198 Trans. Acad. Sci. of St. Louis. 
x*. 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’ is the same 
as in Fig. 2. Their widths have increased from y — & to 
(y? — yx) =y (y— 2) in one case and to yx —a2*=—2& 
(y—«) in the other. 
These rectangles all decrease in width if 2 is made to 
approach y in value. When «= y Eq. 3 shows that 
y* — a* 0 ) 
ee ee) 
sear ae 0 dy 
This is the change in the area of y* per unit of change 
in y. 
In Eq. (1) which is represented in Fig. 1, the values 
y and x which have been increased to y* and #* result- 
ing in Eq. 2 and Fig. 2 are horizontal dimensions. This 
may be done by directly changing Eq. (1) to 
y—ae=y (y—a) +a (y’—2’) 
This is Eq. (2). 
When the vertical dimensions y and x are changed to 
y’ and a’, the above equation becomes 
Yee —ste ySs) 
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 reet- 
angles y* and 2° in Fig. 2 to y® and 2°, and y* and 2%, the 
values of y and # remaining unchanged. 
Two rectangles having areas y® and #° 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 z=3 the rectangle y® would have sides 
y® = 3125 em. and y* = 625 em. if drawn to scale. 
