200 Trans. Acad. Sci. of St. Louis. 



The remaining five terms of Eq. (4) may be written 



^ 4 X y 4 (y — x) or x 4 (y 5 — y 4 x ) 

 x 4 X y 3 x (y — x) or x 4 (y 4 x — y 3 x 2 ) 

 x* X y 2 x 2 (y — x) or x 4 (y 3 x 2 — y 2 x 3 ) 

 x* X y x 3 (y — x) or x 4 (y 2 x 3 — y x 4 ) 

 x 4 X x 4 (y — x) or x 4 (y x 4 — x 5 ) 



These terms represent the five rectangles e, f, g, h, h of 

 Fig. 5. They have a common length x 4 . 



If we now increase the shorter sides of these rectan- 

 gles from y 4 and x 4 to y 5 and x 5 , the difference between 

 the areas of the two squares thus formed will be 



y 10 — x 10 = y 9 (y — x) + y s x (y — x) 

 + y 7 x 2 (y-x)-jr y 6 x 3 (y — x) 

 + y 5 x 4 {y — x)-\- y 4 x 5 (y — x) 

 + y 3 x 6 (y — x) + y 2 x 7 (y — x) 

 + yx 8 (y — x)+ x*(y — x) (5) 



The first five terms may be written 



y 5 Xy 4 (y — x)= y 5 {y 5 — y 4 x ) 

 y 5 X y 3 x (y — x) =y 5 (y 4 x — y 3 x 2 ) 

 y 5 X y 2 x 2 (y — x) = y 5 (y 3 x 2 — y 2 x 3 ) 

 y 5 X y x 3 (y — x) = y 5 (y 2 x 3 — y x 4 ) 

 y 5 X x 4 (y — x)=y 5 (yx 4 — x 5 ) 



The four rectangles a, b, c and d of Fig. 5, have been 

 replaced by five, their lengths being y 5 as in Fig. 5. 

 Their widths are now the same as the widths of the five 

 rectangles e, f, g, h, and h 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 



x 5 X y 4 (y — x) = x 5 (y 5 — y 4 x ) 

 x 5 X y 3 x (y — x) =x 5 (y 4 x — y 3 x 2 ) 

 x 5 X y 2 x 2 (y — x) = x 5 (y 3 x 2 — y 2 x 3 ) 

 x 5 X y x 3 (y — x) = x 5 (y 2 x 3 — y x 4 ) 

 x 5 X x 4 (y — x) = x 5 (y x 4 — x 5 ) 



