198 Trans. Acad. Sci. of St. Louis. 



x 2 . They become the rectangles c and d of Fig. 4. The 

 narrow strip across the top of Fig. 2, shown between 

 dotted lines in Fig. 4, was replaced by two strips marked 

 a and b at the top of Fig. 4. Their length y 2 is the same 

 as in Fig. 2. Their widths have increased from y — x to 

 (y 2 — y%) = y{y — x) in one case and to yx — x 2 = x 

 (y — x) in the other. 



These rectangles all decrease in width if x is made to 

 approach y in value. When x = y Eq. 3 shows that 



/ y i — x* \ 



\y — x ) 



= -7T =*y 3 = 4-(y 4 ) 



This is the change in the area of y* per unit of change 

 my. 



In Eq. (1) which is represented in Fig. 1, the values 

 y and x which have been increased to y 2 and x 2 result- 

 ing in Eq. 2 and Fig. 2 are horizontal dimensions. This 

 may be done by directly changing Eq. (1) to 



y 3 — x 3 = y 2 (y — x) + x (y 2 — x 2 ) 



This is Eq. (2). 



When the vertical dimensions y and x are changed to 

 y 2 and x 2 , the above equation becomes 



y* — x* — y 2 (y 2 — x 2 ) + x 2 (y 2 — x 2 ) 



This is one form of Eq. 3 and the resulting diagram is 

 Fig. 4. It is of exactly the same structure as Fig. 1. 



We may now increase the length of the sides of the rect- 

 angles y 3 and x 3 in Fig. 2 to y 5 and x*, and y* and x*, the 

 values of y and x remaining unchanged. 



Two rectangles having areas y 9 and x 9 are thus pro- 

 duced. They are shown in diagram, but not drawn to 

 scale, in Fig. 5. If the centimeter is taken as the unit 

 and y = 5 and x = 3 the rectangle y 9 would have sides 

 y 5 = 3125 cm. and y 4 = 625 cm. if drawn to scale. 



