<l»b' 



GRAPHICAL ALGEBRA.* 

 Francis E. Nipher. 



In the former paper it was shown that algebraic func- 

 tions of a higher order than the second degree may be rep- 

 resented in diagram upon a plane surface. 



In the present paper the same function will be repre- 

 sented by widely different diagrams. 



The difference between the areas of two squares hav- 

 ing sides the length of which are y 5 and x 5 is 



y ™ _ X " -_ y 9 ( y _ ^ _|_ yS x ^ _ x j 



+ y 7 x 2 (y — x) + y*x 3 (y — x) 

 + y 5 x 4 (y — x) + y 4 x 5 (y — x) 

 + y 3 x« {y — x)-\- y 2 x 7 (y — x) 

 -f y x 3 (y — x) + x 9 (y — x) (5) 



This is Eq. (5) of the former paper. 



This equation may be written : 



y" — x 10 = y 5 (y 5 — y 4 x )+ y 4 x (y s — y 4 x ) 

 -\-y 4 x (y 4 x — y 3 x 2 ) + y 3 x 2 (y*x — y 3 x 2 ) 

 -f- y 3 x 2 (y 3 x 2 — y 2 x 3 ) + y 2 x 3 (y 3 x 2 — y 2 x 3 ) 

 + y 2 x 3 (y 2 x 3 — y x 4 ) + y x 4 ( y 2 x 3 — y x 4 ) 

 + yx 4 {yx 4 — x 5 ) + x 5 (yx 4 — x 6 ) (7) 

 = (y* + y*x ) (y 5 — y 4 x ) 

 + (y 4 x +y 3 x 2 ) (y 4 x — y 3 x 2 ) 

 + (y 3 x 2 -f y 2 x 3 ) (y 3 x 2 — y 2 x 3 ) 

 + (y 2 x 3 + y x 4 ) (y 2 x 3 — y x 4 ) 

 + (yx 4 + x 5 ){yx 4 — x 5 ) (7a) 

 = (y" —y*x 2 ) + (y*x 2 — y«x 4 ) 

 + (y 6 x 4 — y 4 x 6 ) + (y 4 x 6 — y 2 x 3 ) 

 + (y 2 x a — x w ) 



* Continued from No. 6, Vol. XXIII. Presented by title before the 

 Academy of Science of St. Louis, January 20, 1919. 



(205) 



