(y sin a) 2 = *-}- 2 zycos a-j- y'cos'a.-f^sin'a But sin* a + cos 



u = l and R' 



2 z y cos a sin 9 y* sin .. _ 



2 2 z y cos a sin 2 y- sin 0. . 



_| [_ But in 



the summation, owing to the fact that the instrument returns to the same position 

 from which it started, the 2 sin must = and wherever combined with constants 

 only, in the above equation will reduce to 0. hence the first and last terms will dis- 

 appear, but the middle term which contains the variable cos a will remain, hence, 



X = 2 z y cos a sin 



(2) 



It will be seen by reference to Fig. 1 that z sin =pp l when is very small, .-. 

 z sin cos a=p p l cos a=p l q which is the component of the motion of the wheel 

 which is at right angles to its axis, and is therefore the part which represents the rota- 

 tion of the wheel for a small motion t t of the tracing point. And this component 

 multiplied by the arm y gives z y sin cos a which by equation (2) expresses the area 

 of X after summation. But 2 z sin cos a is the resultant rotation of the wheel 

 after the tracing point has completed the circuit of X, hence the area X = distance 

 rolled by the wheel multiplied by the length of the arm y. Calling the circumfer- 

 ence of the wheel c, and the number of resultant number of revolutions made during 

 the measurement n, we have X= y c n (3) 



And if the instrument is graduated so as to record y c n, and we call the record of 

 +he instrument r, we shall have, X=r. 



