200 Trans. Acad. Sci. of St. Louis. 
The remaining five terms of Eq. (4) may be written 
a Xy* (y—a) or a (yy —y*x) | 
atx yx (y—a) or x (ye — y°a") 
at X y’a* (y— x) or a (y’a* — y*a*) 
atx ya (y—a) or x (y’a*?*— y 2) 
atx at (y—a) or x (y a*—2a’) 
These terms represent the five rectangles e, f, g, h, k of 
Fig. 5. They have a common length 2%. 
If we now increase the shorter sides of these rectan- 
gles from y* and «* to y® and 2°, the difference between 
the areas of the two squares thus formed will be 
ee Sy eee ae) 
+ y'x* (y— 2) + ya? (y— a) 
+ ya* (y— a) + y*a* (y— @) 
+ y’a* (y—a) + ya" (y— ax) 
+ya(y—a)+ a (y—a) (95) 
The first five terms may be written 
xy ea yy) 
y X ya (y—az) = y? (y*a — y*a*) 
y’ X ya? (y— x) = y? (y*a* — y’a*) 
y Xy a (y—a2) = y* (y’a* — y a) 
ex we (ya) ee pee) 
The four rectangles a, b, c and d of Fig. 5, have been 
replaced by five, their lengths being: y° as in Fig. 5. 
Their widths are now the same as the widths of the five 
rectangles e, f, g, h, and k in Fig. 5. The widths of the 
four rectangles a, b, c and d have all been multiplied by 
y and the last one above represented has been added. 
The remaining five terms of Eq. 5 may be written 
axy (y—az)—ae (yo —y*a ) 
aX ya (y—ax) =a (ya —y*a*) 
aX yy’? (y—a) =a (ya? — y*x*) 
a” Xy x (y—a) = 2 (y’a*— yz") 
aX at(y—a2)—a2 (yat*— 2-2) 
