Case II. When the anchor point is in- 

 side the figure to be measured, and Mir 

 rotation or the wheel i> forward or plus 

 with reference to the figures on its circum- 

 ference, ft will be seen that there will ho 

 one posit ion nf the arms // and .:, which, 

 while tlie point t describes a circle around 

 a as a center, (see fig. 2) will produce no 

 rotation of the wheel. The condition 

 under which this will occur will be fulfilled 

 when the angle t w a = 90. in which case 



and the area of the circle described' under 

 these conditions = IT 11* = TT z 2 -f- 2 TT // f-\- 



fry- (4) 



Call this the corrective circle. In (fig. 2) 



the area of t a t l = ^/.Q the area of the 







whole figure X = 2 77 R*|^j ; ^s before 

 K 1 = z* + 2 z y cos a + y z and hence X = 



2 7T ggQ + 2 2 7T ?/ COS 3QQ + 2 * ? / 



360 (5) 



But in tlie summation the instrument makes a complete revolution around a, 



n 



the sum of 2 _ =1, and when combined with constants only, will not appear in 

 the result as a factor. Hence we have 



-Try 2 ... (6) 



Now follow out on the diagram in Fig. 3. the motion 

 of the wheel, which corresponds to a motion of the 

 tracing point from t to ^ first dividing that motion into 

 two parts ts ands ^ swinging the arm y around the 

 pointp until it becomes parallel to , p t while z remains 

 fixed, produces the first motion and causes the wheel 

 to to roll backward to s t and as the path of its motion 

 is everywhere perpendicular to its axis, s t to will repre- 

 sent rotation or distance rolled during that motion, but 



C\ 



as s p t = 6 the distance s l w = 2 TT /OQQ which is the 



backward or minus rotation of the wheel. 



The second part of the motion is bv moving the arm 

 y from s s, to J, p t during which the wheel moves from s t to u\ and this motion is part 

 sliding and part rolling, the rolling component is p g, and causes forward or plus 



rotation of the wheel, the value of p q is 2 TTZ ^-^ cos , and the resultant rotation 



of the wheel is on completing the circuit =2 f 2 TT z cos a T^ J -'27r/; and the area 

 expressed by the wheel as by Case I is 



2 zTrscosa ^'2^ (7) 



Comparing the area expressed by the wheel which we will call r with the true 

 area of the figure as given by Eq. 6, we have 



*+ 2 



2 Try/ 



