196 Trans. Acad. Sci. of St. Louis. 
In Fig. 3 we find the lengths in centimeters to be 
Unit = 0.5 en. —~ 10 
gee 1S em: eS 
y= 25 em.— 5 
xy = 7.5 em. — 15 
y’ = 12.5 em. — 25 
The lengths in terms of the chosen unit are given in the 
final column. 
The distance x?y may be laid off upon the axes, by 
drawing a line through the points marked 1 and zy. 
A line parallel to this through the point distant 2 on 
the vertical axis of the diagram, Fig. 3, will determine 
the point distant 2?y from the origin on the horizontal 
axis. 
If we increase the width of the two rectangles y* and 
a* of Fig. 2 to y* and 2, we shall have two squares whose 
areas are y’ X y? = y' and a” X a? = a. 
They are represented in Fig. 4. Figures 1 and 2 will 
also be identified in this diagram. 
The difference between the areas of these two squares 
is 
y— at — (y"+ a4) (y* a4) 
=(y +2) (y+ 2) (y—z) 
=y (y—a2) + ya (y—2a) 
“F ya? (y =a) + a* (y—a). (3) 
The first two terms of the second member may be 
written 
y’ (y* — yx) and y* (yx — 2”) 
They are represented in Fig. 4 by the large rectangles 
a and b. 
The two following terms of Eq. (3) may be written 
a (y*— yx) and a? (ya — a”) 
They are represented by the rectangles marked c and d. 
