618 hersey: an integrating device 



and call the curve the front of the templet. The templet is to 

 be placed on the drawing board right over the (x, y) curve so 

 that it can be slid along with its base on the x-axis. Starting 

 with the back at x x , make a mark where the front of the templet 

 crosses the (x, y) curve. Then slide it along until the back comes 

 to the mark and make a second mark where the front now 

 crosses the curve, and so on. Let n be the number of steps 

 necessary to travel across from x x to x 2 in this manner. An 

 approximate numerical value of the integral I will then be 



I = nC (2) 



in which C is a known constant for a given templet. 



In order to obtain the simple result (2), it is necessary only 

 that the front of the templet be cut to the curve 



f(Y)-X = C (3) 



Here X and Y are respectively the abscissa and ordinate of any 

 point on the templet curve, relative to the back and the base 

 as axes. 



To prove (2), let the variable Ax denote the width of each step 

 along the x-axis. When the templet is in any one of the suc- 

 cessive positions marked on the (x, y) curve, X = Ax and Y = y. 

 Hence by (3) 



f(y) ■ ax = c (4) 



Integrating (1) between x and x + Ax gives for the contribution 

 which the strip of width Ax makes to the integral/, approximately 



Al=f{y)'Ax (5) 



Comparing (4) and (5), 



Al = C (6) 



Thus every strip contributes the same amount C; therefore the 

 whole integral is 



/ = 2AJ = nC (7) 



The accuracy of the result is enhanced if the device be made 

 up of two such templets, back to back in one piece. The work- 

 ing formula will then be 



I =nK (8) 



